On deformations of flag manifolds with b2=1.
Norbert Kuhlmann
Flag manifolds with Picard number 1 and del Pezzo manifolds of degree 5 and dimension $\geq3$\ are (globally) rigid.
math0502573.tex
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\begin{document}
\title[deformations]{On deformations of flag manifolds with b2=1.}
\author{Norbert Kuhlmann}
\address{Universit\"{a}t Duisburg-Essen, Standort Essen, FB 6, Mathematik,
Universit\"{a}tsstr. 2, 45117 Essen, Germany}
\email{norbert.kuhlmann@uni-due.de}
\urladdr{Norbert Kuhlmann}
\thanks{This paper is in final form }
\date{June 3, 2010}
\subjclass[2000]{ Primary 32G05,Secondary 14J99}
\keywords{deformations, flag manifolds, del Pezzo manifolds}
\dedicatory{ }
\begin{abstract}
Flag manifolds with Picard number 1 and del Pezzo manifolds of degree 5 and
dimension $\geq3$\ are (globally) rigid.

\end{abstract}
\maketitle
\tableofcontents


\section{\textbf{Introduction}}

\subsection{\textbf{Statement of the main theorem.}}

A holomorphic, proper, surjective, regular map $\tau:X\rightarrow Y,$ where
$X,Y$ are (connected) complex manifolds, is called a \emph{deformation map}.
Let $F_{1},F$ be compact complex manifolds. $F_{1}$ is called a
\emph{deformation} of $F,$ if both occur as fibers of the same deformation map
$\tau:X\rightarrow Y.$ $F$ is called \emph{(locally) rigid resp. (locally)
nondeformable}, if any deformation map $X\rightarrow Y$ which has $F$ as one
fiber possesses a neighborhood of this fiber, such that all fibers in this
neighborhood are biholomorphic to $F.$ $F\,$is called \emph{(globally) rigid
resp. (globally) non deformable}, if for all deformation maps $X\rightarrow Y$
where one fiber is biholomorphic to $F$, all fibers are biholomorphic to
$F.$If all fibers of the deformation map are k\"{a}hler, we speak of a
k\"{a}hler deformation map and call 2 fibers of a k\"{a}hler deformation map
k\"{a}hler deformations of each other. In the following "k\"{a}hlerian" has
the same meaning as "k\"{a}hler"\textbf{.}\newline\newline The spaces $F$ we
consider in this paper, flag manifolds with $b_{2}=1$ (also called flag
manifolds with Picard number 1, or rational homogeneous manifolds with Picard
number 1)\ and del Pezzo manifolds of degree $5$ and dimension $\geq3$, are
well known to be locally rigid. We show in this paper, that \ $n$-dimensional
flag manifolds with Picard number 1 and del Pezzo manifolds of degree 5 and
dimension $\ n>2$\ are globally rigid. \newline\newline For the rest of this
paper $Y$ will denote the unit disc $(\left\vert t\right\vert <1)$ and
$\tau:X\rightarrow Y$ a\ deformation map.$.$\newline\newline If $\tau
:X\rightarrow Y$ is a deformation map, then all fibers $X_{t}=\tau^{-1}(t)$,
$t\in Y$, are diffeomorphic to the fiber $X_{0}$. But it is not true, that
always\ in such a situation all fibers $X_{t}$ are biholomorphic to $X_{0}$,
as for instance the well known\ example $X_{0}=$\emph{ a smooth complex
elliptic curve }shows. On the other hand, if $X_{0}$ is a smooth rational
curve, i.e. if $X_{0}$ is biholomorphic to the complex projective space
$\mathbf{P}^{1}$, then in such a situation all fibers are biholomorphic to
$X_{0}$, i.e. all deformations of $\mathbf{P}^{1}$ are biholomorphic to
$\mathbf{P}^{1}$. In 1957 F. Hirzebruch, K. Kodaira posed \ in
(\cite{Hirzebruch-Kodaira}, see \cite{Kodaira-Spencer})\ the problem whether a
deformation of $\mathbf{P}^{n}$ is a $\mathbf{P}^{n}$ for $n\in\mathbf{N}$
arbitrary. \newline\newline In this context it suffices to discuss deformation
maps $\tau:X\rightarrow Y$, which are differentiably trivial, i. e. for which
there exists a diffeomorphism $X\rightarrow Y\times X_{0}$, which maps each
fiber $X_{t}$ onto $(t)\times X_{0}$; because $\tau$ is differentiably trivial
over a neighborhood of $0\in Y$. In the following we will assume, that the
occurring deformation maps are differentiably trivial.\newline\newline In the
1970s it was observed\ that a (global)\ deformation of $\mathbf{P}^{n}$ must
be moishezon and that consequently the Hodge decomposition theorem holds, so
some simple consequences could be drawn. This means: Let $\tau:X\rightarrow Y$
be a deformation map with $X_{t}=\mathbf{P}^{n}$ for $t\neq0$; then $X_{0}$ is
moishezon. This allowed to construct a certain bimeromorphic map
$\mu:X\rightarrow X^{^{\prime}}=Y\times\mathbf{P}^{n}$, defined by $\tau$ and
sections $s_{0},...,s_{n}$ in the positive generating line bundle (concering
this notion see below)\ $\mathcal{D\in}H^{1}(X,\mathcal{O}_{X}^{\ast
})=\mathbf{Z}$\ of $X$, with $P\rightarrow(\tau(P);s_{0}(P):...:s_{n}(P))$ for
general $P\in X$.\ $\mu$ maps $X\backslash X_{0}$ biholomorphically onto
$(Y\backslash(0))\times\mathbf{P}^{n}$. $\mu$ and the induced meromorphic map
$\mu_{0}:X_{0}\rightarrow\mu\lbrack X_{0}]\subset(0)\times\mathbf{P}%
^{n}=\mathbf{P}^{n}$\ of $X_{0}$ onto the proper transform $\mu\lbrack X_{0}%
]$\ became the object of study. It was realized\ that $\mu\lbrack
X_{0}]=(0)\times\mathbf{P}^{n}=\mathbf{P}^{n}$ is equivalent to $X_{0}%
=\mathbf{P}^{n}$. - Here $\mu\lbrack X_{0}]$ is the closure of the set of
all$\ (\mu(P)\in(0)\times\mathbf{P}^{n}|P\in X_{0}$ and $\mu$ is holomorphic
in $P)$ in $(0)\times\mathbf{P}^{n}=\mathbf{P}^{n}$; the graph of $\mu_{0}$ is
the closure of $((P,$ $\mu(P))\in X_{0}\times X_{0}|$ $\mu$ $is$ $holomorphic$
$in$ $P)$ in $X_{0}\times X_{0}$; by an abuse of notations we will denote
$\mu_{0}$ also by $\mu|X_{0}$. \newline\newline In order to gain more
information about $\mu$ and$\ \mu_{0}$ one can study the behaviour of
sequences of $\mu^{-1}(l_{t})\subset X_{t}$ for $t\rightarrow0$, where the
$l_{t}$ are lines in $(t)\times\mathbf{P}^{n}$, $t\neq0$. These lines are
orbital curves (i.e. closures of orbits) of Euler vector fields; an Euler
vector field on $\mathbf{P}^{n}$\ can be described the outside $\mathbf{P}%
^{n}$-infinity-hyperplane, i.e. on the remaining $\mathbf{C}^{n}$ with
$z_{1},...,z_{n}$ as affine coordinates, by $%
%TCIMACRO{\dsum }%
%BeginExpansion
{\displaystyle\sum}
%EndExpansion
(z_{i}-a_{i})\frac{\partial}{\partial(z_{i}-a_{i})}$ ($=%
%TCIMACRO{\dsum }%
%BeginExpansion
{\displaystyle\sum}
%EndExpansion
(z_{i}-a_{i})\frac{\partial}{\partial z_{i}}$); the isolated zero
$(a_{1},...,a_{n})$ is called the center of this holomorphic vector field. In
order to study these sequences of the $\mu^{-1}(l_{t})$ and their limits for
$t\rightarrow0$ \ H.\ Tsuji proposed to consider Euler vector fields on
$Y\times\mathbf{P}^{n}$ and their liftings\ to $X$; he pointed out that this
bimeromorphic map $\mu$ is not biholomorphic iff the lifting of Euler vector
fields on $Y\times\mathbf{P}^{n}$ to $X$ delivers meromorphic vector fields on
$X$ with a proper pole along $X_{0}$. Thus Euler vector fields entered the
scene. Finally Siu proved in \cite{Siu1}, \cite{Siu2}, \cite{Siu3}, that any
deformation of the complex projective space $\mathbf{P}^{n}$ is (analytically
isomorphic to) a $\mathbf{P}^{n}$; one major ingredient was the use of Euler
vector fields. Up to now the shortest proofs of the $\mathbf{P}^{n}%
$-nondeformability are those which are based on or related to lemma 1 of
\cite{Hwang1} (see also \cite{Hwang2}), this means proofs, which exploit
properties of (additive) holomorphic vector fields with a divisorial zero. The
global rigidity of smooth quadrics $\mathbf{Q}_{n}$ of dimension $n\geq3$ was
proved in \cite{Hwang1}. Therefore it can be stated:\newline\newline%
\emph{Theorem}$.$ Let $\tau:X\rightarrow Y$ be a deformation map. If for all
$t\neq0$ the fiber $X_{t}$ is biholomorphic to $\mathbf{P}^{n}$, then the
central fiber $X_{0}$ is biholomorphic to $\mathbf{P}^{n}$ too. If for all
$t\neq0$ the fiber $X_{t}$ is biholomorphic to a smooth quadric $\mathbf{Q}%
_{n}$ of dimension $n\geq3$, then the central fiber $X_{0}$ is biholomorphic
to $\mathbf{Q}_{n}$ too.\newline\newline\emph{Remark}. With the additional
assumption $X_{0}$ being k\"{a}hler, the non deformability of $\mathbf{P}^{n}$
is generally looked at as a simple consequence of the stronger result
\cite{Hirzebruch-Kodaira} and the $\mathbf{Q}_{n}$-case, $n\geq3$, is
generally looked at as a simple\ consequence of \cite{Brieskorn}. There are
simpler approaches, if one is only interested in nondeformability, for
instance approaches which essentially employ that the homology ring of $X_{0}$
must be isomorphic to the homology ring of $\mathbf{P}^{n}$ resp.
$\mathbf{Q}_{n}$. - In this context one should point out the following result
of I. Nakamura (here also \ T. Peternell has to be mentioned): For $n=3$ any
moisezon $n$-fold homeomorphic to $\mathbf{P}^{n}$ resp. to the smooth
$\mathbf{Q}_{n}$\ is analytically isomorphic to $\mathbf{P}^{n}$ resp. to
$\mathbf{Q}_{n}$ (see \cite{Kollar}, 5.3.4, 5.3.5, 5.3.6); Nakamura discusses
also results of this type for $n=4$ and gets partial results\ for $n\geq3$
arbitrary, especially he proves the deformation invariance of $\mathbf{P}^{4}$
and the smooth $\mathbf{Q}_{4}$,\ see the references listed in \cite{Parsha},
p.238;$\ $the methods leading to these results are not the subject of this
paper. The present paper is deeply indebted to the Euler vector field
approach. \newline\newline Hwang and Mok proved in \cite{Hwang-Mok},
\cite{Hwang3}, \cite{Hwang-Mok3}, \cite{Hwang-Mok4}:\newline\newline%
\emph{Theorem 1}$.$\emph{ Let }$\tau:X\rightarrow Y$\emph{\ \thinspace be a
deformation map, let all fibers }$X_{t},$ $0\neq$\emph{\ }$t\in Y,$\emph{\ be
(analytically isomorphic to) a fixed \ flag manifold }$F$\emph{\ with }%
$b_{2}=1$.\emph{\ Then the fiber }$X_{0}$\emph{,} $0\in Y,$\emph{\ is
analytically isomorphic to }$F$\emph{, if the central fiber }$X_{0}$\emph{\ is
k\"{a}hler, i.e. projective (in the present context).}\newline\newline In the
present paper we prove a stronger form of this theorem, where the
\emph{k\"{a}hler assumption for the central fiber\ }$X_{0}$ is dropped.
\newline\newline\emph{Theorem 2 (Main theorem). Let }$\tau:X\rightarrow
Y$\emph{\ \thinspace be a deformation map, let all fibers }$X_{t},0\neq
$\emph{\ }$t\in Y,$\emph{\ be (analytically isomorphic to) a fixed flag
manifold }$F$\emph{\ with }$b_{2}=1$.\emph{\ Then the fiber }$X_{0}$\emph{,}
$0\in Y,$\emph{\ is analytically isomorphic to }$F$\emph{. }\newline\newline
As flag manifolds are locally rigid (see for instance \cite{Akhiezer}, p.
131), theorem 2 can be rephrased: \emph{Let }$\tau:X\rightarrow Y$%
\emph{\ \thinspace be a deformation map, suppose that one fiber is a flag
manifold }$F$\emph{\ with }$b_{2}=1$\emph{, Then all fibers are analytically
isomorphic to }$F.$ \newline\newline In theorem 1 and 2 one\ cannot weaken the
regularity condition of $\tau$ by a flatness condition, as the example of
\cite{Timmerscheidt}, p. 274 shows.\newline\newline The assumptions of theorem
1 and 2 imply readily $H^{1}(X,\mathcal{O}_{X}^{\ast})=H^{1}(X_{0}%
,\mathcal{O}_{X_{0}}^{\ast})=\mathbf{Z}$; we denote the positive generating
line bundle $\in H^{1}(X,\mathcal{O}_{X}^{\ast})$\ on $X$ by $\mathcal{D}$ and
the restriction to $X_{0}$ by $\mathcal{D}_{0}$. $\mathcal{D}$ is
characterized by: $\mathcal{D}$ generates $H^{1}(X,\mathcal{O}_{X}^{\ast
})=\mathbf{Z}$; for $t\neq0$ the induced line bundle $\mathcal{D}_{t}$ on
$X_{t}$ is very ample. \newline\newline In the beginning the structure of
proof (say of theorem 1) may approximately and redundantly be indicated
thus:\newline\newline\emph{(*)} \ \emph{We suppose:} \textbf{a. }$F$ is a Fano
manifold of dimension $n\geq3$ with $H^{1}(F,\mathcal{O}_{F}^{\ast})=$
$H^{2}(F,\mathbf{Z})$ $=H_{2}(F,\mathbf{Z})=\mathbf{Z}$.\ The positive
generating line bundle, i.e. the ample generating line bundle $\mathcal{D}%
_{F}\in H^{1}(F,\mathcal{O}_{F}^{\ast})$ is very ample. There are $\widehat
{s}_{0},...,\widehat{s}_{n}\in H^{0}(F,\mathcal{D}_{F})$, hypersurfaces
$H_{F}\subset F$ ,$H^{^{\prime}}\subset\mathbf{P}^{n}$, and $s_{0}^{^{\prime}%
},...,s_{n}^{^{\prime}}$\ $\in$ $H^{0}(\mathbf{P}^{n},\mathcal{D}%
_{\mathbf{P}^{n}})$\ with the following properties: $H_{F}=(\widehat{s}%
_{0}=0)$, $H^{^{\prime}}=(s_{0}^{^{\prime}}=0)$. The meromorphic map $\mu
_{F}:F\rightarrow\mathbf{P}^{n}$ , defined by $(\widehat{s}_{0}:...:\widehat
{s}_{n})=(s_{0}^{\prime}:...:s_{n}^{^{\prime}})$ in general points, maps
$F\backslash H_{F}$ biholomorphically onto $\mathbf{C}^{n}=\mathbf{P}%
^{n}\backslash H$. Translations of $\mathbf{C}^{n}=\mathbf{P}^{n}/H^{^{\prime
}}$ extend to biholomorphic maps $F\rightarrow F.$\newline\newline\textbf{b.
}Denote $z_{i}:=\frac{\widetilde{s}_{i}}{\widetilde{s}_{0}}$, $i=1,...,n.$ For
every point $P\in F\backslash H_{F}=\mathbf{C}^{n}$ with $a_{i}:=\ z_{i}(P)$,
$i=1,...,n$, there is a holomorphic vector field $\eta$\ on $F$ (called an
Euler vector field on $F$ with center $P$ resp. $(a_{1},...,a_{n})$) with the
following property: On $F\backslash H_{F}=\mathbf{C}^{n}$ the vector field
$\eta$ has the form $\sum_{i=1}^{n}(z_{i}-a_{i})\cdot\frac{\partial}{\partial
z_{i}}$. \newline\newline c\textbf{.} $F$ is covered by $F$-lines, .i.e.
rational curves of $\mathcal{D}_{F}$-degree one. Any two points in $F$ can be
connected by a chain of\ $F$-\ lines. \newline\newline d. If $F^{^{\prime
\prime}}$ is an analytic subset\ of $F$ of at least codimension 2, then
general lines in $F$ do not meet\ $F^{^{\prime\prime}}$, i.e. there is a
nonempty open subset in the space of lines on $F$, such that the lines of this
subset evade $F^{^{\prime\prime}}$\newline\newline e\emph{. }For all points
$P\in F\backslash H_{F}$ the corresponding\ Euler vector field $\eta$\ with
center $P$\ possesses at least one orbital curve through $P$, which is a line
in $F$. General lines on $F$ do not meet the set of indeterminancy of $\mu
_{F}$. \newline\newline\textbf{f. }\ $F$ is homogeneous.\newline%
\newline\emph{Conclusion}: Let $\tau:X\rightarrow Y=(|t|<1)$ be a holomorphic,
proper, regular, surjective map, such that all fibers $X_{t}$, $t\neq0$, are
biholomorphic to $F$.\ \emph{ }Then the fiber $X_{0}$ is biholomorphic to $F$,
if $X_{0}$ is projective. \newline\emph{End of (*).}\newline\newline%
\emph{Remark}. The aim of \emph{(*)} is to list some of the properties of $F$
which will become important in this paper. Certainly the list is redundant,
the assumptions are by far too strong. Nor will our proof of theorems 1, 2,
3\ be conducted exactly along these lines,but only roughly, as there are
steps, which have to be handled differently for different types of spaces $F$.
\newline\ \newline$D_{F}$ and its trivial extension to $Y\times F$ are also
called the hyperplane bundle on $F$ resp. on $Y\times F$, although the zero
sets of\ sections in the hyperplane bundle of $F$ (also called
hyperplanes)\ are\ for $F\neq\mathbf{P}^{n}$ not necessarily complex
hyperplanes in the usual sense. - Here an irreducible\ curve in $F$ is a line
if it represents a (the positive) generating element of $H_{2}(F,\mathbf{Z}%
)=\mathbf{Z}$. If $F$ is embedded into a $\mathbf{P}^{N}$ via a base of
$H^{0}(F,D_{F})$, an irreducible curve in $F$ is a line in $F$, if it is a
line in $\mathbf{P}^{N}$, because a general hyperplane meets it just
once;\ especially lines in $F$ are rational curves. \newline\newline%
\emph{Theorem 3. Any (global)\ deformation of a del Pezzo manifold }$F$\emph{
of degree 5\ and dimension n }$\geq3$\emph{\ is analytically isomorphic to
}$F$\emph{.}\newline\newline In subsection 4.1 of chapter 4 we start the proof
with the construction of certain\ bimeromorphic maps $\mu:X\rightarrow
X^{^{\prime}}:=Y\times\mathbf{P}^{n}$, $\widehat{\mu}:X\rightarrow\widehat
{X}^{^{\prime}}:=Y\times F$, where $\widehat{\mu}$ is biholomorphic outside
$X_{0}$ and where $\mu$ induces a bimeromorphic map $\mu_{t}$\ of each $X_{t}%
$, $t\neq0$, onto $(t)\times\mathbf{P}^{n}$; these bimeromorphic maps are
defined by $\tau$ and sections $s_{0},...,s_{n}$, resp. $s_{0},...,s_{n}%
,...,s_{N}$ in the positive generating line bundle $\mathcal{D}$ of $X$. In
order to show the biholomorphy of the by $\widehat{\mu}$ induced meromorphic
map $\widehat{\mu}_{0}:X_{0}\rightarrow\widehat{X}_{0}^{^{\prime}}=(0)\times
F$ it suffices to prove the surjectivity of the induced meromorphic map
$\widehat{\mu}_{0}:X_{0}\rightarrow\widehat{X}_{0}^{^{\prime}}$ resp.
the\ surjectivity of the induced meromorphic map $\mu_{0}:X_{0}\rightarrow
X_{0}^{^{\prime}}$ $=$ $(0)\times\mathbf{P}^{n};$ here surjectivity means
$\widehat{\mu}[X_{0}]=\widehat{X}_{0}^{^{\prime}}$, resp. $\mu\lbrack
X_{0}]=X_{0}^{^{\prime}}$; at first sight this surjectivity\ assertion
possibly looks\ astonishing, but one has to take into account the special
situation one is dealing with, compare 2.3. . \newline\newline Suppose that
$\mu_{0}$ and $\widehat{\mu}_{0}$ are not surjective. Under this assumption we
study in chapter 5 and 7 the lifting and the holomorphic lifting of Euler
vector fields on $\widehat{X}^{^{\prime}}=Y\times F$ to $X$ and to the graph
$\widehat{G}$\ of$\ \widehat{\mu}$ and can after these preparations give a
proof of theorem 2, i.e. can disprove $X_{0}^{^{\prime\prime}}:=\mu\lbrack
X_{0}]\neq X_{0}^{^{\prime}}$ resp. $\widehat{X}_{0}^{^{\prime\prime}%
}:=\widehat{\mu}[X_{0}]\neq\widehat{X}_{0}^{^{\prime}}$. If $\widehat{X}%
_{0}^{^{\prime\prime}}\neq\widehat{X}_{0}^{^{\prime}}$, $\widehat{G}_{0}$ has
two\ components $\widehat{G}_{0}^{\ast}$, $\widehat{X}_{0}^{\ast}$, which do
not coincide, where $\widehat{G}_{0}^{\ast}$ is a modification of $\widehat
{X}_{0}^{^{\prime}}$ and where $\widehat{X}_{0}^{\ast}$ is a modification of
$X_{0}$, there may be more components. Of importance is the "attractor" resp.
the "limit set" (Hwang)$\ S^{\ast}$, the image of $\widehat{G}_{0}^{\ast}$ in
$X_{0};$ for practical purposes one may look at $S^{\ast}$ as the
indeterminate set of $\widehat{\mu}$. \newline\newline$\mu$ and $\widehat{\mu
}$ have the following property: Both maps are holomorphic outside $(s_{0}=0)$,
$\widehat{\mu}$ is biholomorphic outside $X_{0}$; $Y\times F\backslash
((s_{0}^{^{\prime}}=0)$ equals $Y\times\mathbf{C}^{n}$. \newline\newline
Formally all this leads finally to an application of the adjunction formula.
For this purpose it is essential to prove the existence of lines, i.e. of
minimal rational curves in general $\mu_{0}$-fibers, i.e. of irreducible,
rational curves in general $\mu_{0}$-fibers, which generate $H_{2}%
(X_{0},\mathbf{Z)=Z}$. \newline\newline On the first go theorem 1 is proved in
chapter 6, the proof of\ which is simpler than the proof in the general case
of theorem 2; because in the case of theorem 1 for instance irreducible curves
in $X_{0}$ are positive multiples of the positive generating element of
\ $H_{2}(X_{0},\mathbf{Z)=Z},$as $X^{^{\prime}}$ is k\"{a}hler resp.
projective. \newline\newline That the k\"{a}hler assumption for $X_{0}$ in
theorem 1 is superfluous as an assumption, can be explained (with a bit more
than just cum grano salis) thus: Critical is the case $\dim S^{\ast}\geq1$.
One can discard $\dim S^{\ast}\geq1$ in the case of theorem 1\ by employing a
certain line\ connectedness property of flag manifolds with $b_{2}=1$; the
k\"{a}hler assumption guarantees \ that\ curves in $S^{\ast}$ are positive
multiples of the positive generating element of $H_{2}(X_{0},\mathbf{Z}%
\emph{)=}\mathbf{Z}$, if $\dim S^{\ast}\geq1$. But this can also be guaranteed
with the help of Siu's method proving the non deformability of $\mathbf{P}%
^{n}$. This method assures that any general fiber $fib$ of $\widehat{\mu}_{0}$
rsp. $\mu_{0}$\ is a $\mathbf{P}^{m}$ naturally with $m=\dim fib$; one can
show that $S^{\ast}$ is of dimension $m-1$ and is\ contained in any general
fiber $fib$, $S^{\ast}$ is a hyperplane in $fib=\mathbf{P}^{m}$. The
homogenity of $fib$ delivers the wanted positivity result for curves.
\newline\newline An underlying basic line of thought of this paper to start
with can be described thus\textbf{: }We can assume $F\neq\mathbf{P}^{n}$.
Suppose that $X_{0}\neq F$, i.e. that $\widehat{X}_{0}^{^{\prime\prime}%
}=\widehat{\mu}_{0}[X_{0}]\neq\widehat{X}_{0}^{^{\prime}}$ $=(0)\times F$,
i.e. $m\geq1$,\ i.e.$\ \dim\widehat{X}_{0}^{^{\prime\prime}}=\dim
X_{0}^{^{\prime\prime}}$ $=$ $n-m\leq n-1$. If $\dim\widehat{X}_{0}%
^{^{\prime\prime}}=n-1$ (We call this the hypersurface case because in this
case $X_{0}^{^{\prime\prime}}$ is a hypersurface in\textbf{ }$\mathbf{P}^{n}%
$), i.e. $m=1$, $\dim S^{\ast}=0$, we get by holomorphic liftings of
certain\ Euler vector fields on $Y\times F$ to $X_{0}$ (additive) holomorphic
vector fields on $X_{0}$ with divisorial zeros, as is shown by direct
computation. Then \cite{Hwang1} or \cite{Hwang2} immediately deliver
$F=\mathbf{P}^{n}$, the wanted contradiction. - Now suppose $m\geq2$, i.e.
$\dim$ $\widehat{X}_{0}^{^{\prime\prime}}\leq n-2$, i.e. $\dim S^{\ast}>0.$
One possibility to proceed is: Show that general (suitable) lines
$l^{^{\prime}}$ on $(0)\times F=\widehat{X}_{0}^{^{\prime}}$ do not meet
$\widehat{X}_{0}^{^{\prime\prime}}\subset\widehat{X}_{0}^{^{\prime}}$ and that
general fibers of $\widehat{\mu}_{0}$ resp. $\mu_{0}$ are $\mathbf{P}^{m}$s
such that the adjunction formula is applicable delivering $X_{0}%
=\mathbf{P}^{n}$, the wanted contradiction. Another shorter possibility
(giving a weaker result) is: Suppose $X_{0}$ to be k\"{a}hler. Let
$l^{^{\prime}}$ be a line on $F$. Then $(\widehat{\mu}^{-1}[Y\times
l^{^{\prime}}])_{0}$ generates $H_{2}(X_{0},\mathbf{Z)=Z}$ and thus has to be
irreducible as $X_{0}$ is k\"{a}hler. Show that there are lines$\,l^{^{\prime
}}$ such that $(\widehat{\mu}^{-1}[Y\times l^{^{\prime}}])_{0}$ has at least 2
components, a contradiction.$\newline$\newline I have to thank Gregor Fels for
a discussion on one of my attempts (many years ago) to prove theorem 2 and 3
with inadequate means. Furthermore I am grateful to him for calling my
attention to the readable "Diplomarbeit" of A. Radtke (Bremen, 1996), where
\cite{Hwang2} was reworked.\newline

\subsection{More on the overall structure of the proof of theorem 1 and 2.}

\textbf{a. On the construction of }$\mu:X\rightarrow Y\times\mathbf{P}^{n}%
$\textbf{, }$\widehat{\mu}:X\rightarrow Y\times F$. Let\ $n\geq3$ and let for
$t\neq0$ the fiber $X_{t}$ be analytically isomorphic to $F$, a compact
irreducible, hermitian symmetric manifold (or more general, a flag manifold
with $b_{2}=1$) of dimension $n$,\ we may suppose $X$ to be diffeomorphic to
$Y\times X_{0}$. If nothing else is agreed upon we suppose for the rest of the
introduction $X_{0}\neq F$, i.e. $X_{0}^{^{\prime\prime}}=\mu\lbrack
X_{0}]\neq X_{0}^{^{\prime}}=(0)\times\mathbf{P}^{n}$, $\widehat{X}%
_{0}^{^{\prime\prime}}=\widehat{\mu}[X_{0}]\neq\widehat{X}_{0}^{^{\prime}%
}\ =(0)\times F$, $i.e.X_{0}\neq F$. We want to disprove the assumption
$X_{0}\neq F$. \newline\newline A theorem of Iitaka (see 2.1) implies that
$X_{0}$ is at least moishezon and this in turn gives $H^{1}(X,\mathcal{O}%
_{X})=H^{2}(X,\mathcal{O}_{X})=0,$ $H^{1}(X,\mathcal{O}_{X}^{\ast})\cong
H^{2}(X,\mathbf{Z)}=H^{2}(X_{t},\mathbf{Z)}\cong\mathbf{Z}$ for all $t\in Y.$
We denoted by $\mathcal{D}$ the positive generating line bundle of
$H^{1}(X,\mathcal{O}_{X}^{\ast})$, which induces for $t\neq0$ on $X_{t}=F$ the
hyperplane bundle, i.e. the positive generating line bundle on $F$, which is
very ample.\newline\newline In the following Grauert's image sheaf theorem
\cite{Grauert1} and Chow varieties (\cite{Barlet}, \cite{Fujiki2}%
,\cite{Shafarevich}) are applied. - We will show:\ Let $\mathcal{D}^{^{\prime
}}$ be the hyperplane bundle on $X^{^{\prime}}=Y\times\mathbf{P}^{n}$, i.e.
the line bundle which induces for all $t\in Y$ on $(t)\times\mathbf{P}%
^{n}=\mathbf{P}^{n}$ the hyperplane bundle. There are $s_{0},...,s_{n}%
\in\Gamma(X,\mathcal{D)}$ and $s_{0}^{^{\prime}},...,s_{n}^{^{\prime}}%
\in\Gamma(X^{^{\prime}},\mathcal{D}^{^{\prime}}\mathcal{)}$, such that the
restrictions $s_{i}\mid X_{t},\,i=0,...,n,$ are linearly independent over
$\mathbf{C}$ for all $t\in Y$ and give a bimeromorphic map $\mu:X\rightarrow
Y\times\mathbf{P}^{n},$ $P\rightarrow(\tau(P);s_{0}(P):...:s_{n}(P))$ $=$
$(\tau(P);s_{0}^{^{\prime}}(\mu(P)):...:s_{n}^{^{\prime}}(\mu(P)))$ for a
general $P\in X$, which induces for all $t\neq0$ a bimeromorphic map
$X_{t}\rightarrow(t)\times\mathbf{P}^{n};$ let $G\subset X\times
Y\times\mathbf{P}^{n}$ be the graph of $\mu$ with the projections
$\varphi:G\rightarrow X,$ $\psi:G\rightarrow Y\times\mathbf{P}^{n}$
$=:X^{^{\prime}}$. $G_{0}$ (= the fiber of $G$ over $0\in Y$) contains exactly
one component $G_{0}^{\ast}$ mapped by $\psi$ bimeromorphically onto
$(0)\times\mathbf{P}^{n}$ and exactly one component $X_{0}^{\ast}$ mapped by
$\varphi$ bimeromorphically onto $X_{0}.$ Denote $X_{0}^{^{\prime\prime}%
}:=\psi(X_{0}^{\ast})\subset X_{0}^{\prime}$. There are $s_{n+1},...,s_{N}%
\in\Gamma(X,\mathcal{D)}$ (with $N>n$, if $F\neq\mathbf{P}^{n}$), such that
$\tau$ and $s_{0},...,s_{n},s_{n+1},...,s_{N}$ (these are linearly independant
along $X_{0}$) define a bimeromorphic map $\widehat{\mu}:X\rightarrow Y\times
F$, which is biholomorphic outside $X_{0}$. Let $\widehat{G}\subset X\times
Y\times F$ be the graph of $\widehat{\mu}$ with the projections $\widehat
{\varphi}:\widehat{G}\rightarrow X,$ $\widehat{\psi}:\widehat{G}\rightarrow
Y\times F$ $=:\widehat{X}^{^{\prime}}.$ $\widehat{G}_{0}$ (= the fiber of
$\widehat{G}$ over $0\in Y)$ contains exactly one component $\widehat{G}%
_{0}^{\ast}$ mapped by $\widehat{\psi}$ bimeromorphically onto ($0)\times F$
and exactly one component $\widehat{X}_{0}^{\ast}$ mapped by $\widehat
{\varphi}$ bimeromorphically onto $X_{0}$. Denote $\widehat{X}_{0}%
^{^{\prime\prime}}:=\widehat{\psi}(\widehat{X}_{0}^{\ast})$. Let
$\widehat{\mathcal{D}}^{^{\prime}}$ be the positive generating line bundle on
$\widehat{X}^{^{\prime}}=Y\times F$; there are $\widehat{s}_{0}^{^{\prime}%
},...,\widehat{s}_{n}^{^{\prime}},\widehat{s}_{n+1}^{^{\prime}},...,\widehat
{s}_{N}^{^{\prime}}\in H^{0}(\widehat{X}^{^{\prime}}$, $\widehat{\mathcal{D}%
}^{^{\prime}})$ such that for all general $P\in X$ the relation $\widehat{\mu
}(P)=(\tau(P);s_{0}(P):...:s_{n}(P):s_{n+1}(P):...:s_{N}(P))$ $=(\tau
(P);\widehat{s}_{0}^{^{\prime}}(\widehat{\mu}(P)):...:\widehat{s}%
_{n}^{^{\prime}}(\widehat{\mu}(P)):\widehat{s}_{n+1}^{^{\prime}}(\widehat{\mu
}(P))$ $:...:\widehat{s}_{N}^{^{\prime}}(\widehat{\mu}(P))$ holds.
\newline\newline Behind $\mu$ and the choice of $s_{0}^{^{\prime}}%
,...,s_{n}^{^{\prime}}$ resp. $\widehat{s}_{0}^{^{\prime}},...,\widehat{s}%
_{n}^{^{\prime}}$ is the Bruhat embedding $\mathbf{C}^{n}=(\mathbf{P}%
^{n}\backslash(\infty-hyperplane))$ $\rightarrow F$ resp. the
related\ embedding $Y\times\mathbf{C}^{n}\rightarrow Y\times F$, which is
defined via a root space decomposition. $\widehat{s}_{0}^{^{\prime}%
},...,\widehat{s}_{n}^{^{\prime}}$ can be chosen such that there exists a
hypersurface $H_{F}$ in $F$ with $F\backslash H_{F}$ =$\mathbf{C}^{n}$,
$H_{F}=(\widehat{s}_{0}^{^{\prime}}=0)$ and \ such that the biholomorphism
$Y\times(F\backslash H_{F})\rightarrow Y\times\mathbf{C}^{n}\subset
Y\times\mathbf{P}^{n}$ can be extended to a bimeromorphic map $Y\times
F\rightarrow Y\times\mathbf{P}^{n}$. We may suppose that this bimeromorphic
map is defined by $P^{^{\prime}}\rightarrow(\tau(P^{^{\prime}});\widehat
{s}_{0}^{^{\prime}}(P^{^{\prime}}):...:\widehat{s}_{n}^{^{\prime}}%
(P^{^{\prime}}))$ for general $P^{^{\prime}}\in Y\times F=\widehat
{X}^{^{\prime}}$. This implies that for a general point $P\in X_{0}$ (with
$\mu_{0}(P)\in X_{0}^{^{\prime\prime}}\cap(\mathbf{P}^{n}\backslash
(\infty-hyperplane))=X_{0}^{^{\prime\prime}}$ $\cap((0)\times\mathbf{C}%
^{n})=X_{0}^{^{\prime\prime}}$ $\cap\mathbf{C}^{n}$) the $\mu_{0}$- fiber and
the $\widehat{\mu}_{0}$- fiber through $P$ coincide$.$\newline\newline We can
suppose $Y\times H_{F}=(s_{0}^{^{\prime}}=0)$. \newline\newline One can see
(compare 2.3 in chapter 2): $\widehat{\mu}$ is biholomorphic iff $G_{0}^{\ast
}=X_{0}^{\ast}$, i.e. iff $\widehat{G}_{0}^{\ast}=\widehat{X}_{0}^{\ast},$
i.e. iff $\mu_{0}:=$ $\mu\mid X_{0}:X_{0}\rightarrow$ $(0)\times\mathbf{P}%
^{n}$ is bimeromorphic, i.e. (in our case) iff $X_{0}^{^{\prime\prime}}=$
$\mu\left[  X_{0}\right]  $= $(0)\times\mathbf{P}^{n}$, i.e. iff $\widehat
{X}_{0}^{^{\prime\prime}}=$ $\widehat{\mu}\left[  X_{0}\right]  $= $(0)\times
F$; see 2.3. \newline\newline Concerning the notation $\mu|X_{0}$ and the
notion of the "restriction"$\ \mu_{0}:=\mu|X_{0}$: $X_{0}\rightarrow$
$\mu\left[  X_{0}\right]  $ compare the subsections 1.4 and 9.1: The graph of
$\mu_{0}$ is by definition $X_{0}^{\ast}=\overline{((P,\mu(P))|P\in X_{0}%
,\mu\text{ }is\text{ }holomorphic\text{ }in\text{ }P)}$; it is always $\mu
_{0}(P)\subset\mu(P);$ if $\mu$ is holomorphic in $P\in X_{0},$ then $\mu
_{0}(P)=\mu(P).$ The image in $X_{0}$ of a fiber of $X_{0}^{\ast}\rightarrow
X_{0}^{^{\prime\prime}}$ is a $\mu_{0}$-fiber; any general $\mu_{0}$-fiber is
the topological closure in $X$ of a fiber of $\mu|(\mu(P)|P\in X_{0},\mu$ is a
holomorphic map in an $X-$neighborhood of $P)$. The meromorphic map $\mu
|X_{0}$ is defined and exists if there exists a point $P\in X_{0}$ in which
$\mu$ is holomorphic.\ Analogously the notion of a $\widehat{\mu}_{0}$-fiber
can be introduced. $X_{0}^{\ast}$ is the graph of $\mu_{0}$, $\widehat{X}%
_{0}^{\ast}$ is the graph of $\widehat{\mu}_{0}$, we have $X_{0}%
^{^{\prime\prime}}=\psi(X_{0}^{\ast})=\mu\left[  X_{0}\right]  $, $\widehat
{X}_{0}^{^{\prime\prime}}=\widehat{\psi}(\widehat{X}_{0}^{\ast})=\widehat{\mu
}\left[  \widehat{X}_{0}\right]  $.\ Let $m$ ($=n-\dim X_{0}^{^{\prime\prime}%
}$) be the dimension of a general $\mu_{0}$-fiber. By a $\mu_{0}$-fiber we
understand a fiber of the projection $X_{0}^{\ast}\rightarrow X_{0}%
^{^{\prime\prime}}$ resp. its image in $X_{0}$, and by a $\widehat{\mu}_{0}%
$-fiber a fiber of the projection $\widehat{X}_{0}^{\ast}\rightarrow
\widehat{X}_{0}^{^{\prime\prime}}$ resp. its image in $X_{9}$.\newline\newline
We suppose $X_{0}^{^{\prime\prime}}\neq(0)\times\mathbf{P}^{n}$, i.e.
$\widehat{X}_{0}^{^{\prime\prime}}\neq(0)\times F$. On an introductory level
one may assume $G_{0}=X_{0}^{\ast}$ $\cup G_{0}^{\ast}$, $\widehat{G}%
_{0}=\widehat{X}_{0}^{\ast}$ $\cup\widehat{G}_{0}^{\ast}$; what counts is,
that $G_{0}^{\ast}$ cuts any general fiber of $X_{0}^{\ast}\rightarrow
X_{0}^{^{\prime\prime}}$ in codimension 1, resp. that $\widehat{G}_{0}^{\ast}$
cuts any general fiber of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$ in codimension 1. Of importance for our purposes is
the "attractor set" $S^{\ast}:=\varphi(G_{0}^{\ast})$ ($=\widehat{\varphi
}(\widehat{G}_{0}^{\ast})$). \newline\newline If $X_{0}^{^{\prime\prime}}%
=\psi(X_{0}^{\ast})=\mu\left[  X_{0}\right]  \neq(0)\times\mathbf{P}^{n}$,
then the common zero set $\widehat{S}$ of $s_{0},...,s_{N}$ is not empty;
naturally $\widehat{S}\subset X_{0}$, $\dim\widehat{S}\leq n-2$, and $S^{\ast
}\subset\widehat{S}$. As the $s_{i}\mid X_{0}$, $i=0,...,n,$ are linearly
independent over $\mathbf{C}$, $X_{0}^{^{\prime\prime}}$ cannot lie in a
proper linear subspace of $(0)\times\mathbf{P}^{n}=\mathbf{P}^{n}$ and is
especially of degree $d>1$.\ $H_{2}(X_{0},\mathbf{Z)=Z}$ and 2.2.b. imply
$\dim X_{0}^{^{\prime\prime}}=\dim\widehat{X}_{0}^{^{\prime\prime}}\geq2$.
\newline\newline If $X_{0}^{^{\prime\prime}}=(t=0,f_{d}^{^{\prime}}=0)$ with
an irreducible homogeneous polynomial $f_{d}^{^{\prime}}$ $\in\Gamma
(X^{^{\prime}},d\cdot\mathcal{D}^{^{\prime}}\mathcal{)}$ of degree $d$, then
$X_{0}^{^{\prime\prime}}$ is a hypersurface in $(0)\times\mathbf{P}%
^{n}=\mathbf{P}^{n}$ (we call this case the \textquotedblright%
\textit{hypersurface case}\textquotedblright), necessarily of degree $d\geq2$
and of dimension $\geq2$. In this situation the (complex)\ dimension of a
general $\mu_{0}$-fiber is $m=$ $1$. \newline\newline In this paper we suppose
most of the time\ for reasons of simple notations, if nothing else is agreed
upon, that general $\mu_{0}$-fibers (i.e. here\ general fibers of $X_{0}%
^{\ast}\rightarrow X_{0}^{^{\prime\prime}}$ too)\ are irreducible; thus
naturally sometimes\ we are talking about general\ fibers, where we should
talk about irreducible\ components of general fibers. The same shall hold for
general $\widehat{\mu}_{0}$-fibers; these are naturally general $\mu_{0}$-
fibers. \newline\newline Suppose we replace $\widehat{X}_{0}^{\ast}$ by a
canonical desingularization $\widehat{X}_{0}^{\ast\ast}$\ (being in this case
the normalization of $\widehat{X}_{0}^{\ast\text{ }}$, as it turns out)\ and
thus may\ assume without doing harm, that $\widehat{X}_{0}^{\ast}$ and
$\widehat{X}_{0}^{^{\prime\prime}}$ are smooth. Consider the set of points
$\in\widehat{X}_{0}^{\ast}$ where $rank(\widehat{\psi}|\widehat{X}_{0}^{\ast
})<\dim\widehat{X}_{0}^{^{\prime\prime}}$, here $rank$ can be looked at as
being\ defined by the rank of the functional matrix; its $\psi$-image is a
proper analytic subset of $\widehat{X}_{0}^{^{\prime\prime}}$. Let
$\widehat{X}_{0}^{\ast}\rightarrow\widetilde{Z}\rightarrow X_{0}%
^{^{\prime\prime}}$ be the Stein factorization (\cite{Grauert-Remmert}, p.
213). All fibers of $\widehat{X}_{0}^{\ast}\rightarrow\widetilde{Z}$\ are
connected, \ general fibers of $\widehat{X}_{0}^{\ast}\rightarrow\widetilde
{Z}$ are irreducible. - This tells: If we assume that general fibers of
$\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are
irreducible, we are talking of general fibers of such maps $\widehat{X}%
_{0}^{\ast}\rightarrow\widetilde{Z}$ . In the same way we can deal with
general $\widehat{\mu}_{0}$-fibers. Compare 5.2.c and 8.1.c. But if we deal
(in chapter 5 and 8) with normal bundles and with the adjunction formula, it
is not enough to have at disposal only statements on irreducible components of
general fibers, the irreducibility of general fibers can be proved, this is
done in 5.2.c and 8.1.e,\ and in this context we should not always replace
just without some discussion $\widehat{X}_{0}^{\ast}$ by $\widehat{X}%
_{0}^{\ast\ast}$\newline\newline\textbf{b. On k\"{a}hler deformations of
}$\mathbf{P}^{n}$ and $\mathbf{Q}_{n}$.\textbf{\ }The statements of this
subsection b. are known since long.- It is worthwhile to look briefly at\ the
most simple cases of the theorem 1 and\ to note, that the preceding subsection
can already\ be used in order to prove, that k\"{a}hler (i.e. in our case
projective) deformations of $\mathbf{P}^{n}$ are $\mathbf{P}^{n}$s, i.e. can
be used to prove theorem 1 for $F=\mathbf{P}^{n}$. One simple approach is: We
employ that $H_{2\cdot\nu}(\mathbf{P}^{n},\mathbf{Z)=Z}$ for\ $\nu=0,...,n$
and that all the other homology groups over $\mathbf{Z}$ are $0$, and that any
complex\ linear $\mathbf{P}^{n}$-subspace of complex dimension $\nu$ (resp.
its homology class) generates $H_{2\cdot\nu}(\mathbf{P}^{n},\mathbf{Z)}$. Any
irreducible (complex)$\ \nu$-dimensional analytic subset of \ $\mathbf{P}^{n}$
of degree $d$ \ is homologous to the $d$-fold multiple of the
positive\ generating element $\in H_{2\cdot\nu}(\mathbf{P}^{n},\mathbf{Z)}$.
The fiber$\ X_{0}$ is diffeomorphic to $\mathbf{P}^{n}$ and has the same
homology ring as $\mathbf{P}^{n}$, $X$ can assumed to be diffeomorphic to
$Y\times\mathbf{P}^{n}$. It suffices to discuss the case $n=3$. We give a
sketch in order to describe the means which are used.:\newline\newline Let
$\tau:X\rightarrow Y$ be a differentiably trivial deformation map with
$X_{t}=\mathbf{P}^{3}$ for $t\neq0$ and a projective central fiber $X_{0}$.
Let $\mu:X\rightarrow Y\times\mathbf{P}^{3}$be a bimeromorphic map as defined
above (for $F=\mathbf{P}^{3}$), which maps $X\backslash X_{0}$
biholomorphically onto $(Y\backslash(0))\times\mathbf{P}^{3}$. There are
$s_{0},...,s_{3}\in\Gamma(X,\mathcal{D)}$ such that the$\ s_{i}\mid
X_{t},\,i=0,...,3,$ are linearly independent over $\mathbf{C}$ for all $t\in
Y$ and define the bimeromorphic map $\mu:X\rightarrow Y\times\mathbf{P}^{3}$
with $P\rightarrow(\tau(P);s_{0}(P):...:s_{3}(P))$ for general $P\in X$. For
$t\neq0$ the bimeromorphic map $\mu$ induces a biholomorphic map
$X_{t}\rightarrow X_{t}^{^{\prime}}=\mathbf{P}^{3}$ and thus an isomorphism of
the corresponding homology rings. Suppose that $\mu$ is not biholomorphic,
i.e. that $\mu_{0}$ is not holomorphic. Then $2=\dim X_{0}^{^{\prime\prime}}$
(see 2.3 of the present paper) and $d=\deg X_{0}^{^{\prime\prime}}\geq2,$ here
$m=1$ is the dimension of a general $\mu_{0}$-fiber. Let $S$ be the common
zero set of $s_{0}$, .., $s_{3}$; it can easily be seen (compare 2.2.b) that
$S$ $\subset X_{0}$ is of dimension $0$. We may suppose $S$ to be irreducible.
\newline\newline Now let $L_{0}^{^{\prime}}$ be a general linear complex
$1$-dimensional subspace of $\mathbf{P}^{3}$, \ let $L^{^{\prime}}:=Y\times
L_{0}^{^{\prime}}$, $L:=\varphi(\psi^{-1}[L^{^{\prime}}])$. For all $t$ \ the
complex $1$-dimensional curve $L_{t}$ generates $H_{2}(X_{t},\mathbf{Z}%
)=\mathbf{Z}$, for reasons of continuity. But $L_{0}^{^{\prime}}$ meets
$X_{0}^{^{\prime\prime}}$ in $d$ general points. The fiber $L_{0}$ of
$L=\varphi(\psi^{-1}[L^{^{\prime}}])$ decomposes on the one hand into at least
$d\geq2$ components of complex dimension $m=1$, but\ on the other hand $L_{0}$
must be irreducibel, as $L_{0}$ generates $H_{2\cdot m}(X_{0},\mathbf{Z}%
)=\mathbf{Z}$. Consequently $\mu$ is biholomorphic. \newline\newline Briefly
with other words: The essential tool in this procedure is the$\ $equality of
the degree $c_{1}(\mathcal{D}_{t}\mathcal{)}^{n}$ $=1$ for all $t$; this
degree is a\ topological invariant, indepedent of $t$. One makes use of the
fact, that general irreducible $m$-dimensional analytic sets in $X_{0}$ induce
in $H_{2\cdot m}(X_{0},\mathbf{Z)=Z}$ positive multiples of the (positive)
generating element.\newline\newline With little effort this method can be made
work for the smooth quadric $\mathbf{Q}_{n}$ of dimension $n\geq3$. We will
briefly indicate the sort of reasoning for $n=3$.\ Suppose there is a
differentiably trivial\ deformation map $\tau:X\rightarrow Y$ with a fiber
preserving bimeromorphic map $\widehat{\mu}:X\rightarrow Y\times\mathbf{Q}%
_{3}\subset Y\times\mathbf{P}^{4}$ as described above, esp. with the following
properties: $\widehat{\mu}$ is not biholomorphic, only biholomorphic outside
$X_{0}$, .i.e. $\widehat{\mu}[X_{0}]=\widehat{X}_{0}^{^{\prime\prime}}$ is a
proper analytic subset of $(0)\times\mathbf{Q}_{3}$; the central fiber $X_{0}$
is k\"{a}hler resp. projective; $Y\times\mathbf{Q}_{3}\subset Y\times
\mathbf{P}^{4}$ is an embedding; $\widehat{\mu}$ is defined by $\tau$ and
elements $s_{0},...,s_{4}\in\Gamma(X,\mathcal{D)}$, the $s_{i}\mid
X_{t},\,i=0,...,4,$ are linearly independent over $\mathbf{C}$ for all $t\in
Y$ \ As the degrees $c_{1}(\mathcal{D}_{t}\mathcal{)}^{3}$ for all $t$ are $2$
(because we are dealing with deformations of smooth quadrics and because the
homology ring of $X_{0}$ is isomorphic to the one of $\mathbf{Q}_{3}$), it is
easy to see that $\widehat{\mu}[X_{0}]=\widehat{X}_{0}^{^{\prime\prime}%
}\subset(0)\times\mathbf{P}^{4}=\mathbf{P}^{4}$ is of dimension $2$ and\ has
to be a quadric too; have in mind that these degrees are topological
invariants But then $\widehat{X}_{0}^{^{\prime\prime}}$ has to lie in a linear
$\mathbf{P}^{4}$-subspace of dimension $\dim\widehat{X}_{0}^{^{\prime\prime}%
}+1=3$ and $s_{0}|X_{0},...,s_{4}|X_{0}$ are linearly dependent, which is not
the case by construction. This shows that smooth quadrics of dimension $\geq3$
are locally and globally non deformable under k\"{a}hler deformations.This
approach does not only work for the smooth quadric $\mathbf{Q}_{n}$ with
$n\geq3$ uneven, but also for $n\geq4$ even\ -\ This is not true for the
smooth quadric $\mathbf{Q}_{2}$ of dimension $2$; $\mathbf{Q}_{2}%
=\mathbf{P}^{1}\times\mathbf{P}^{1}$ is locally non deformable, but (global)
k\"{a}hler deformations of $\mathbf{Q}_{2}$ can be different from
$\mathbf{Q}_{2}$ (see \cite{Kodaira}, p.54 and p. 71-75)..\newline\newline A
definition: Let $\widetilde{F}$ be an $n-$dimensional moishezon manifold with
$H^{2}(\widetilde{F},\mathbf{Z)=Z}$ and let $\mathcal{D}_{\widetilde{F}}$ be a
line bundle on $\widetilde{F}$. Define the $\Delta-$genus of $\ \mathcal{D}%
_{\widetilde{F}}$ to be$\ \Delta(\mathcal{D}_{\widetilde{F}}):=n+c_{1}%
(\mathcal{D}_{\widetilde{F}})^{n}-\dim H^{0}(\widetilde{F},\mathcal{D}%
_{\widetilde{F}})$. Suppose $n\geq2$; then $H^{2}($ $\widetilde{F}%
,\mathbf{Z)=Z}$ implies $H^{1}(\widetilde{F},\mathcal{O}_{\widetilde{F}}%
^{\ast})=\mathbf{Z}$. Let $\mathcal{D}_{\widetilde{F}}$ $\in H^{1}%
(\widetilde{F},\mathcal{O}_{\widetilde{F}}^{\ast})$\ on $\widetilde{F}$ be the
positive generating line bundle\ on $\widetilde{F}$.\newline\newline As
$\widetilde{F}$ is moishezon, there exists a proper analytic subset $T$ of
$\widetilde{F}$ with the following property: If $A$ is an irreducible
$k-$dimensional analytic subset of $\widetilde{F}$ with $A\nsubseteqq T$, then
always $A\circ c_{1}(\mathcal{D}_{\widetilde{F}})^{k}>0$. But we have to be
aware that for $A\subset T$ possibly $A\circ c_{1}(\mathcal{D}_{\widetilde{F}%
})^{k}<0$. \newline\newline Let us call $\mathcal{D}_{\widetilde{F}}$
numerically semipositive, if always in such a situation$\ A\circ
c_{1}(\mathcal{D}_{\widetilde{F}})^{k}\geq0$ holds. In our case
numerical\ semipositivity is equivalent to nefness, as can be seen by an
adaption of the proof in chap. I, \S 6 of the book \cite{Hartshorn}, here
nefness of $\mathcal{D}_{\widetilde{F}}$ means that for all irreducible curves
$A\subset\widetilde{F}$ the condition $A\circ c_{1}(\mathcal{D}_{\widetilde
{F}})\geq0$ holds. \newline\newline The deformation\ reasoning with
$\mathbf{P}^{n}$, $\mathbf{Q}_{n}$ above also works, if the assumption, that
$X_{0}$ is k\"{a}hler is replaced by the assumption, that the positive
generating line bundle $\mathcal{D}_{0}$ on $X_{0}$ is numerically
semipositive. \newline\newline\emph{Remark}: K. Timmerscheidt improved and
sharpened these statements\ on k\"{a}hler deformations of $\mathbf{P}^{n}$,
$\mathbf{Q}_{n}$, which had been discussed in a seminar, and the related
method\ in his "Diplomarbeit" (Essen,1979). - Several people discussed the
rigidity of $\mathbf{P}^{n}$and the smooth $\mathbf{Q}_{n}$(for $n\geq3$)
under k\"{a}hler deformations from different angles in the 1970s. - K.
Timmerscheidt established in this respect in his "Diplomarbeit" the following
characterization of $\mathbf{P}^{n}$ and of odd-dimensional smooth quadrics
$\mathbf{Q}_{n}$ . Have in mind that here $H^{2}(\widetilde{F},\mathbf{Z)=Z}$
implies $H^{1}(\widetilde{F},\mathcal{O}_{\widetilde{F}}^{\ast})=\mathbf{Z}$:
\newline\newline\emph{Theorem. }Let $\widetilde{F}$ be an $n-$dimensional
moishezon manifold with $H^{2}($ $\widetilde{F},\mathbf{Z)=Z}$, $n\geq3$ .
Suppose that the positive generating line bundle $\mathcal{D}_{\widetilde{F}}$
$\in H^{1}(\widetilde{F},\mathcal{O}_{\widetilde{F}}^{\ast})$\ on
$\widetilde{F}$ is numerically semipositive.\textbf{ a}. If $c_{1}%
(\mathcal{D}_{\widetilde{F}})^{n}=1$ and $\Delta(\mathcal{D}_{\widetilde{F}%
})\leq0$, then $\mathcal{D}_{\widetilde{F}}$ is very ample and $\widetilde
{F}=\mathbf{P}^{n}$. \textbf{b}\emph{. }Assume $n\geq3$ uneven$,$
$c_{1}(\mathcal{D}_{\widetilde{F}})^{n}=2$, $\Delta(\mathcal{D}_{\widetilde
{F}})\leq0$ and that the homology ring of $\widetilde{F}$ is isomorphic to the
one of the smooth $\mathbf{Q}_{n}$. Then $\mathcal{D}_{\widetilde{F}}$ is very
ample and $\widetilde{F}=\mathbf{Q}_{n}$\newline\newline Concerning statement
\textbf{a}: Here the assumption$\ c_{1}(\mathcal{D}_{\widetilde{F}})^{n}=1$
already\ implies that\ the homology ring of $\widetilde{F}$ is isomorphic to
the one of $\mathbf{P}^{n}$.\newline\newline\emph{Remark. }Let $\widetilde{F}$
be a deformation of $\mathbf{P}^{3}$ and let $\mathcal{D}_{\widetilde{F}}$
be\ the positive generating line bundle on $\widetilde{F}$ \emph{. }If
$\widetilde{F}\neq\mathbf{P}^{3}$, then\ there must exist an irreducible curve
$A$ of $\widetilde{F}$ with $A\circ c_{1}(\mathcal{D}_{\widetilde{F}})<0$. If
such a curve $A$ does not exist, we would be done in the $\mathbf{P}^{3}$-
deformation case. The same holds in the $\mathbf{Q}_{3}$-deformation case.\ -
In order to decide this question, whether such curves $A$ exist, one has to
look out for additional tools..\newline\newline All efforts to eliminate at
this stage the k\"{a}hler condition for $X_{0}$ without a new\ powerful tool
failed. It turned out that Euler vector fields delivered these tools.\newline%
\newline\textbf{c. On Euler vector fields. }After this interruption let us
continue with \textbf{a}. We need a good description of general$\ \widehat
{\mu}_{0}$-fibers resp. of the components of general $\widehat{\mu}_{0}%
$-fibers as certain orbital varieties; if $m=1$ these are orbital curves of
certain additive holomorphic vector fields $\chi_{0}$ on $X_{0}$, i.e. the
closures of integral $\chi_{0}$-curves; if $m>1$, roughly speaking, by
combining suitable several (i.e. $m$) independant vector fields of this type
one gets integral manifolds the closures of which\ are called orbital
varieties. \newline\newline Let $F$ be a flag manifold with Picard number one.
Let $\chi^{^{\prime}}$ be an Euler vector field\ on $\mathbf{P}^{n}$ resp. on
$Y\times\mathbf{P}^{n}$, this type of\ vector field\ can be described in
a\ suitable $\mathbf{C}^{n}=\mathbf{P}^{n}\backslash\infty-hyperplane$ \ resp.
$Y\times\mathbf{C}^{n}$ as $\sum(z_{i}-a_{i})\frac{\partial}{\partial z_{i}}$
where$(t$,$a)=(t,a(t))=(t,(a_{1}(t),...,a_{n}(t)))$ is a holomorphic section
$\sigma$ over $Y$; the point $a(0)$ resp. $(t$, $a(t))$\ is called\ the center
of $\chi_{0}^{^{\prime}}$, resp.$\ \chi_{t}^{^{\prime}}$ resp. (by abuse of
notation) $\chi^{^{\prime}}$; here in general we suppose that $(0)\times
a(0)\in$\ $Y\times\mathbf{C}^{n}\subset Y\times\mathbf{P}^{n}$ is a general
point of $X_{0}^{^{\prime\prime}}$.\ Most of the time it will not do any harm
to think of $a(t)$ as constant and to identify the graph of the function
$a(t)$ with $Y\times a$ with $a\in\mathbf{C}^{n}$.\ We can look at
$\mathbf{C}^{n}$ as the complement of a certain hypersurface $H_{F}$\ on $F$
and can extend $\chi^{^{\prime}}|\mathbf{C}^{n}$ to a holomorphic vector field
$\widehat{\chi}^{^{\prime}}$\ on $Y\times F$. As the by $\widehat{\mu}$
induced map $X\backslash X_{0}\rightarrow\widehat{X}^{^{\prime}}%
\backslash\widehat{X}_{0}^{^{\prime}}$ is biholomorphic, we can lift
$\widehat{\chi}^{^{\prime}}$ to $X\backslash X_{0}$; this lifted vector field
can be extended to a meromorphic vector field $\widetilde{\chi}$\ on $X$
(called the \emph{lifting} or the \emph{meromorphic lifting}\ of
$\widehat{\chi}^{^{\prime}}$ to $X$), which has $X_{0}$ as\ a pole set of a
certain pole order $k>0$. By multiplying $\widetilde{\chi}$ with $t^{k},$ one
gets a holomorphic vector field $\chi$ on $X,$ such that the restriction
$\chi_{0}$ of $\chi$ to $X_{0}$ is a holomorphic vector field on $X_{0}$,
which\ is not identically zero; $\chi_{0}$ turns out to be \emph{additive}; we
call $\chi$ and$\chi_{0}$\ a \emph{holomorphic lifting} of $\chi^{^{\prime}}$
resp. $\widehat{\chi}^{^{\prime}}$. \newline\newline As $\widetilde{\chi}$ has
$X_{0}$\ as pole divisor, general\ \emph{orbital curves }$\gamma_{0}^{\ast
(1)}$, i.e. closures of integral curves of such vector fields $\chi_{0}$,\ lie
in general fibers of $\mu_{0}$. In the same fashion one can lift
$\chi^{^{\prime}}$ resp. $\widehat{\chi}^{^{\prime}}$ to a meromorphic vector
field $\widetilde{\chi}_{\widehat{G}}$ on $\widehat{G}$, such that
$\widehat{\chi}_{\widehat{G}}:=t^{k}\cdot\widetilde{\chi}_{\widehat{G}}$ is
holomorphic on $\widehat{G}$ and $\widehat{\varphi}_{\ast}\widehat{\chi
}_{\widehat{G}}=\chi$ holds. We call also $\widehat{\chi}_{\widehat{G}}$
(instead of $\widehat{\chi}_{\widehat{G}}$ we also use the notation
$\widehat{\chi}^{\ast}$) and $\widehat{\chi}_{0}^{\ast}:=\widehat{\chi}^{\ast
}|\widehat{X}_{0}^{\ast}$\ a holomorphic lifting. - Have in mind that we
distinguish between liftings $\widetilde{\chi}$, $\widetilde{\chi}%
_{\widehat{G}}$, which may be meromorphic, and holomorphic liftings $\chi$,
$\chi_{0}$, $\widehat{\chi}_{\widehat{G}}$, $\widehat{\chi}^{\ast}$,
$\widehat{\chi}_{0}^{\ast}$; these holomorphic liftings will also be called
Euler vector fields.\newline\newline$\chi_{0}|\gamma_{0}^{\ast(1)}$ is
additive, is set theoretically only zero in $\gamma_{0}^{\ast(1)}\cap S^{\ast
}$, analytically this point is a double zero of $\chi_{0}|\gamma_{0}^{\ast
(1)}$, see \cite{Hwang1}, p. 329. \newline\newline We show nearly right at the
beginning of chapter 5 and 7, that the lifted vector fields $\widetilde{\chi}$
mentioned above have proper poles along $X_{0}$ of an order $k\geq1$; this
means that the (irreducible components of) general fibers of $\mu_{0}$ contain
the orbital varieties of the corresponding holomorphc liftings $\chi$;\ one
can handle this by calculation. - It is no restriction to suppose, that that
$X_{0}^{^{\prime\prime}}$ does not lie in a proper linear subspace of
$X_{0}^{^{\prime}}=(0)\times\mathbf{P}^{n}$, i.e. that $s_{0}|X_{0}%
,...,s_{n}|X_{0}$ are linearly independent.\newline\newline So far we dealt
with the notion of Euler vector fields on $\mathbf{P}^{n}$. Now a few words on
the definition of Euler vector fields $\widehat{\chi}^{^{\prime}}$ on
arbitrary flag manifolds $F$ \ with Picard number $1$. How should Euler vector
fields be defined ? This is clear for $F=\mathbf{P}^{n}$, $F=\mathbf{Q}_{n}$
with $n\geq3$ or $F=$ a grassmannian. For arbitrary $F$ we can proceed thus:
Embed $\mathbf{C}^{n}$ suitably in $F$ (Bruhat embedding), let $\widehat{\chi
}$ be an Euler vector field on $\mathbf{C}^{n}$, this vector field on
$\mathbf{C}^{n}$ can be extended holomorphically to $\mathbf{P}^{n}$ (clear).
Show that $\widehat{\chi}$ can be extended to a holomorphic vector field
$\widehat{\chi}^{^{\prime}}$ on $F$, for instance using 3.1, 3.2, 3.3, esp.
3.1.d. Such a vector field has all the properties which are needed for our
purposes. Let us call it an Euler vector field on $F$. \newline\newline One of
the first tasks is to show the following assertions: Let $\widehat{\chi
}^{^{\prime}}$ be a general Euler vector field on $\widehat{X}^{^{\prime}%
}=Y\times F$ such that its restriction $\widehat{\chi}_{0}^{^{\prime}}$ to
$\widehat{X}_{0}^{^{\prime}}=(0)\times F$\ has as center a general point of
\ $\widehat{\mu}[X_{0}]=\ \widehat{X}_{0}^{^{\prime\prime}}$. Then any general
orbital curve of $\widehat{\chi}_{0}^{^{\prime}}$ on $(0)\times F$%
\ $=\widehat{X}_{0}^{^{\prime}}$\ meets $\widehat{X}_{0}^{^{\prime\prime}}$ at
most in a finite set of points, i.e. $\widehat{X}_{0}^{^{\prime\prime}}$
cannot be covered completely by compact orbital curves of $\widehat{\chi}%
_{0}^{^{\prime}}$ lying in $\widehat{X}_{0}^{^{\prime\prime}}$. If$\ \widehat
{X}_{0}^{^{\prime\prime}}$ is a hypersurface in $\widehat{X}_{0}^{^{\prime}}$,
i.e. if $m=1,$ i.e. if $\dim S^{\ast}=0$, then the holomorphic lifting
$\chi_{0}$ of $\widehat{\chi}^{^{\prime}}$ to $X_{0}$ is an additive vector
field with a divisorial zero; in such a situation $X_{0}=\mathbf{P}^{n}$ holds
(\cite{Hwang1};theorem 1;\cite{Hwang2}). \newline\newline Especially these
calculations give that the vector fields $\widetilde{\chi}$ we constructed
above have proper poles along $X_{0}$. According to \cite{Hwang1}, p. 318,
this occurrence of poles in this context has originally been observed by T.
Mabuchi. \newline\newline A scalar\ ($\neq0$)-multiple of an Euler vector
field will also be called an Euler vector field. Sometimes, when speaking of
Euler vector fields, we have a certain standardization in mind (this will be
clear from the context), so that an Euler vector field is (up to certain
automorphisms)\ determined by its "center", a discussion is conducted in
chapter 3.. In the following (mainly) 2 standardizations for flag
manifolds\ will be used, which differ by the factor $-1$. We will call both
expressions, $\ \sum z_{i}\cdot\frac{\partial}{\partial z_{i}}$ and $-\sum
z_{i}\cdot\frac{\partial}{\partial z_{i}}$, Euler vector fields with center
zero. Replacing an Euler vector field by a ($\neq0$)-multiple scalar\ only
implies a reparametrization of the corresponding orbital curves and does not
change the geometry of the system of orbital curves.\newline\newline We are
performing calculations with Euler vector fields on $Y\times\mathbf{C}%
^{n}\subset Y\times\overline{\mathbf{C}^{n}}=Y\times\mathbf{P}^{n}$,
therefore\ we will besides $\widehat{\mu}:X\rightarrow\widehat{X}^{^{\prime}%
}=Y\times F$ also investigate $\mu:X\rightarrow X^{^{\prime}}=Y\times
\mathbf{P}^{n}$.\newline\newline Euler vector fields on these spaces $F$
depend on $F$ and on certain hypersurfaces (hyperplanes, i.e. hypersurfaces
generating $H_{2n-2}(F,\mathbf{Z)=Z}$) $H_{F}$ of $F$ with $F\backslash
H_{F}=\mathbf{C}^{n}$, here we use very often in such a situation an equality
sign "$=$" although $F\backslash H_{F}$being biholomorphic to $\mathbf{C}^{n}$
is meant. For $F=\mathbf{P}^{n}$ all hyperplanes are possible choices $H_{F}$.
For $F=\mathbf{P}^{n}$ such Euler vector fields are zero in all points of
$H_{F}$ (with order one in all general points of $H_{F}$). In such a situation
we call $H_{F}$ also a (reduced) divisor of such an Euler vector field.
\newline\newline Another simple\ example. In the case of smooth quadrics $F=$
$\mathbf{Q}_{n}\subset\mathbf{P}^{n+1}$ with $n\geq3$, possible choices for
such $H_{F}$ are the intersections $H_{F}=\mathbf{Q}_{n}\cap T_{P}$ of
$\mathbf{Q}_{n}$ and the tangent hyperplanes $T_{P}\subset\mathbf{P}^{n+1}$ to
$\mathbf{Q}_{n}$ in points $P\in\mathbf{Q}_{n}$. The projection $\mu
_{F}:F\rightarrow\mathbf{P}^{n}$\ of $F=\mathbf{Q}_{n}\subset\mathbf{P}^{n+1}$
from $P$ onto a $\mathbf{P}^{n}$ can be described by: $\mu_{F}$\ is
bimeromorphic and is holomorphic in all points $\neq P$, i.e. $P$ is the only
point of indeterminancy of $\mu_{F}$; $\mu_{F}$\ maps $\mathbf{C}%
^{n}=\mathbf{Q}_{n}\backslash(T_{P}\cap\mathbf{Q}_{n})$ biholomorphically onto
the complement of a hyperplane $H^{^{\prime}}\subset\mathbf{P}^{n}$, a
$\mathbf{C}^{n}$, blows down $H_{F}=T_{P}\cap\mathbf{Q}_{n}$ to a smooth
quadric $\mathbf{Q}_{n-1}\subset H^{^{\prime}}$ of dimension $n-1$ and blows
up $P$ to a $\mathbf{P}^{n-1}$ which we may identify with $H^{^{\prime}}$,
$\mathbf{Q}_{n-1}$ is the indeterminancy set of $\mu_{F}^{-1}$ . -\ Let
$\chi^{^{\prime}}$ be a $\mathbf{P}^{n}$-\ Euler vector field with center
$\in\mathbf{P}^{n}\backslash H^{^{\prime}}$ and vanishing (with order one)\ in
all general\ points of $H^{^{\prime}}$, i.e. $H^{^{\prime}}$ is a reduced
divisor of $\chi^{^{\prime}}$. We can restrict $\chi^{^{\prime}}$ to
$\mathbf{P}^{n}\backslash H^{^{\prime}}$, lift via $\mu_{F}$ this restriction
to $\mathbf{Q}_{n}\backslash(T_{P}\cap\mathbf{Q}_{n})$ and can extend this
lifted restriction to a holomorphic vector field $\widehat{\chi}^{^{\prime}}$
on $\mathbf{Q}_{n}$; the corresponding computations use (besides Riemann's
extension theorem) that the graph of $\mu_{F}$ is a monoidal modification of
$\mathbf{P}^{n}$ with blowing up $\mathbf{Q}_{n-1}$ and decisively that
$H^{^{\prime}}$ is a (reduced) divisor of $\chi^{^{\prime}}$. Because of this
set up we can lift $\chi^{^{\prime}}$ to a holomorphic vector field on the
graph of $\mu_{F}$ and then map this vector field onto a holomorphic vector
field $\widehat{\chi}^{^{\prime}}$\ on $F=$ $\mathbf{Q}_{n}$, the wanted Euler
vector field on $\mathbf{Q}_{n}$. \newline\newline\emph{Remark. }In the
following we will sometimes fix one possible suitable $H_{F}$ as $\infty
-$hyperplane without pointing out this explicitely, we will speak just of
(holomorphic) Euler vector fields $\widehat{\chi}^{^{\prime}}$\ on $F$\ and
will perform computations with one or several Euler vector fields all
corresponding to the same $H_{F}$. The following properties of the flag
manifold $F$ with Picard number $1$, of $H_{F}$\ and of $\widehat{\chi
}^{^{\prime}}$ should be fulfilled: It is$\ F\backslash H_{F}=\mathbf{C}^{n}$,
the orbital curves of $\widehat{\chi}^{^{\prime}}$ in $F$ exist and meet
$H_{F}$ in a finite set of points or are contained in $H_{F}$. (A non trivial)
Euler vector field$\ \widehat{\chi}_{0}^{^{\prime}}$ on $(0)\times F$ \ has
just one zero in $\mathbf{C}^{n}$.There are sections $s_{0},...,s_{n}$in the
line bundle $\mathcal{D}_{F}$ defined by $H_{F}$ with $H_{F}=(s_{0}=0)$ which
define a bimeromorphic map $\mu_{F}:F\rightarrow\mathbf{P}^{n}$, such that
$F\backslash H_{F}$ is mapped biholomorphically onto the complement of a
$\mathbf{P}^{n}$-hyperplane. Let $z_{1}=\frac{s_{1}}{s_{0}},...,z_{n}%
=\frac{s_{n}}{s_{0}}$ and let the zero point $O=(z_{1}=...z_{n}=0)$ of
$\mathbf{C}^{n}$ be the isolated zero of $\chi_{F}$ in $\mathbf{C}^{n}$. Then
$\widehat{\chi}^{^{\prime}}|\mathbf{C}^{n}$ has (up to a scalar multiple
$\neq$ $0$) the form $\sum z_{i}\cdot\frac{\partial}{\partial z_{i}}$. - So
when speaking of Euler vector fields on $F$ one should have (at least in mind)
a specific possible $H_{F}$. \newline\newline\textbf{d.} \textbf{The
adjunction formula is applied.} Above we introduced the attractor set
$S^{\ast}=$ $\varphi(G_{0}^{\ast})$\ $=$ $\widehat{\varphi}(\widehat{G}%
_{0}^{\ast})$. - Each general $\mu_{0}$-fiber meets $S^{\ast}$ in dimension
$m-1$, therefore $\dim S^{\ast}\geq m-1$; if $\dim S^{\ast}=m-1$ all general
fibers of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$
are of pure dimension $m$ and all general$\ \widehat{\mu}_{0}$-fibers (and all
general$\ \mu_{0}$-fibers)\ contain $S^{\ast}$. We may want $S^{\ast}$ as
small as possible; we cannot hope for an easy proof of $\dim S^{\ast}=0$, i.e.
$m=1$. We have to be content with $\dim S^{\ast}=m-1$. \newline\newline The
case $m=1$ bears no difficulties, the existence of minimal rational curves of
degree one in $X_{0}$ can be proved with the help of \cite{Hwang1}, lemma 1
resp. theorem 1. One can finish the ($\dim S^{\ast}=0$) - case by showing that
general $\mu_{0}$ - fibers are lines (i.e. \ irreducible rational\ curves of
degree one) in $X_{0}$; if $F=\mathbf{P}^{n}$ one can show $\dim S^{\ast}=0$
following Hwang, and can consider the case $F=\mathbf{P}^{n}$ to be dealt
with. Suppose $F\neq\mathbf{P}^{n}$. Consider the normal bundle of such a
general $\mu_{0}$ - fiber in $X_{0}$; as $\dim S^{\ast}=0$, the adjunction
formula implies $c_{1}(X_{0})=n+1$, the wanted contradiction, as on the other
hand $c_{1}(F)=c_{1}(X_{0})<n+1$. If $X_{0}$ is k\"{a}hler, one too can show
$\dim S^{\ast}=0$. $\ $For $F\neq\mathbf{P}^{n}$ and not necessarily
k\"{a}hler we are left with a discussion of $m\geq2$. \newline\newline The
procedure giving 10.2 of \cite{Siu2} delivers, that over general points of
$\widehat{X}_{0}^{^{\prime\prime}}$ the graph $\widehat{X}_{0}^{\ast}$\ of
$\widehat{\mu}_{0}:\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}%
^{^{\prime\prime}}$ is a $\mathbf{P}^{m}-$bundle and that this allows to prove
$\dim S^{\ast}=m-1$, if $\widehat{X}_{0}^{\ast}$ is smooth, one proves even
more, namely that $S^{\ast}$ lies in the intersection of all $\widehat{\mu
}_{0}$-fibers; if $\widehat{X}_{0}^{\ast}$ is not smooth, replace $\widehat
{X}_{0}^{\ast}$ by a canonical desingularization $\widehat{X}_{0}^{\ast\ast}$,
which amounts in our case de facto to move from $\widehat{X}_{0}^{\ast}$ to
its normalization. \newline\newline Now we can apply\ again\ the adjunction
formula\ and produce a contradictory statement in a similar fashion as in the
case $\dim S^{\ast}=0$, if general $\mu_{0}-$fibers contain orbital curves,
which are minimal rational curves (of degree one) in $X_{0}$, but this is the
case. \newline\newline\textbf{e. Further remarks on the rough pattern of
proof.} So summarizing,\ the basic pattern of the proof of theorem 1 can
briefly\ be described thus: Let $F$ be an $n$-dimensional flag manifold$\ F$
with Picard number 1 which\ is $\neq\mathbf{P}^{n}$ with $n\geq3$,
consequently $c_{1}(F)<n+1$: Let $\tau:X\rightarrow Y$ $=(|t|<1)$ be a
deformation map with $X_{t}=F$ for $t\neq0$ and $X_{0}\neq F$. This leads to a
bimeromorphic map $\widehat{\mu}:X\rightarrow Y\times\mathbf{F}$ which has the
following properties: Let $\widehat{G}$ be the graph of $\widehat{\mu}.$For
$t\neq0$ the fiber $\widehat{G}_{t}$ of $\widehat{G}\rightarrow Y$ describes a
biholomorphic map $X_{t}\rightarrow(t)\times F$. The fiber $\widehat{G}_{0}$
$\subset X_{0}\times F$ decomposes into at least 2 different components, among
them a component $\widehat{X}_{0}^{\ast}$ which is a modification of $X_{0}$,
a component $\widehat{G}_{0}^{\ast}$ which is a modification of $(0)\times F$,
$\widehat{X}_{0}^{\ast}$ $\neq\widehat{G}_{0}^{\ast}$. Let $S^{\ast}$ be the
image of $\widehat{G}_{0}^{\ast}$ in $X_{0}$. The image $\widehat{X}%
_{0}^{^{\prime\prime}}$\ of $\widehat{X}_{0}^{\ast}$ in $\widehat{X}%
_{0}^{^{\prime}}=(0)\times F=F$ under the projection $\widehat{X}_{0}^{\ast
}\rightarrow F$ is a proper analytic subset of $F$. If $n-m=\dim\widehat
{X}_{0}^{^{\prime\prime}}$, then a general fiber $fib$\ of $\widehat{X}%
_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ has dimension $m$;
$\ fib$ can also be considered as an analytic subset of $X_{0}$. Now one
invokes\ holomorphic liftings $\chi$ of suitable\ holomorphic Euler vector
fields\ on $Y\times F$\ (these depend on $F$)\ to $X$ and studies their
orbital curves $l$ through general points in $fib$; they are rational, smooth
and\ stay in $fib$, and their images in $X_{0}$ generate $H_{2}(F,\mathbf{Z)}%
=\mathbf{Z}$. One examines the normal bundle of $l$ in $X_{0}$ or in $fib$.
Suppose $m=1$, which is equivalent to $\dim S^{\ast}=0$, which means that
$S^{\ast}$ is a point. The adjunction formula delivers $c_{1}(X_{0})=n+1$; but
$c_{1}(X_{0})$ equals $c_{1}(F)$, therefore $c_{1}(X_{0})=c_{1}(F)=n+1$, a
contradiction to $c_{1}(F)<n+1$. In the case of theorem 1 (where $X_{0}$ is
supposed to be k\"{a}hler) the case $m>1$ can be dismissed immediately (see
chapter 5 and 6). In the case of theorem 2 (the k\"{a}hler assumption is
dropped) this is not so and we have to discuss the possibility $m>1$ more
extensively. A procedure of \cite{Siu2} \ delivers, that any general
$\widehat{\mu}_{0}$-fiber $fib$ is a $\mathbf{P}^{m}$. Considering the normal
bundle of $l$ in $X_{0}$ again the adjunction formula delivers the
contradictory statement $c_{1}(X_{0})=c_{1}(F)=n+1$. \newline\newline This is
also\ (more or less) the underlying\ rough type\ of pattern of \cite{Siu1},
\cite{Siu2}, \cite{Siu3}, \cite{Hwang2} (where the nondeformability of
$\mathbf{P}^{n}$ is proved) and of \cite{Hwang1} (where the non deformability
of smooth quadrics $\mathbf{Q}_{n}$ of dimension $n\geq3$ is proved). -
Formally Siu and Hwang argue with certain maximal orbital varieties lying in
$\mu_{0}$-fibers. The insight that these orbital varieties are (components of)
whole $\mu_{0}$-fibers is useful.\newline\newline The approach towards theorem
1 above in the series \cite{Hwang-Mok}, \cite{Hwang3}, \cite{Hwang-Mok3},
\cite{Hwang-Mok4} looks different.\newline\newline\emph{Remark}: For
computational purposes it is useful to realize that (as mentioned above) in
$X_{0}$ general $\mu_{0}$-fibers and $\widehat{\mu}_{0}$-fibers coincide. -
More precise: Over $\mathbf{C}^{n}=F\backslash H_{F}$ $=(0)\times F\backslash
H_{F}=(0)\times\mathbf{P}^{n}\backslash H_{\mathbf{P}^{n}}=(0)\times C^{n}$
the graphs $X_{0}^{\ast}$ of $\mu_{0}$ and $\widehat{X}_{0}^{\ast}$ of
$\widehat{\mu}_{0}$ are biholomorphic, $\mu_{0}$-fibers and $\widehat{\mu}%
_{0}$-fibers over \ points of \ $(0)\times\mathbf{C}^{n}$are biholomorphic and
can naturally be identified. - The map $\mu$ has been introduced, because it
is easier to perform computations with $\mu$ than with $\widehat{\mu}$.
\newline\newline\textbf{f. More on the use of the assumption that }$X_{0}$
\textbf{is k\"{a}hler and how to bypass the\ k\"{a}hlerness assumption}. In
the hypersurface case $m=1$, i.e$.$the case $\dim S^{\ast}=0$ i.e the case
$S^{\ast}$= a point, (cum grano salis) it does not necessarily make such\ a
big difference, whether we assume $X_{0}$ to be k\"{a}hler or not. We run into
the hypersurface case if $n=3$ -\ But there is a big difference in the case
$m\geq2$. If we do not assume $X_{0}$ to be k\"{a}hler, we first have to show,
that general $\mu_{0}-$fibers are $\mathbf{P}^{m}$'s; here a non
trivial\ method of \ Siu is employed. This implies $\dim S^{\ast}=m-1\geq1$
and that every general $\mu_{0}$-fiber $fib=\mathbf{P}^{m}$ contains $S^{\ast
}$ as a hyperplane and that curves lying in $S^{\ast}$ represent positive
multiples of the positive generating element of $H_{2}(X_{0},\mathbf{Z)=Z}$.
Consequently general $\mu_{0}$-fibers $fib$ contain minimal rational curves of
$X_{0}$, orbital curves of Euler vector fields on $X_{0}$; thus we can bypass
the k\"{a}hler assumption for $X_{0}$ \ and are able to apply the adjunction
formula in order to arrive at the wanted contradiction $c_{1}(F)=c_{1}%
(X_{0})=n+1$ for $c_{1}(F)<n+1$, compare chapter 8.\newline\newline%
\emph{Remark}. We call the image $S^{\ast}=\widehat{\varphi}(\widehat{G}%
_{0}^{\ast})=\varphi(G_{0}^{\ast})$ an attractor even if $\dim S^{\ast}>0$ has
to be reckoned with. In \cite{Hwang1} attractors are by definition points,
otherwise $S^{\ast}$ is called a limit set by Hwang.\newline

\subsection{On the rigidity of $\mathbf{P}^{3}$.}

Naturally we could just cite \cite{Hwang1}, theorem 1.\ \ But it is useful to
describe with more details\ some features of the proof, that $\mathbf{P}^{n}$
for $n=3$\ is (globally) nondeformable, as this can serve as an example of how
to employ Euler vector fields. This special use of\ Euler vector fields goes
back to Hwang and Siu.\ Let us sketch the type of geometric\ picture we
encounter in this situation within our framework. We employ the set up above
for $F=\mathbf{P}^{n}$, $n=3$, and we study in this case\ the \ bimeromorphic
map $\mu:X\rightarrow X^{^{\prime}}=Y\times\mathbf{P}^{n}$ with $X_{t}%
=\mathbf{P}^{n}$ for $t\neq0$, i.e. where $\mu$ is biholomorphic outside
$X_{0}$.\ We denoted the graph of $\mu$ by $G$. Suppose that $X_{0}$ is not
biholomorphic to $\mathbf{P}^{n}$, i.e. that the fiber of $G$ over $0\in Y$
\ is reducible. For the purpose of the introduction we may suppose that this
fiber$\ G_{0}$ decomposes into 2 different\ components $G_{0}^{\ast}$ and
$X_{0}^{\ast}$, where $G_{0}^{\ast}$ is a modification of $X_{0}^{^{\prime}}$
and where $X_{0}^{\ast}$ is a modification of $X_{0}$, where the "attractor"
set $S^{\ast}$equals $\varphi(G_{0}^{\ast})$, $\ $where $\mu$ and $\mu_{0}$
are indeterminate in each point of$\ S^{\ast}$. Let $\chi^{^{\prime}}$ be an
Euler vector field on $X^{^{\prime}}$, such that the center on $X_{0}%
^{^{\prime}}$ is a general point $\in$ $X_{0}^{^{\prime\prime}}$;
$\chi^{^{\prime}}$ on $X^{^{\prime}}$ has a zero divisor $(s^{^{\prime}}=0)$
of degree one, i.e. where $s^{^{\prime}}\in H^{0}(X^{^{\prime}},\mathcal{D}%
^{^{\prime}})$. Here $\mathcal{D}^{^{\prime}}$ is the line bundle on
$X^{^{\prime}}=Y\times\mathbf{P}^{n}$ defined by the $\mathbf{P}^{n}%
$-hyperplanes; we may identify $s^{^{\prime}}$ with a homogeneous polynomial
of degree one. Let $s$ be the corresponding associated\ section $\in
H^{0}(X,\mathcal{D)}$. Let $\chi$ be the holomorphic lifting of $\chi
^{^{\prime}}$ to $X$ and let $\chi^{\ast}$ be the holomorphic lifting of
$\chi$ to $G$, $\chi^{\ast}$ vanishes identically along $G_{0}^{\ast}$.
$\varphi^{\ast}s$ vanishes along $G_{0}^{\ast}$ with order one because
$\frac{1}{s}\chi_{0}$ is a line bundle valued holomorphic vector field on
$X_{0}$ which\ for geometric reasons must have a zero on general orbital
curves $\gamma_{0}^{\ast(1)}$\ of $\frac{1}{s}\chi_{0}$. Because these orbital
curves meet\ $S^{\ast}$ in points where they encounter other orbital curves of
$\frac{1}{s}\chi_{0}$. These intersection points are zeros of $\frac{1}{s}%
\chi_{\text{0}}$. \ Have in mind:\newline\newline Because of $n=3$ only the
"hypersurface case" $\dim X_{0}^{^{\prime\prime}}=n-1(=2)$ has to be
discussed; $S^{\ast}$ is a point;\ (the components of) general fibers of
$\mu_{0}$ are rational\ orbital curves $\gamma_{0}^{\ast(1)}$ of $\chi_{0}$
and $\frac{1}{s}\chi_{0}$. $\chi_{0}$ is additive.\ \newline\newline Direct
calculation (in chapter 5) of $\chi_{0}$ shows that besides $(s=0)|X_{0}$ the
vector field $\chi_{0}$ has still another divisorial zero $(\widetilde{s}=0)$,
here is the place where\ we really make use of the hypersurface situation,
\ because of $b_{2}(X_{0})=1$ we can arrange matters such that $(\widetilde
{s}=0)$ meets $\gamma_{0}^{\ast(1)}\cap S^{\ast}$, we even can arrange
$\widetilde{s}=s|X_{0};$ $s|X_{0}$ is sometimes\ abbreviated by $s.$ Let
$\gamma_{0}^{\ast(1)}$\ be a general orbital curve of $\chi_{0}$ on $X_{0}$ as
introduced above. $\gamma_{0}^{\ast(1)}$ meets $S^{\ast}$ in a point. One does
not get rid of these other orbital curves through the point $\gamma_{0}%
^{\ast(1)}\cap S^{\ast}$ by constructing $\frac{1}{\widetilde{s}\cdot s}%
\chi_{0}$, which is holomorphic in the point $\gamma_{0}^{\ast(1)}\cap
S^{\ast}$ and must have a zero here because of the other orbital curves of
$\chi_{0}$ meeting the point $\gamma_{0}^{\ast(1)}\cap S^{\ast}$. Have in mind
that multiplication of holomorphic vector fields with a meromorophic function
does not lead (outside a certain zero set whose points cannot be moved) to new
orbital curves. But this implies that on the rational curve $\gamma_{0}%
^{\ast(1)}$ the holomorphic\ vector field\ $\chi_{0}|$ $\gamma_{0}^{\ast(1)}%
$\ has in the point $\gamma_{0}^{\ast(1)}\cap S^{\ast}$ at least 3 zeros which
cannot be as exactly 2 zeros occur. - This contradiction proves $X_{0}%
=\mathbf{P}^{n}$ for $n=3$.\newline\newline In order to prove the (global)
nondeformability of $\mathbf{P}^{n}$, $n\in\mathbf{N}$ arbitrary, one can do
the following: Construct the holomorphic lifting $\chi^{\ast}$\ to $G$ of such
a general Euler vector fields $\chi^{^{\prime}}$ on $X^{^{\prime}}%
=Y\times\mathbf{P}^{n}$, where the center of $\chi_{0}^{^{\prime}}$ is a
(general) point on $X_{0}^{^{\prime\prime}}$. It is\ easy to show, that
$\chi^{\ast}$ has a divisorial zero $\varphi^{\ast}(s=0)$ with $s\in
H^{0}(X,\mathcal{\ D)}$. Show that $S^{\ast}$ is a point (\cite{Hwang1}) and
the proof indicated\ above can be applied. - In retrospect this proof can
roughly be described thus: Let $\tau:X\rightarrow Y$ be a deformation map with
$X_{t}=\mathbf{P}^{3}$ for $t\neq0$ and $X_{0}\neq\mathbf{P}^{3}$. Replacing
possibly $Y$ by a smaller neighborhood of $0\in Y$ we construct a certain
bimeromorphic map $\mu:X\rightarrow Y\times\mathbf{P}^{3}$, fiberwise
biholomorphic outside $X_{0}$, such that $\mu\lbrack X_{0}]$ is a hypersurface
in $(0)\times\mathbf{P}^{3}$. By "holomorphic liftings" of Euler vector fields
on $Y\times\mathbf{P}^{3}$ we can produce holomorphic vector fields on $X_{0}$
with divisorial zeros, implying immediately $X_{0}=\mathbf{P}^{3}$.
\newline\newline This procedure can be adapted to deformations of the smooth
quadric $\mathbf{Q}_{3}$ roughly like this: Let $\tau:X\rightarrow Y$ be a
deformation map with $X_{t}=\mathbf{Q}_{3}$ for $t\neq0$ and $X_{0}%
\neq\mathbf{Q}_{3}$. Replacing possibly $Y$ by a smaller neighborhood of $0\in
Y$ we construct a certain bimeromorphic map $\mu:X\rightarrow Y\times
\mathbf{P}^{3}$, fiberwise bimeromorphic outside $X_{0}$, such that
$\mu\lbrack X_{0}]$ is a hypersurface in $(0)\times\mathbf{P}^{3}$. Besides
$\mu$ we consider a bimeromorphic map $\widehat{\mu}:X\rightarrow
Y\times\mathbf{Q}_{3}\subset Y\times\mathbf{P}^{4}$,\ fiberwise biholomorphic
outside $X_{0}$, such that $\widehat{\mu}[X_{0}]$ is a hypersurface in
$(0)\times\mathbf{Q}_{3}$.One gets $\mu$ from $\widehat{\mu}$ by projecting
$\mathbf{Q}_{3}$ from one point onto $\mathbf{P}^{3}$ \ By "holomorphic
liftings" of Euler vector fields on $Y\times\mathbf{Q}_{3}$ (with center on
$\widehat{X}_{0}^{^{\prime}}=$ a general point of $\widehat{X}_{0}%
^{^{\prime\prime}}$\ $=\widehat{\mu}[X_{0}]$ ) to $X_{0}$,\ we can produce via
$\mu$ holomorphic vector fields on $X_{0}$ with divisorial zeros, as
$\mu\lbrack X_{0}]$ is a hypersurface in $(0)\times\mathbf{P}^{3}$, implying
according to Hwang \cite{Hwang1} $X_{0}=\mathbf{P}^{3}$, the wanted
contradiction. \newline

\subsection{Some conventions and notations used throughout this paper.}

Elementary or generally well known facts on degeneration sets of
\ holomorphic\ maps (in the sense of Remmert) and Remmert's mapping theorem
and related facts, on meromorphic maps, openness of flat maps, on
normalizations, on the fiber\ connectedness of bimeromorphic holomorphic
maps\ etc. are generally\ used without comment. In this paper strict and
proper transform mean the same; sometimes we use in this context the
expression proper inverse image. As the symbols and notations (not the
notions) which are used in the context of meromorphic maps differ
sometimes\ from author to author (a little bit), let us add some remarks in
this respect, in order to avoid misunderstandings). By a meromorphic map
$\rho_{1}$ of a (reduced) complex space $Z_{1}$ into a (reduced) complex space
$Z_{2}$\ we understand a meromorphic map\ in the sense of Remmert, i.e. a map
$\rho_{1}$\ of $Z_{1}$ into the power set of $Z_{2}$, such that its graph
$G_{1}\subset Z_{1}\times Z_{2}$ is a (reduced) analytic subspace of
$Z_{1}\times Z_{2}$ such that the projection $G_{1}\rightarrow Z_{1}$ is a
proper\ modification map. Concerning the corresponding\ notations we follow
for instance \cite{Ueno} and/or Math.Ann.139, p. 217ff. It is convenient to
write $\rho_{1}:Z_{1}\rightarrow Z_{2}$, although $\rho_{1}$ is a certain map
of $Z_{1}$ into the power set of $Z_{2}$. Related convenient notations: If
$\varrho_{1}:Z_{1}\rightarrow Z_{2},$ $\varrho_{2}:Z_{2}\rightarrow Z_{3}$ are
meromorphic maps of reduced spaces and if for any general point $P_{1}\in
Z_{1}$ of holomorphy for $\rho_{1}$ the meromorphic map $\varrho_{2}$ is
holomorphic in the point $P_{2}=\varrho_{1}(P_{1}),$ there exists a unique
meromorphic map $\varrho_{3}:Z_{1}\rightarrow Z_{3}$ with $\varrho_{3}%
(P_{1})=\varrho_{2}(\varrho_{1}(P_{1}))$ for general $P_{1}\in Z_{1}.$ This so
called "composition of $\varrho_{1}$ and $\varrho_{2}$\textquotedblright\ is
named sometimes $\varrho_{2}\bigtriangleup\varrho_{1};$ we prefer\ the
notation $\varrho_{2}\circ\varrho_{1},$ which cannot be misunderstood; all
this may be looked at as an abuse of language - Let $G_{1}\subset Z_{1}\times
Z_{2}$ be the graph of $\rho_{1}$ with the projection $\pi_{1}:G_{1}%
\rightarrow Z_{1}$; if no component of the reduced analytic\ subspace
$A\subset Z_{1}$ lies in the indeterminancy set of $\rho_{1}$, then by
$\rho_{1}|A$ we mean the meromorphic map $A\rightarrow Z_{2}$ with the graph
$\pi_{1}^{-1}[A]$; $\rho_{1}(P)$ and $(\rho_{1}|A)(P)$ coincide generally only
for general points $P\in A$; we call $\rho_{1}|A$ the restriction\ of
$\rho_{1}$ to $A$ or the by $\rho_{1}$ induced meromorphic map; if $A$ is
irreducible, $\pi_{1}^{-1}[A]$ is the unique component of $\pi_{1}^{-1}(A)$
which is a modification of $A$. For instance in the above the meromorphic
map$\ \mu_{0}:X_{0}\rightarrow X_{0}^{^{\prime\prime}}\subset X^{^{\prime}}$
induced by $\mu$ is the meromorphic map with the graph $X_{0}^{\ast}$. This
again may be looked upon as an abuse of language; if we understand
$\varrho_{1}$ as a map of $Z_{1}$ into the powerset of $Z_{2}$, then the
meromorphic map $\rho_{1}|A$ in our sense is not\ the restriction of
$\varrho_{1}$ to $A$ in the set theoretical sense. These brackets $(..a...)$,
$(a)$ can indicate a point with coordinates ...,$a,$.., a set of elements $a$,
a list of elements $a$, a zero set \thinspace$(f=0)$.\newline\newline We will
sometimes give proofs in special cases if these are general enough such that
the handling of these special cases makes clear how to deal with the general
case; so it can happen, that we only\ discuss $m=1,2,3$ or $m=1,2$, if the
proof of the general case\ can be read from the pattern of proof of these
special cases. \newline\newline Without mentioning it and without changing
notations we will sometimes replace\ $G$\ and other spaces (if feasible and
allowable)\ by their normalizations. - For instance one must be aware, that it
only makes sense to speak of a vanishing order of a holomorphic function along
an irreducible hypersurface (say $t$ on $G$ along a component of $G_{0}),$ if
the complex space in question (say $G$) is normal (at least in general points
of the hypersurface), otherwise we have to replace $G$ by its normalization.
\newline\newline From the context it will become clear, whether during a proof
we are working\ with vector fields\ over $\mathbf{R}$ or $\mathbf{C,}$i.e.
whether we are dealing with a real form of a complex Lie algebra; this applies
for instance when we are dealing with the Cartan-Killing form $($ \ $,$ \ $)$,
need symmetry, work over the reals or with a real basis without mentioning
this explicitly. \newline\newline If nothing else is explicitely agreed upon,
we use in the sequel the following notations: $Y$ is the unit disc $\left\vert
t\right\vert <1$, $F$ is mostly an $n-$ dimensional, irreducible, hermitian,
symmetric complex manifold of compact type,\ resp. more general, an
$n-$dimensional flag manifold with $b_{2}=1,$ if $F$ means a special flag
manifold or\ something different, it will be stated and will be clear from the
context. $\tau:X\rightarrow Y$ is a deformation map with $X_{t}=F$ for
$t\neq0$. Holomorphic vector fields $\chi$\ on $Y\times F$ are called Euler
vector fields, if the restrictions $\chi_{t}$ (exist and) are Euler vector
fields on $(t)\times F=F$. \newline\newline If $\widetilde{F}$ is a compact
complex manifold with $H_{2}(\widetilde{F},\mathbf{Z)=Z}$, any irreducible,
rational curve generating $H_{2}(\widetilde{F},\mathbf{Z)}$ will be called a
minimal rational curve (or a curve of degree one or a line) in $\widetilde{F}%
$. The spaces $F$ we encounter in this paper (flag manifolds with $b_{2}=1$,
del Pezzo manifolds of dimension $\geq3$ and degree $5$) can be embedded into
projective spaces $\mathbf{P}^{\widetilde{n}}$ via sections in the positive
generating line bundle, such that the lines on $F$ are exactly those curves on
$F$, which are lines of the surrounding projective spaces $\mathbf{P}%
^{\widetilde{n}}$. \newline\newline As already mentioned in this paper we
often tacitly assume general fibers of $X_{0}^{\ast}\rightarrow X_{0}%
^{^{\prime\prime}}$, $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$, i.e. general $\mu_{0}$-fibers to be irreducibel, for
reasons of simpler notations. Sections $s$\ in line bundles will often be
identified with local functions. By $(s=0)$ we mean in general the reduced
zero set. - $F(z)\neq a$ can mean, that $F(z)$ does not have in the point $z$
the value $a$; it can also mean, depending on the context, that $F(z)$ is not
the constant function with value $a$,; similarly the equality sign $=$ has to
be interpreted. \newline\newline It is convenient to use the notion of a
general point, line etc.. The meaning of "general"or "generic"\ should emerge
from the context, mostly it means "belonging to a certain non empty Zariski
open subset", i.e. certain proper analytic subsets, for instance singular
points, could be excluded (without specifying them explicitely), or for
instance\ "belonging to a certain open, dense subset" or even "belonging to
the complement of a union of a certain\ sequence of proper analytic subsets"
without specifying these subsets explicitely or belonging to the intersection
of countably many open and dense sets of the space in question; in the last
two cases one speaks also of very general points.- Quite a lot of symbols are
introduced in the course of this paper. In order to reduce the leafing through
the pages, searching certain definitions, some notations are defined more than
once.\newline\newline When handling vector fields it will (hopefully) be clear
from the context, whether we are working over the reals, i.e. are dealing with
real forms or not.,\newline\newline A reference to (subsection) i.j, resp.
i.j.k,... means a reference to chapter i, subsection j (of i), resp. to
chapter i, subsection j (of i), subsection k (of j),....

\section{\textbf{Some tools and prerequisites.}}

\subsection{A result of Iitaka, some consequences.}

\textbf{a. Iitaka's theorem (}\cite{Ueno}$,$\emph{\ theorem 8.1). }For any
line bundle $\mathcal{H}_{F}$ of a complex space $F$ let $l(\mathcal{H}%
_{F}):=\dim H^{0}(F,\mathcal{H}_{F})$. Suppose $F$ to be irreducible, reduced,
compact,$\mathcal{D}_{F}$ to be a line bundle on $F$ with $l(\mathcal{D}%
_{F})>0.$ Denote by $\nu^{\ast}:F^{\ast}\rightarrow F$ the normalization map
and $l_{m}:=l(m\cdot\nu^{\ast}\mathcal{D}_{F})$ for $m\in\mathbf{N.}$ Any base
$\widetilde{s}_{0},...,\widetilde{s}_{l_{m}-1\text{ }}$ of $\Lambda_{m}:=$
$H^{0}(F^{\ast},m\cdot\nu^{\ast}\mathcal{D}_{F})$ defines a meromorphic map
$\Phi_{\Lambda_{m}}:F\rightarrow\mathbf{P}^{l_{m}-1}$, which maps any general
point $P\in F$ to $\Phi_{\Lambda_{m}}(P)=(\widetilde{s}_{0}(\nu^{\ast
}P):...\,:\widetilde{s}_{l_{m}-1}(\nu\ast P))$. $\varkappa.:=\,\varkappa
(F,\mathcal{D}_{F}):=\max_{m}(\dim\Phi_{\Lambda_{m}}(F))$ is called the
$\mathcal{D}_{F}-$line\ bundle\ $\dim$ension. Iitaka proved: \newline%
\newline\emph{There are }$0<\alpha<\beta$\emph{\ with }$\beta\cdot
m^{\varkappa}\geq l_{m}\geq\alpha\cdot m^{\varkappa}$\emph{\ ( }\cite{Ueno}%
$,$\emph{\ theorem 8.1).\newline}\newline We say $l_{m}$ is growing in $m$
like a polynomial of degree $\varkappa.$\newline\newline\textbf{b. A
consequence of a. }Let $X$ be an irreducible, reduced complex space, let
$\tau:X\rightarrow Y=(\left\vert t\right\vert <1)$ be a proper, surjective,
holomorphic map with irreducible, reduced fibers of the same dimension $n$.
Let $\mathcal{D}$ be a line bundle on $X$ with $l(\mathcal{D}\mathcal{)>\,}1.$
Subsection 2.1.a above, Grauert's semicontinuity theorem
\cite{Grauert-Remmert}, p. 210 imply (compare also \cite{Timmerscheidt}, p.
273)\newline\newline\emph{There are a }$k\in N\cup(0)$\emph{\ and a countable
set of points in }$Y$\emph{\ such that for the complement }$W$\emph{\ of this
set the following holds For all }$t\in W$\emph{\ we have }$\varkappa
(X_{t},\mathcal{D})=k$\emph{\ and for all }$t\notin W$\emph{\ there is
}$\varkappa(X_{t},\mathcal{D}_{t})\geq k;$\emph{\ for all }$t_{1},t_{2}\in
W$\emph{\ and all }$m\in N$\emph{\ the identity }$l(m\cdot\mathcal{D}_{t_{1}%
})=l(m\cdot\mathcal{D}_{t_{2}})$\emph{\ holds. If }$k=n,$\emph{\ then all
fibers of }$\tau$\emph{\ are moishezon. }\newline\newline\textbf{c. }$X_{0}%
$\textbf{\ is moishezon.} Now let $\tau:X\rightarrow Y$ be a deformation map,
i.e. let $X$ be a complex manifold, let $\tau$ be a proper, regular,
surjective, holomorphic map; especially all fibers are smooth, compact
manifolds. Assume that for $t\neq0$ the anticanonical bundle $-\mathcal{K}%
_{t}$ of the fiber $X_{t}$ has line bundle dimension $n$. The previous
subsection 2.1.\textbf{a.} and Grauert's semicontinuity theorem
\cite{Grauert-Remmert}, p. 210\ imply:\newline\newline\emph{The central fiber
}$X_{0}$\emph{\ is moishezon.}\newline\newline\textbf{d. An extension of b.
}Let us mention that there are stronger forms of subsection 2.1.b above.
\newline\newline$\mathbf{\alpha}$\ Let us continue to use the notations and
assumptions of the preceding subsection b. For every $\widetilde{m}%
\in\mathbf{N}$ there are $\widetilde{s}_{0},...,\widetilde{s}_{N}\in
H^{0}(X,\widetilde{m}\cdot\mathcal{D}),$ such that the images $\tau
_{(0)}\widetilde{s}_{0},...,\tau_{(0)}\widetilde{s}_{N}$ generate the image
sheaf $\tau_{(0)}(\widetilde{m}\cdot\mathcal{D})$ over $Y$. Let $\widetilde
{\mu}:X\rightarrow Y\times\mathbf{P}^{N}$ be the meromorphic map defined by
$P\rightarrow(\tau(P);\widetilde{s}_{0}(P):...:\widetilde{s}_{N}(P))$ for a
general $P\in X.$ For sufficiently large $\widetilde{m}$ the proper transform
$\widetilde{\mu}\left[  X_{t}\right]  $ is of dimension $k$ for all $t\in Y$
outside a proper analytic subset of $Y.$ Subsection \textbf{a }above does not
imply $\dim\widetilde{\mu}\left[  X_{t}\right]  \geq k$ for large
$\widetilde{m}$.$-$We can prove in this respect:\newline\newline\emph{Let
}$t_{0}\in Y$\emph{\ and suppose the fiber }$X_{t_{0}}$\emph{\ to be
moishezon. Then the proper transform }$\widetilde{\mu}\left[  X_{t_{0}%
}\right]  $\emph{\ is of dimension }$k$\emph{\ for sufficiently large
}$\widetilde{m}.$\newline\newline Proof: Let $\lambda:\overset{\sim}{X}%
^{(0)}\rightarrow X_{t_{0}}$ be a holomorphic, bimeromorphic map with a
smooth, projective $\overset{\sim}{X}^{(0)}.$ Let $\mathcal{C}$ be a very
ample line bundle on $\overset{\sim}{X}^{(0)}$ , $\overset{\sim}{\mathcal{D}%
}^{(0)}$ :=$\lambda^{\ast}\mathcal{D}_{t_{0}}$, $\widetilde{\Lambda
}_{\widetilde{m}}:=H^{0}(X,\widetilde{m}\cdot\mathcal{D)}$, $\Lambda
_{\widetilde{m}}:=image(\widetilde{\Lambda}_{\widetilde{m}}%
\mathcal{\rightarrow}H^{0}(\overset{\sim}{X}^{(0)},\widetilde{m}\cdot
\overset{\sim}{\mathcal{D}}^{(0)})),$ $\gamma:=c_{1}(\mathcal{C}%
\mathcal{)}^{n},$ $\overset{\sim}{t}=c_{1}\mathcal{(}\overset{\sim
}{\mathcal{D}}^{(0)})\circ(c_{1}(\mathcal{C}\mathcal{))}^{n-1}/c_{1}%
(\mathcal{C}\mathcal{)}^{n},$ $i=\dim\widetilde{\mu}\left[  X_{t_{0}}\right]
.$ The proof of lemma 2.1 on p. 515 of \cite{Liebermann-Mumford} gives
$i+\gamma\cdot\overset{\sim}{t}\cdot\widetilde{m}^{i}\geq\dim\Lambda
_{\widetilde{m}}.$ But as the dimension of $\Lambda_{\widetilde{m}}$ grows at
least like $\alpha\cdot\widetilde{m}^{k}$ with $\alpha>0$ we must have $i=k$
for large $\widetilde{m}.$ This finishes the proof.\newline\newline The
special $k=n$ case\ is also a consequence of \ Hironaka's Chow lemma
(\cite{Hironaka}).\newline\newline$\mathbf{\beta.}$A related result is: Let us
weaken in $\mathbf{\alpha}$ the assumption, that the $\tau_{(0)}\widetilde
{s}_{0},...,\tau_{(0)}\widetilde{s}_{N}$ generate the image sheaf $\tau
_{(0)}(\widetilde{m}\cdot\mathcal{D})$ over $Y$. Let $\widetilde{\Lambda
}_{\widetilde{m}}$ be the maximal subspace of $H^{0}(X,\widetilde{m}%
\cdot\mathcal{D)}$ generated over a Zariski open subset of $Y$ by the
$\widetilde{s}_{0},...,\widetilde{s}_{N}\in H^{0}(X,\widetilde{m}%
\cdot\mathcal{D})$. Suppose $\dim\widetilde{\mu}\left[  X_{t}\right]  =k$\ all
large $\widetilde{m}$ and all general $t\neq t_{0}$, where $k$ is independent
of $\widetilde{m}$, and that the dimensions of the$\ \Lambda_{\widetilde{m}}$
grow at least like $\alpha\cdot\widetilde{m}^{k}$ with $\alpha>0$ independent
of $\widetilde{m}$. The proof in $\mathbf{\alpha}$ delivers for large
$\widetilde{m}$: \newline\newline\emph{If }$X_{t_{0}}$\emph{\ is moishezon,
then }$\widetilde{\mu}[X_{t_{0}}]$\emph{\ has dimension }$k$\emph{.}\newline

\subsection{On line bundles on $X$.}

\textbf{a. On line bundles and corresponding cohomology groups on }%
$X.$\textbf{\ }Deformation maps are locally differentiably trivial
(\cite{Kodaira}, p. 67). In the following we always assume that the
deformation maps $X\rightarrow Y$ we consider are differentiably trivial, i.
e. that there is a diffeomorphism $X\rightarrow Y\times X_{0}$ with $\tau$
being the composition of this diffeomorphism and the projection $Y\times
X_{0}\rightarrow Y.$\newline\newline\emph{Let }$\tau:X\rightarrow Y$\emph{\ be
a deformation map with all fibers }$X_{t}$\emph{\ being moishezon and with
}$H^{2}(X_{t},\mathbf{Z})=\mathbf{Z},$\emph{\ }$H^{1}(X_{t},\mathbf{Z}%
)=0.$\emph{\ Then }$H^{2}(X,\mathbf{Z})=H^{2}(X_{0},\mathbf{Z})=\mathbf{Z}%
,$\emph{\ }$\ H^{1}(X,\mathbf{Z})\ =\ H^{1}(X_{0},\mathbf{Z})=0,$%
\emph{\ }$\ H^{1}(X_{0},\mathcal{O}_{X_{0}})=H^{1}(X,\mathcal{O}_{X})=$
$H^{2}(X_{0},\mathcal{O}_{X_{0}})=H^{2}(X,\mathcal{O}_{X})=0;$\emph{\ the
Chern map }$c_{1}:H^{1}(X,\mathcal{O}_{X}^{\ast})\rightarrow H^{2}%
(X,\mathbf{Z})=\mathbf{Z}$\emph{\ is an isomorphism.} \newline\newline Proof:
$\tau$ being differentiably trivial, $X_{t}$ and $X$ have the same
$\mathbf{Z}-$cohomology groups. As $X_{t}$ is moishezon, we have the Hodge
decomposition $\mathbf{C}$=$H^{2}(F,\mathbf{C})=H^{1,1}\oplus H^{0,2}\oplus
H^{2,0},$ $0=H^{1}(F,\mathbf{C)\ }=H^{1,0}\oplus H^{0,1}$ with $\dim
H^{0,2}=\dim H^{2,0}.$ Our assertions follow from well known results to be
found in \cite{Ueno}, \cite{Grauert-Remmert}. This finishes the proof.\newline%
\newline Although not really\ needed, let us remark, that for all moishezon
manifolds $F$ of dimension $\geq2$ the equality $H^{2}(F,\mathbf{Z)=Z}$
implies $H^{1}(F,\mathbf{Z)=}0$ ( if one replaces $\mathbf{Z}$ by
$\mathbf{C,}$ one gets a theorem due to Remmert and van de Ven; have in mind,
that $H^{1}\left(  F,\mathbf{Z}\right)  $ has no torsion). \newline%
\newline\emph{Remark}. As because of $H^{1}(X,\mathcal{O}_{X}^{\ast}%
)=H^{2}(X,\mathbf{Z)}$ the sequence $0\rightarrow H^{0}(X,\mathcal{O}%
_{X}^{\ast})\rightarrow H^{0}(X,\mathcal{M}_{X}^{\ast})\rightarrow$
$H^{0}(X,\mathcal{M}_{X}^{\ast}/\mathcal{O}_{X}^{\ast})\rightarrow
H^{1}(X,\mathcal{O}_{X}^{\ast})=H^{2}(X,\mathbf{Z)}$ is exact, any element of
$H^{0}(X,\mathcal{M}_{X}^{\ast}/\mathcal{O}_{X}^{\ast}),$ the image of which
in $H^{2}(X,\mathbf{Z)}$ is zero, is represented by a meromorphic function on
$X.$\newline\newline\emph{Suppose additionally that all fibers }$X_{t}%
,t\neq0,$\emph{\ have positive Chern class }$c(-\mathcal{K}_{t}$%
\emph{)}$\,>0.$\emph{\ Then there exists exactly one line bundle (the so
called positive generating line bundle) }$\mathcal{D}\in H(X,\mathcal{O}%
_{X}^{\ast})=Z,$\emph{\ such that the analytic restriction }$\mathcal{D}_{t}%
$\emph{\ of }$\mathcal{D}$\emph{\ to }$X_{t}$\emph{\ for all }$t\neq
0$\emph{\ is the positive generating line bundle. Suppose that for all }%
$t\neq0$\emph{\ the dimension of }$H^{0}(X_{t},\mathcal{D}_{t})$\emph{\ is
independent of }$t.$\emph{\ Then the restriction }$H^{0}(X,\mathcal{D}%
)\rightarrow H^{0}(X_{t},\mathcal{D}_{t})$\emph{\ is surjective for all
}$t\neq0.$\newline\emph{\newline}Proof:\emph{\ }Let $\dim H^{0}(X_{t}%
,\mathcal{D}_{t})$ be constant for $t\neq0.$ Because of Grauert's image sheaf
theorem and the Nakayama lemma there are $s_{0},...,s_{N}\in H^{0}%
(X,\mathcal{D}),$ such that for $t\neq0$ the analytic restrictions $s_{0}\mid$
$X_{t},...,s_{N}\mid X_{t}$ form a base of $H^{0}(X_{t},\mathcal{D}_{t})$ and
are linearly independent for $t=0;$ because a holomorphic vectorbundle over a
polycylinder is holomorphically trivial. If we do not want to apply this
result on vector bundles, we get trivially this surjectivity statement for all
$t\neq0$ of a sufficiently small neighbourhood of $0\in Y.$ This will be
sufficient for all later purposes. This finishes the proof. \newline%
\newline\textbf{b.}\textit{\ \textbf{On the irreducibility of certain
divisors}. }\emph{Let }$F$\emph{\ be a moishezon manifold with }%
$H^{2}(F,\mathbf{Z})=\mathbf{Z},$\emph{\ }$H^{1}(F,\mathbf{Z})=0,$\emph{\ let
}$\mathcal{D}_{F}$\emph{\ be a line bundle on }$F,$\emph{\ such that }%
$c_{1}(\mathcal{D}_{F})$\emph{\ generates }$H^{2}(F,\mathbf{Z}).$\emph{\ Let
}$D$\emph{\ be the zeroset of a nontrivial }$s\in H^{0}(F,\mathcal{D}_{F}%
).$\emph{\ Then }$D$\emph{\ is irreducible and }$s$\emph{\ vanishes along }%
$D$\emph{\ with order 1}. \newline\newline Proof: We have $c_{1}(D)=1.$
Suppose $D=\rho_{1}\cdot D_{1}+\rho_{2}\cdot D_{2}$ with $\rho_{1}>0,\rho
_{2}>0$ and effective, irreducible divisors $D_{1},D_{2}$ and $c_{1}%
(D_{1})>0,$ $c_{1}(D_{2})\leq0.$ There are integers $\lambda_{1}>0,$
$\lambda_{2}>0$ with $c_{1}(\lambda_{1}\cdot D_{1}+\lambda_{2}\cdot D_{2})=0.$
$F$ being moishezon, the sequence $H^{0}(F,\mathcal{M}_{F}^{\ast}%
)\rightarrow\ H^{0}(F,\mathcal{M}_{F}^{\ast}/\mathcal{O}_{F}^{\ast
})\rightarrow\ H^{1}(F,\mathcal{O}_{F}^{\ast})=H^{2}(F,\mathbf{Z}%
)\rightarrow0$ is exact. (\cite{Ueno}, p. 32). Have in mind that
$H^{1}(F,\mathcal{O}_{F})=0$ $.$ Therefore the Chern map $H^{1}(F,\mathcal{O}%
_{F}^{\ast})\rightarrow H^{2}(F,\mathbf{Z})$ is injective. Because of
$H^{2}(F,\mathbf{Z})=\mathbf{Z}$ and $F$ being moishezon the Chern map has to
be surjective. Consequently there exists a nonconstant meromorphic function
$f$ with $\left(  f\right)  =\lambda_{1}\cdot D_{1}+\lambda_{2}\cdot D_{2}.$
As $f$ has no poles $f$ has to be constant. Contradiction. Thus $D$ is
irreducible and $s$ vanishes with order 1 along $D.$ The general case is
handled similarly. Actually we proved: The Chern class of every effective
divisor is positive. This finishes the proof. \newline\newline Let us recall
that $F$ moishezon, $H^{1}(F,\mathbf{Z)=}0,\,H^{2}(F,\mathbf{Z)=Z}$ imply an
isomorphism $H^{1}(\mathbf{F,}\mathcal{O}_{\mathbf{F}}^{\ast})=H^{2}%
(F,\mathbf{Z)=Z.}$ In this case there is exactly one generating line bundle
$\mathcal{D}_{F}\in H^{1}(F,\mathcal{O}_{F}^{\ast})$ with positive line bundle
dimension $\varkappa(\mathcal{D}_{F},F)>0,$ called the \emph{positive
generating line bundle}.- By extending $\mathcal{D}_{F}$ trivially to
$\widehat{X}^{^{\prime}}=Y\times F$ one gets the positive generating line
bundle $\widehat{\mathcal{D}}^{^{\prime}}\in$ $H^{1}(\widehat{X}^{^{\prime}%
},\mathcal{O}_{\widehat{X}^{^{\prime}}}^{\ast})$\ of $\widehat{X}^{^{\prime}}%
$.\newline

\subsection{A case where surjectivity resp. bimeromorphy implies
biholomorphy.}

Let $\tau:X\rightarrow Y$ be a deformation map, $F$ a compact complex
manifold, $\pi:Y\times F\rightarrow Y$ the projection, $\widehat{\mu
}:X\rightarrow Y\times F$ a bimeromorphic map with $\tau=\pi\circ\widehat{\mu
},$ i. e. with $\widehat{\mu}(X_{t})\subset(t)\times F$ for all $t\in Y,$
$\widehat{S}$ the set of indeterminancy of $\widehat{\mu}.$ As $\widehat{S}$
is at least of codimension $2$ in $X$ and as the fibers of $\tau$ are of
\ codimension 1, $X_{0}$ cannot lie in $\widehat{S}.$ Therefore the closure of
$((P,\widehat{\mu}(P))\mid P\in X_{0}\setminus\widehat{S})$ in $X_{0}%
\times(0)\times F=X_{0}\times F$ is a proper modification of $X_{0}$ and the
graph of a meromorphic map $X_{0}\rightarrow F,$ denoted by (abuse of
language) $\widehat{\mu}\mid X_{0}$ and called the restriction of
$\widehat{\mu}$ to $X_{0}.$ We may characterize $\widehat{\mu}_{0}%
:=\widehat{\mu}\mid X_{0}$ as the meromorphic map $X_{0}\rightarrow F,$
defined by $\widehat{\mu}_{0}(P)=\widehat{\mu}(P)$ \ for a general $P\in
X_{0}.$ But have in mind that only for $P\in X_{0}\setminus\widehat{S}$ \ we
are sure of $\widehat{\mu}_{0}(P)=\widehat{\mu}(P),$ for $P\in\widehat{S}\cap
X_{0}$ we only assert $\widehat{\mu}_{0}(P)\subset\widehat{\mu}(P).$ The
proper transform $\widehat{\mu}\left[  X_{0}\right]  $ equals $\overline
{\widehat{\mu}_{0}(X_{0}\backslash S)}.$For some more remarks on the
definition of $\widehat{\mu}_{0}$ see chap. 1. \newline\newline\emph{Assume
}$\widehat{S}\subset X_{0}$\emph{\ and that }$\widehat{\mu}$\emph{\ maps
}$X\setminus X_{0}$\emph{\ biholomorphically onto (}$Y\setminus(0))\times
F.$\emph{\ If }$\widehat{\mu}\left[  X_{0}\right]  =(0)\times F,$\emph{\ then
}$\widehat{\mu}$\emph{\ maps }$X$\emph{\ biholomorphically onto }$Y\times
F.$\newline\newline Proof: The assumptions imply that $\widehat{\mu}_{0}$ maps
$X_{0}$ bimeromorphically onto $(0)\times F=F.$ The zero set of the Jacobian
of $\mu\mid X\setminus S$ is empty or of pure codimension 1. As $\widehat{\mu
}_{0}$ is bimeromorphic, this zero set is empty and $\widehat{\mu}$ is locally
biholomorphic on $X\setminus S.$ The assumptions imply that $\widehat{\mu}$
maps $X\setminus S$ biholomorphically onto its image. Consequently
$\frac{\partial}{\partial t}$ can be lifted to a holomorphic vector field on
$X\setminus S$ (the use of $\frac{\partial}{\partial t}$ is important)\ and
can be extended to a holomorphic vector field without zeroes\ on $X$, because
of Riemann's removable singularity theorem. Therefore $\tau$ is
holomorphically trivial and $\widehat{\mu}$ maps $X$ biholomorphically onto
$Y\times F$; construct the orbital curves of $\frac{\partial}{\partial t}$
starting in the points of $(0)\times F=F$, this amounts to solve systems of
differential equations similar to the type discussed in \cite{Kodaira}, p. 63,
65; one has only\ to take into consideration, that\ in our case there is no
need\ (and no possibility) to work with partitions of unity as in the proof of
\cite{Kodaira}, p. 64, theorem 2.4. Thus we get a biholomorphic map
$X\rightarrow Y\times F$. This finishes the proof. A variant of this proof is:
As \emph{If }$\widehat{\mu}\left[  X_{0}\right]  =(0)\times F$ the by
$\widehat{\mu}$\ induced meromorphic map $\widehat{\mu}_{0}:X_{0}%
\rightarrow(0)\times F=F$ is bimeromorphic and the projections $\widehat
{G}\rightarrow X$, $\widehat{G}\rightarrow Y\times F$ of the graph
$\widehat{G}$ are biholomorphic according to a well known theorem on the
purity of the branch and the degeneration locus, as the set of points where
these projections are not locally biholomorphic are not of codimension one
(\cite{Kerner}, compare also \cite{Kuhlmann}). q.e.d.\newline

\subsection{Some further remarks.}

\textbf{a. On constructing a set of sections which are linearly independent
along the central fiber.}In chapter 4 we will employ the following procedure:
Let $\tau:X\rightarrow Y=\{\left\vert t\right\vert <1\}$ be a deformation map,
$\mathcal{D}$ be a line bundle on $X,$ $s_{0},$...,$s_{n}$ sections in
$\mathcal{D},$ the restrictions of which to $X_{t}$ are linearly independent
for $t\neq0.$ Suppose that $s_{0},...,s_{n}$ are linearly dependent along
$X_{0},$ i.e. that we have a relation of the type $\alpha_{0}\cdot
s_{0}+\alpha_{1}\cdot s_{1}+...+\alpha_{n}\cdot s_{n}=t^{\sigma}\cdot s$ ,with
$s\mid X_{0}$ not $\equiv0,\sigma>0,$ where $\alpha_{1},...,\alpha_{n}$ are
holomorphic functions on $Y,$ not all of value 0 in the point $t=0.$ Outside
$t=0,$ $s$ is a linear combination of $s_{0},...,s_{n}.$ Consider the subsheaf
generated over $\mathcal{O}_{Y}$ by $\tau_{\left(  0\right)  }s_{0}%
,...,\tau_{\left(  0\right)  }s_{n}$ and all $\tau_{\left(  0\right)  }s$ of
this type. As $\tau_{\left(  0\right)  }\mathcal{D}$ is coherent, this
subsheaf is coherent and free over $\mathcal{O}_{Y};$ we may therefore replace
$s_{0},...,s_{n}$ by $n+1$ sections in $\mathcal{D}$ , the images of which in
$\tau_{\left(  0\right)  }\mathcal{D}$ generate this subsheaf and are already
independent along $X_{0}$; compare \cite{Grauert1}$.$\textit{\ }%
\newline\newline\emph{We can therefore choose already s}$_{0},...,s_{n}%
$\emph{\ as sections which are linearly independent along }$X_{0}%
.$\emph{\newline}\newline The same procedure applies, if we are dealing with
sets of elements $\in\Gamma(X,\widetilde{d}\cdot\mathcal{D)},$ which are
linearly independent along the $X_{t}$ for $t\neq0$..\newline\newline%
\textbf{b. On vector fields of complex spaces.}Let us recall some results of
\cite{Kaup1}, \cite{Kaup2}, see \cite{Akhiezer}$.$ The set of singular points
of a (reduced) complex space $R$ is denoted by $N\left(  R\right)  .$ The
chain $R=:N_{0}\left(  R\right)  \supset N_{1}\supset N_{2}\supset...\supset
N_{i+1}\left(  R\right)  :=N\left(  N_{i}\left(  R\right)  \right)
\supset...$ is finite; let $J_{0}=0\subset J_{1}\subset J_{2}\subset...\subset
J_{i+1}\subset...$ be the corresponding chain of ideal sheaves. A holomorphic
vector field $\chi$ on $R$ is a continuous sheaf homomorphism $\mathcal{O}%
_{R}\rightarrow\mathcal{O}_{R}$ satisfying the product rule $\chi\left(
f\cdot g\right)  =f\cdot\chi\left(  g\right)  +g\cdot\chi\left(  f\right)  .$
On manifold coordinate neighborhoods $U\left(  z_{1},...,z_{n}\right)  \subset
R$ we can write $\chi\mathcal{=}\sum a_{i}\left(  z_{1},...,z_{n}\right)
\cdot\frac{\partial}{\partial z_{i}}.$ Let $\omega=\omega\left(  R\right)  $
be the Lie algebra of holomorphic vector fields on $R.$ We say $\chi\in\omega$
is respecting the chain of ideal sheaves above if $\chi\left(  J_{i}\right)
\subset J_{i}$ for all i. $\nu\left(  R\right)  :=(\chi\in\omega
|\mathcal{\ }\chi$ is respecting the chain of ideal sheaves above$)$ is a lie
subalgebra of $\omega$. $\newline$Now let $Q\subset\mathbf{C\times}R$ be open
with $(0)\times R\subset Q.$ A holomorphic map $\Phi:Q\rightarrow R$ is called
a local group of local transformations of $R,$ if $\Phi\left(  0,P\right)  =P$
for all $P\in R$ and $\Phi\left(  \omega,\Phi\left(  \omega^{\prime},P\right)
\right)  =\Phi\left(  \omega+\omega^{\prime},P\right)  $ if all arguments lie
in $Q.$ Any local group $\Phi$ of local transformations on $R$ induces on $R$
a holomorphic vector field $\chi_{\Phi}\in\nu\left(  R\right)  .$ One
identifies two such vector fields if they agree in a neighborhood of $\left\{
0\right\}  \times R$ as maps. Each holomorphic vector field $\chi\mathcal{\in
}\nu\left(  R\right)  $ is induced by exactly one local group of local
transformations. From this Siu infers on \cite{Siu2}, p. 262.\newline%
\newline\emph{Let }$C$ \emph{be a complex curve }$\subset U=U\left(  0\right)
\subset\mathbf{C}^{n}$ \emph{with }$0$ \emph{as the only singular point and
with }$p:\tilde{C}\rightarrow C$ \emph{as normalization map, let }$p\left(
P_{0}\right)  =0.$ \emph{Let }$T\emph{\ }$\emph{be a holomorphic vector field
on }$U$ \emph{tangential to }$C$ \emph{in all} \emph{nonsingular points of}
$C.$ \emph{Then there is exactly one holomomorphic vector field}
$\widetilde{T}$ \emph{on} $\tilde{C}$ \emph{with } $\left(  dp\right)
\widetilde{T}\left(  P\right)  =T\left(  p\left(  \emph{P}\right)  \right)  $
\emph{for all} $P\in\tilde{C}.$ \emph{Especially }$P_{0}$ \emph{is a zero of}
$\tilde{T}.$\newline\newline We will apply this theorem (possibly without
mentioning it explicitely) in the sequel in the following situation: We have a
family of curves $\gamma^{\ast}\subset X$ over $Y$ together with a holomorphic
vector field $\chi^{\ast}|\gamma^{\ast};$ we can lift this vector field to a
holomorphic vector field on the normalization of $\gamma^{\ast}$. Therefore we
may in later chapters without loss of generality suppose sometimes
$\gamma^{\ast}$ to be normal. - Similarly one can see, that holomorphic vector
fields on (reduced) irreducible complex spaces $G$ can be lifted to
holomorphic vector fields on the normalization of $G.$ \newline\newline Let
$\mathbf{P}^{n}$ be the complex projective space with the homogeneous
coordinates $s_{0}^{\prime},...,s_{n}^{\prime},$ let $H^{^{\prime}}$ be the
infinite hyperplane $s_{0}^{\prime}=0,$ provide $\mathbf{C}^{n}=\mathbf{P}%
^{n}\backslash H^{^{\prime}}$with the inhomogeneous coordinates $z_{1}%
=\frac{s_{1}^{\prime}}{s_{0}^{\prime}},...,z_{n}=\frac{s_{n}^{\prime}}%
{s_{0}^{\prime}}.$ The \emph{Euler vector} field $\chi=\chi\left(
z_{1},...,z_{n}\right)  =z_{1}\cdot\frac{\partial}{\partial z_{1}}%
+...+z_{n}\cdot\frac{\partial}{\partial z_{1}}$ is extendible to a holomorphic
vector field on $\mathbf{P}^{n};$ this extended vector field vanishes along
$H^{\prime}$with order one and is (in generell points) along $H^{\prime}$of
the form $-y_{0}\cdot\frac{\partial}{\partial y_{0}},$ where $H$ is given by
$y_{0}=0.$ The integral curves of $\chi$ resp. their closures (i.e. the
orbital curves of $\chi)$ in $\mathbf{P}^{n}$ are complex lines $l$ through
$0\in\mathbf{C}^{n}=\mathbf{P}^{n}\backslash H.$ Let $i:l\rightarrow
\mathbf{P}^{n}$ be the injection. There is exactly one holomorphic vector
field $\chi_{l}$ on $l$ with $i_{\ast}\left(  \chi_{l}\left(  P\right)
\right)  =\chi\left(  P\right)  $ for all $P\in l.$ Instead of $\chi_{l}$ we
too write $\chi\left\vert l\text{. }\chi\right\vert l$ has exactly two zeroes,
namly in $0\in\mathbf{C}^{n}$ and in $l\cap H.$ Denote by $\mathfrak{D}%
^{\prime}$ the positive generating line bundle of $\mathbf{P}^{n}.$
$\tilde{\chi}:=\left(  s_{0}^{\prime}\right)  ^{-1}\chi$ is a $\mathfrak{D}%
^{\prime-1}-$valued holomorphic vector field on $\mathbf{P}^{n}.$ Analogously
to $\chi|l$ we may define $\tilde{\chi}\left\vert l.\text{ }\tilde{\chi
}\right\vert l$ has exactly one zero, namely $0\in\mathbf{C}^{n}$.\newline

\subsection{The adjunction formula.}

$\alpha$.\textbf{\ }Let $F$ be an $n-$dimensional Fano manifold with $b_{2}=1$
and with $c_{1}(F)\geq n.$ Then $F$ is one of the following: \ If
$c_{1}(F)\geq$ $n+1,$ then $c_{1}(F)=n+1$ and $F=\mathbf{P}^{n}.$ If
$c_{1}(F)=n\geq3,$ then $F$ is the smooth quadric $\mathbf{Q}_{n}.-$ See
\cite{Parsha}, p. 52. - This theorem is due to Kobayashi and Ochiai. Further
information on these spaces $F$ can be found in chapter 3.\newline%
\newline$\beta.$ Let $\widetilde{F}$ be a (compact) moishezon manifold of
dimension $n$ with $b_{2}=1.$ $C\subset\widetilde{F}$ be a smooth rational
curve of degree $d_{C}$ in $\widetilde{F}$. Suppose $N_{C\backslash
\widetilde{F}}=\overset{n-1}{\oplus}\mathcal{O}\mathfrak{(r}_{i})$ with all
$r_{i}\geq1.$ The adjunction formula $\mathcal{K}_{C}=\mathcal{K}%
_{\widetilde{F}}|C\otimes\overset{n-1}{\bigwedge}\mathcal{N}_{C\backslash
\widetilde{F}}$ implies $c_{1}(\overset{n-1}{\bigwedge}\mathcal{N}%
_{C\backslash\widetilde{F}})$ = $\sum_{i=1}^{n-1}c_{1}(\mathcal{O}%
(r_{i}))=\sum_{i=1}^{n-1}r_{i}=$ $c_{1}(\mathcal{K}_{C})-$ $c_{1}%
(\mathcal{K}_{\widetilde{F}}|C)=-2-d_{1}\cdot c_{1}(\mathcal{K}_{\widetilde
{F}}).$ \newline\newline Suppose for instance $r_{1}=...=r_{n-1}\geq1$ and
$d_{l}=1$, then we have the equality$\ -c_{1}(\mathcal{K}_{\widetilde{F}%
})=c_{1}(\widetilde{F})=n+1,$ as is the case for $\widetilde{F}=$(deformation
of) $\mathbf{P}^{n}$; the case $c_{1}(\widetilde{F})\leq n$ is not possible.
\newline

\section{\textbf{Euler vector fields on flag manifolds with }$b_{2}%
=1$\textbf{\ and on some related spaces.}}

\subsection{ Generalities on flag manifolds $F$ with $b_{2}=1$.Euler vector
fields on $n$-dimensional flag manifolds $F$\ with Picard number one\ and
their description on Bruhat $n$-cells $B$\ identified with $\mathbf{C}^{n}%
$\newline}

\textbf{a. The topic of this chapter. }Let $F$ be an $n$-dimensonal flag
manifold (i.e. a rational, homogeneous, compact, complex manifold)\ with
$b_{2}=1$, let $\mathcal{D}_{F}$ be the positive generating line bundle on
$F$, also called the hyperplane bundle on $F$. We need that there are
hyperplanes $H_{F}$ in $F$ (i.e. zero sets of a sections in $\mathcal{D}_{F}$)
with complements $F\backslash H_{F}$ which are mapped by sections in the line
bundle $\mathcal{D}_{F}$\ biholomorphically onto the complement $\mathbf{C}%
^{n}$\ of a\ hyperplane $H^{^{\prime}}$\ in $\mathbf{P}^{n}$; we also collect
some generalities on \emph{Euler} \emph{vector fields\ }living on flag
manifolds with $b_{2}=1$\ This is the main theme of this chapter 3. I suppose
that there is nothing new in this chapter,but I do not always know a
suitable\ reference for the assertions of this chapter. \newline%
\newline$H^{^{\prime}}$, $H_{F}$ are supposed to be fixed with the properties
above, if nothing else is agreed upon. A holomorphic vector field on the
complex projective space $\mathbf{P}^{n}$ is called an \emph{Euler vector
field \ (with center }$O=(0.,,,.0)\in\mathbf{C}^{n}$\emph{)} if in the
inhomogeneous $\mathbf{C}^{n}$-coordinates $z_{1},...,z_{n}$ it can be
described by
\[
\sum_{i=1}^{n}z_{i}\cdot\frac{\partial}{\partial z_{i}}.
\]
It has an isolated zero in the origin $z_{1}=...=z_{n}=0$, the "center"; the
infinity hyperplane $H^{^{\prime}}$ of $\mathbf{P}^{n}$ is a divisorial zero
of degree one. Multiplication with a complex number $\neq0$ delivers a complex
vector field with the same orbital curves. \ Here we\ suppose that
$\mathbf{P}^{n}$ carries the homogeneous coordinates $p_{0},...,p_{n}$ with
$z_{i}=\frac{p_{i}}{p_{0}},i=1...,n.$ Let $a=$ $(a_{1},...,a_{n})\in
\mathbf{C}^{n},$ introduce the homogeneous coordinates $p_{0}^{^{\prime}%
}=p_{0},\,p_{i}^{^{\prime}}=p_{i}-a_{i}\cdot p_{0},\,i=1,...,n,$ and the
inhomogeneous coordinates $z_{i}^{^{\prime}}=\frac{p_{i}^{^{\prime}}}%
{p_{0}^{^{\prime}}}.$ For \ the Euler vector field on $\mathbf{C}^{n}$\ with
\textquotedblright center\textquotedblright\ $a=(a_{1},...,a_{n})$ we get
\newline%
\[
\sum_{i=1}^{n}z_{i}^{^{\prime}}\cdot\frac{\partial}{\partial z_{i}^{^{\prime}%
}}=\sum_{i=1}^{n}(z_{i}-a_{i})\cdot\frac{\partial}{\partial(z_{i}-a_{i})}%
=\sum_{i=1}^{n}(z_{i}-a_{i})\cdot\frac{\partial}{\partial z_{i}}.
\]
\newline The infinity hyperplane $H^{^{\prime}}=$ $(p_{0}=0)$ \ is a
divisorial zero of degree one of the into $\mathbf{P}^{n}$ extended vector
field, in points $p_{1}\neq0$ of the infinity hyperplane this Euler vector
field on $\mathbf{P}^{n}$ is\ of the form $-w_{0}\cdot\frac{\partial}{\partial
w_{0}}$ , where the coordinates are $w_{0}=\frac{p_{0}}{p_{1}}$, $w_{2}%
=\frac{p_{2}}{p_{1}}$, ... \newline\newline We will see that there exists a
hypersurface $H_{F}$ in $F$, a hyperplane $H^{^{\prime}}$in $\mathbf{P}^{n}$
and a bimeromorphic map $\mu_{F}:F\rightarrow\mathbf{P}^{n}$ with the
following properties: $\mu_{F}$ maps $F\backslash H_{F}$ biholomorphically
onto $\mathbf{C}^{n}=\mathbf{P}^{n}\backslash H^{^{\prime}}$; any Euler vector
field on $\mathbf{C}^{n}$ (as described above, naturally with center
$a\in\mathbf{C}^{n}$) can be extended holomorphically to a holomorphic vector
field on $F$; this vector field is called (in this paper) an Euler vector
field on $\ F$ with center $a$; it depends an $F$ and $a$, see also the
subsection 1.1.c. - Before proceeding let us look at some easy examples and
discuss some generalities.\newline\newline\textbf{b. Explicit Euler vector
field computations for grassmannians}. In special cases the existence of
holomorphic Euler vector fields on flag manifolds $F$\ can be shown by simple
computation, as is the case for $F=$ a smooth quadric $\mathbf{Q}_{n}$ of
dimension $\geq3$ ( \cite{Hwang1}), or $F=$ a grassmannian $G(m,r),$ the space
of all $m-$dimensional subvector spaces of $\mathbf{C}^{r}.$ In order to have
a concrete example, which is general enough, let us take a look at $F=G(3,5):$
\newline\newline The Pl\"{u}cker coordinates $p_{0}:....:p_{N}$ with
$N=\binom{5}{3}-1=9$\ describe sections in the $H^{2}(F,\mathbf{Z)=Z}$
generating line bundle $\mathcal{D}_{F}$ of $F$\ (being very ample). These
sections form a basis of $\Gamma(F,\mathcal{D}_{F})$ and give an embedding
$F\hookrightarrow\mathbf{P}^{N}$. \newline\newline Let $V$ be the affine
subspace of $F,$ the points of which are the subvector spaces of
$\mathbf{C}^{5},$ which are generated by the rows of the following matrix;
here $\left(  u,v,w,x,y,z\right)  \in\mathbf{C}^{6}$:%
\[
\left(
\begin{array}
[c]{ccccc}%
1 & 0 & 0 & u & v\\
0 & 1 & 0 & w & x\\
0 & 0 & 1 & y & z
\end{array}
\right)  .
\]
\newline The Pl\"{u}cker coordinates of $V$ are $\left\vert
\begin{array}
[c]{ccc}%
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}
\right\vert :\left\vert
\begin{array}
[c]{ccc}%
0 & 0 & u\\
1 & 0 & w\\
0 & 1 & y
\end{array}
\right\vert :\left\vert
\begin{array}
[c]{ccc}%
1 & 0 & u\\
0 & 0 & w\\
0 & 1 & y
\end{array}
\right\vert :...:\left\vert
\begin{array}
[c]{ccc}%
0 & u & v\\
0 & w & x\\
1 & y & z
\end{array}
\right\vert =1:u:...$ .\newline$V$ is analytically isomorphic to
$\mathbf{C}^{6}$, $F\setminus V$ is the irreducible hypersurface
$(p_{0}=0)\mathbf{.}$ \newline\newline Similarly, another open affine subset
$W$ of $F$ is generated by the rows of the matrix
\[
\left(
\begin{array}
[c]{ccccc}%
1 & 0 & u_{1} & 0 & v_{1}\\
0 & 1 & w_{1} & 0 & x_{1}\\
0 & 0 & y_{1} & 1 & z_{1}%
\end{array}
\right)
\]
with $\left(  u_{1},v_{1},w_{1},x_{1},y_{1},z_{1}\right)  \in\mathbf{C}^{6}%
;$in $W\cap V$ we have $u_{1}=-\frac{u}{y},v_{1}=v-u\cdot\frac{z}{y}%
,w_{1}=-\frac{w}{y},$ $x_{1}=x-w\cdot\frac{z}{y},$ $y_{1}=\frac{1}{y}%
,z_{1}=\frac{z}{y}.\,W\cap V$ is characterized by $y\neq0$, $y_{1}\neq0$. -
One gets these equations, if one tranforms the last matrix by row operations
into the form of the first matrix. \newline\newline One gets the form of the
second matrix just by interchanging two special columns of the first matrix,
namely $(0,0,1)^{Transose}$ and $(u_{1},w_{1},y_{1})^{Transpose}$
\newline\newline Identify $V$ with $\mathbf{P}^{6}\setminus\,$hyperplane$.$
The corresponding bimeromorphic map $\nu:W\rightarrow\mathbf{P}^{6}$ is given
by\newline\newline$\left(  u_{1}:v_{1}:w_{1}:x_{1}:y_{1}:z_{1}:1\right)
\,\rightarrow\left(  u:v:w:x:y:z:1\right)  \,=$\newline$((-u_{1}):(v_{1}%
y_{1}-u_{1}z_{1}):(-w_{1}):(x_{1}y_{1}-w_{1}z_{1}):1:z_{1}:y_{1}).$%
\newline\newline Let $\chi$ in $V$ \thinspace be the Euler vectorfield
\[
u\frac{\partial}{\partial u}+v\frac{\partial}{\partial v}+w\frac{\partial
}{\partial w}+x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}%
+z\frac{\partial}{\partial z}%
\]
In $W\cap V$ the vector field $\chi$ can be expressed as
\[
v_{1}\frac{\partial}{\partial v_{1}}+x_{1}\frac{\partial}{\partial x_{1}%
}-y_{1}\frac{\partial}{\partial y_{1}}.
\]
Thus because of Riemann's removable singularity theorem $\chi$ is extendible
to a holomorphic vectorfield on $F.$ Let us mention that one can deal\ with
arbitrary Grassmannians $G(n,m)$ in the same way by interchanging 2 special
columns. From the above one can read the proof that such $\chi$'s can be
extended to holomorphic vector fields on Grassmannians, thus securing the
existence of Euler vector fields on Grassmannians. \newline\newline$\chi$ is
an Euler vector field on $F=$ $G(2,5$) with $H_{F}=$zero divisor of $y_{1}$
and $F\backslash H_{F}=V=\mathbf{C}^{6}$ .$\chi$ is not an Euler vector field
on $F$ with $H_{F}=$ zero divisor of $y$ and $F\backslash H_{F}=W=\mathbf{C}%
^{6}$, therefore one should not be too surprised that the signs of $v_{1}$,
$x_{1},y_{1}$ in $v_{1}\frac{\partial}{\partial v_{1}}+x_{1}\frac{\partial
}{\partial x_{1}}-y_{1}\frac{\partial}{\partial y_{1}}$\ above differ.\newline%
\newline\textbf{c. Generalities on flag manifolds (with }$b_{2}=1$). The best
known flag manifolds $F$ with $b_{2}=1$ are the complex projective space
$\mathbf{P}^{n}$, the smooth quadric $\mathbf{Q}_{n}$ of dimension $n\geq3$,
the grassmannian\ $G(p,q)$, i.e. the space of all $p$-dimensional subvector
spaces of $\mathbf{C}^{q}$. - Flag manifolds $F$ are locally rigid
(\cite{Bott}). These spaces $F$ satisfy $H^{1}(F,\mathcal{O}_{F}$)
$=H^{2}(F,\mathcal{O}_{F})=0.$ The positive generating line bundle
$\mathcal{D}_{F}$ of $H^{1}(F,\mathcal{O}_{F}^{\ast})$ $=H^{2}(F,\mathbf{Z})$
$=\mathbf{Z}$ (we suppose $b_{2}=1$) is very ample. Especially the
anticanonical line bundle $-\mathcal{K}_{F}$ is very ample. \newline\newline
We refer first to \cite{BT}, p. 193-196, and \cite{Goto}; our main reference
is \cite{Akhiezer}. - Let $F$ be a rational, homogeneous, compact, complex
manifold with $b_{2}=1$, $Aut(F)$ the group of biholomorphic maps
$F\rightarrow F,$ $Aut_{0}(F)$ the identity component of $Aut(F)$,
$r=\dim_{\mathbf{C}}Aut_{0}(F)$. Take an arbitrary $P\in F,$ let
$\mathfrak{P}$ be the subgroup of the isotropy group of $P$ lying in
$Aut_{0}(F)$. $F$ is analytically isomorphic to $Aut_{0}(F)/\mathfrak{P}.$
$Aut_{0}(F)$ is semisimple. Let $\mathit{g}$ be the Lie algebra of (left
invariant) holomorphic vector fields on $\mathfrak{G:=}Aut_{0}(F).$ As
$\mathit{g}$ is semisimple, even simple, the map $\mathit{g\rightarrow}ad$
$\mathit{g}$ with $x\rightarrow ad$ $x$ is a Lie algebra isomorphism.\ We may
identify $\mathfrak{G=}Aut_{0}(F)$ and $Int(\mathfrak{G)}$, and both are
isomorphic to the group $Ad(\mathit{g})$\ of inner automorphisms of the Lie
algebra $\mathit{g}$ (\cite{Akhiezer}, p. 75 and \cite{Helgason}, p. 126), the
latter (also called the adjoint group of $\mathit{g}$) can be described as the
analytic subgroup of $Gl(\mathit{g})$ with Lie algebra $ad$ $\mathit{g}$ or as
the subgroup of $Gl(\mathit{g})$\ generated by $\exp$ $ad$ $\mathit{g}$; we
can identify these spaces (as in \cite{Goto}, p. 814); this allows to describe
$exp:ad$ $\mathit{g\rightarrow}\mathfrak{G}$, especially $exp:ad$
$\mathit{N}_{1}\mathit{\rightarrow}\mathfrak{G}$ and $log:exp(ad$
$\mathit{N}_{1}\mathit{)}\rightarrow\mathit{N}_{1}$ $\subset ad$ $\mathit{g}$
via exponential matrices as in \cite{Goto}, p. 816, also $\exp(adh_{1})$.
\newline\newline\emph{Remark}. \cite{Goto} thus gets an embedding of $F$\ in a
grassmannian $G(r,n)$ with $r=\dim\mathit{g}$ and $n=\dim F$. One cannot
expect that the restriction of the positive generating line bundle of $G(r,n$)
to the embedded $F$ is the positive generating line bundle on $F$. \ - The
exponential map $\exp$ depends on $\mathfrak{G}$, $\exp=\exp_{\mathfrak{G}}$.
There may exist unbranched covering spaces $\widetilde{\mathfrak{G}}$ of
$\mathfrak{G}$ with a corresponding Lie group homomorphism $\widetilde
{\mathfrak{G}}\rightarrow\mathfrak{G}$, $\widetilde{\mathfrak{G}}%
\neq\mathfrak{G}$ ; the Lie algebras of $\mathfrak{G}$ and $\widetilde
{\mathfrak{G}}$ coincide. For local computations it does not matter whether we
work with $\exp_{\widetilde{\mathfrak{G}}}:g\rightarrow\widetilde
{\mathfrak{G}}$ or with $\exp_{\mathfrak{G}}$. In general we will just write
$\exp$ without specifying $\widetilde{\mathfrak{G}}$ resp. $\mathfrak{G}$.
\newline\newline There is a subgroup $\mathfrak{N}_{1}$ of $Aut_{0}(F)$,
generated by ad($\mathit{N}_{1})$, with the following properties: Let
$\mathit{N}_{1}$ be the Lie algebra of $\mathfrak{N}_{1},$ $\dim\mathit{N}%
_{1}=n,$ i.e. $\mathit{N}_{1}\approx\mathbf{C}^{n}.$ The exponential map
ad($\mathit{N}_{1})\rightarrow\mathfrak{N}_{1}$ and its inverse can locally be
considered as biholomorphic polynomial maps, at least locally, take a look at
\cite{Goto}, one is dealing with nilpotent matrices. Let us identify $F$ and
$Aut_{0}(F)/\mathfrak{P},$ the corresponding map $\varkappa:\mathfrak{N}%
_{1}\rightarrow F,$ $\gamma\rightarrow\gamma(P),$ maps $\mathfrak{N}_{1}$
biholomorphically onto an open subset, a Bruhat cell of complex dimension $n$,
of $F.$ $H_{F}:=F\backslash\varkappa(\mathfrak{N}_{1})$ is necessarily\ a
hypersurface in $F$, which generates $H_{2\cdot n-2}(F,\mathbf{Z)=Z}$. In the
following we will often identify $\mathit{N}_{1}$ and $\mathfrak{N}_{1}%
$\ \newline\newline(*)\ \ \emph{If }$N_{1}\approx C^{n}$\emph{\ is (suitably)
identified with }$P^{n}\backslash H^{\prime}\approx C^{n},H^{\prime}%
=$\emph{\ a hyperplane, the composition of the exponential map\ }%
$N_{1}\rightarrow\mathfrak{N}_{1}$\emph{\ and }$\kappa:\mathfrak{N}%
_{1}\rightarrow F$\emph{\ can be extended to a bimeromorphic map }%
$P^{n}\rightarrow F,$\emph{\ such that the inverse bimeromorphic map }$\mu
_{F}:F\rightarrow P^{n}$\emph{\ is given by }$n+1$\emph{\ elements }%
$s_{0},...,s_{n}\in H^{0}(F,\mathcal{D}_{F}),$\emph{\ where }$\mathcal{D}_{F}%
$\emph{\ is the positive generating line\ bundle in }$H^{1}(F,O_{F}^{\ast
})=\mathbf{Z.}$\newline\newline Let us first postpone further discussion of
(*).\newline\newline We \ continue to suppose $F$ to be a flag manifold of
dimension $n$ with $b_{2}=1$. We denoted $\mathfrak{G}=Aut_{0}(F),$
$\mathfrak{P\subset G}$ the isotropy group of a point $P$ in $F$. We can
identify $F$ with the quotient $\mathfrak{G/P}.$ $\mathit{g}$ was introduced
as the Lie algebra of (left invariant) holomorphic vector fields on
$\mathfrak{G},$ let be $\mathit{h}$\textit{\ }a Cartan subalgebra with a
(real) basis $h_{1},...,h_{l}.$ Let $\alpha(1)>...>\alpha(n)>\beta
(1)>...>\beta(m)>$ $0>-\beta(m)>...>-\beta(1)>-\alpha(n)>...>-\alpha(1)$ be an
ordered root system with respect to $\mathit{h}\mathfrak{,}$ $e_{\alpha(i)},$
$e_{-\alpha(i)},$ $e_{\beta(j)},$ $e_{-\beta(j)}$ corresponding root
(sub)spaces with $\left[  e_{\alpha(i)},e_{-\alpha(i)}\right]  \in\mathit{h}$
etc., compare for instance \cite{Goto}.We may assume that the Lie
algebra\textit{\ }$\mathit{p}$\textit{\ }corresponding to $\mathfrak{P}$ is
generated by $e_{\beta(1)},...,e_{\beta(m),\text{ }}h_{1},...,h_{l},$
$e_{-\beta(m)},...,e_{-\beta(1)}$, $e_{-\alpha(1)},...,e_{-\alpha(n)}$, and
that $N_{1}$ is generated by $e_{\alpha(1)},...,e_{\alpha(n)}$. Here we have
chosen $e_{\alpha(i)},$ $e_{-\alpha(i)}$ such that for $h_{\alpha(i)}:=$
$[e_{\alpha(i)},e_{-\alpha(i)}]$ the relation $\alpha(i)(h_{\alpha(i)})=2$,
i.e.$\ \left[  h_{\alpha(i)},e_{\pm\alpha(i)}\right]  =\pm2\cdot e_{\pm
\alpha(i)}$ holds; the same choices are fixed for the $\beta(i)$.\ For all
$\widetilde{h}\in\mathit{h}$ and all roots $\alpha$ we have $[$ $\widetilde
{h},e_{\alpha}]=\alpha(\widetilde{h})\cdot e_{\alpha}$. For all roots
$\alpha,\beta$ the term $[e_{\alpha},e_{\beta}]$ is a multiple of
$e_{\alpha+\beta}$, if $\alpha+\beta$ is a root, otherwise $[e_{\alpha
},e_{\beta}]$ is $0$. \newline\newline Take a set $\Pi$ of independent simple
roots (these shall be positive) $\alpha_{1},...,\alpha_{l}$, such that every
positive root can be written in the form $\sum_{\varkappa=1}^{l}n_{\kappa
}\cdot\alpha_{\kappa}$ with nonnegative integer coefficients $n_{\varkappa}%
.$There exists a $k,1\leq k\leq l,$ such that the roots $\beta(1),...,\beta
(m)$ are exactly those positive roots which can be written as linear
combinations of the simple roots $\alpha_{\kappa},$ $\kappa\neq k.$ The roots
$\alpha(1)$ ,..., $\alpha(n)$ can be written as a linear combination of these
simple roots with nonnegative integer coefficients, the coefficient of
$\alpha_{k}$ is always $\geq$1. \newline\newline The $n$-dimensional,
irreducible, hermitian symmetric compact\ manifolds are those $n$-dimensional
flag manifolds with $b_{2}=1$, where one can arrange matters such that\ above
the coefficient of $\alpha_{k}$ always equals 1, look up the lists in
\cite{Helgason}, chap. X and/or check\ \cite{Knapp}, p.684 ff. In this paper
we may take this as the definition of an irreducible, hermitian, symmetric
compact complex\ manifold\newline\newline\textbf{d. More generalities on flag
manifolds with }$b_{2}=1$\textbf{, construction and definition of Euler vector
fields on these spaces. }.\newline\newline Let $\left(  ,\right)  $ denote the
Killing form on $\mathit{h,}$ which is nondegenerate, $\mathit{g}$ being
semisimple. See \cite{Akhiezer}, \S \ 4.6, and/or \cite{Helgason}, esp. the
table on p. 476, giving information on some highest roots $\alpha(1).$This is
in the hermitian, symmetric case, as already mentioned, of the form
$\alpha_{k}+$ a linear combination of the $\alpha_{\kappa},\kappa\neq k$
\newline\newline Let $\Delta$ be the root system of the Lie algebra $g$, we
consider $h_{\mathbf{R}}:=\underset{\alpha\in\Delta}{\Sigma}\mathbf{R\cdot
}h_{\alpha}$. $(,)|h_{\mathbf{R}}\times h_{\mathbf{R}}$ is a positive definite
scalar product, each root is real-valued on $h_{\mathbf{R}}$. For any
$\beta\in h_{\mathbf{R}}^{\ast}$ let $t_{\beta}\in h_{\mathbf{R}}$ be defined
by $(t_{\beta},h)=\beta(h)$ for all $h\in h_{\mathbf{R}}.\mathit{.}$For any
root $\alpha$ we have $t_{\alpha}=\frac{2}{(h_{\alpha},h_{\alpha})}h_{\alpha
}.$\ Let $\omega_{k}\in\mathit{h}^{\ast}$, the fundamental weight
corresponding to $\alpha_{k},$ be defined by $\omega_{k}(h_{\alpha_{\kappa}%
})=0$ for $\varkappa\neq k,$ $\omega_{k}(h_{\alpha_{k}})=1$. Let
$\widetilde{h}_{1}:=\frac{(h_{\alpha_{k}},h_{\alpha_{k}})}{2}\cdot
t_{\omega_{k}}$. \ $\widetilde{h}_{1}$ lies in the center of the Lie algebra
generated by $e_{\beta(1)},...,e_{\beta(m),\text{ }}h_{1},...,h_{l},$
$e_{-\beta(m)},...,e_{-\beta(1)}$. \newline\newline All $\alpha(i)(\widetilde
{h}_{1})=(\alpha_{k}+$a linear combination of the $\alpha_{\varkappa
},\varkappa\neq k)(\widetilde{h}_{1})$ coincide, one can show that
$\alpha_{\kappa}(\widetilde{h}_{1})=0$ for $\kappa\neq k$. Because $a_{\kappa
}(\widetilde{h}_{1})=(t_{a_{\kappa}},\widetilde{h}_{1})$ is for $\kappa\neq k$
a multiple of $(h_{a_{\kappa}},t_{\omega_{k}})=\omega_{k}(h_{a_{\kappa}}%
)=0$.\ Thus $\left[  \widetilde{h}_{1},e_{\pm\alpha(1)}\right]  =$ $\pm
\alpha_{k}(\widetilde{h}_{1})\cdot e_{\pm\alpha(1)}=\pm e_{\pm\alpha(1)}$,
$\left[  \widetilde{h}_{1},e_{\pm\alpha(i)}\right]  =\pm\alpha_{k}%
(\widetilde{h}_{1})\cdot e_{\pm\alpha(i)}=\pm$ $e_{\pm\alpha(i)}$for $i\neq1$
too, which gives (look at the transformation matrix defind by the linear map
$\mathit{g\rightarrow g,}$ defined by $x\rightarrow\lbrack\widetilde{h}%
_{1},x]$, which is a very special diagonal matrix, where the entries
corresponding to the $e_{\alpha(i)}$ all coincide, and look\ at
$exp(ad\mathit{N}_{1})$\textit{. Considering only relevant entries,
}$ad\mathit{N}_{1}$ consists in the hermitian symmetric case of diagonal
$(n+l)\times(n+l)-$\ matrices, among the diagonal entries\textit{. are the
}$e_{\alpha(i)}$, $i=1,...,n$.\textit{ }It should be expected, that this
simple set up could be handled by computation in order to define and construct
Euler vector fields. But there is a possibly simpler, even more general way,
we will take.. \newline\newline Let $F$ be an arbitrary flag manifold with
$b_{2}=1$, not necessarily of the hermitian, symmetric type. We work on
$\mathbf{C}^{n}$ resp. $\ $on $\mathit{N}_{1}$\ with\ the base $e_{a(i)}$,
$i=1,...,n$;$.$ We do not change the geometry of orbital curves if we multiply
all basis elements with the same (real) scalar $\neq0$.\ We can identify
$\mathfrak{N}_{1}=\exp(ad\mathit{N}_{1})$ with a \emph{cell} in $F,$ where
$F\backslash$\emph{cell} is a hypersurface $H_{F}$\ of degree 1 in $F.$ The
Euler vector field $\widehat{\chi}$ on $\mathbf{C}^{n}$ can be extended to a
holomorphic vector field $\chi^{^{\prime}}$\ on the naturally given\ closure
$\mathbf{P}^{n}$ of $\mathit{N}_{1}\emph{=}\mathbf{C}^{n}$ (shown by easy
computation). We have to make sure that this Euler vector field $\widehat
{\chi}$\ on $\mathbf{C}^{n}$ resp. on $\overline{\mathbf{C}^{n}}%
=\mathbf{P}^{n}$\ can be extended to a holomorphic vector field $\widehat
{\chi}_{0}^{^{\prime}}$\ on $F$. $H_{F}$ is irreducible. Therefore it suffices
to show, that $\widehat{\chi}$ on $\mathbf{C}^{n}$ can be extended
holomorphically into at least one point of $H_{F}$, because of Riemann's
removable singularity theorem. \newline\newline Let $\mu_{F}:F\rightarrow
\mathbf{P}^{n}$ be the bimeromorphic map which is (at least) biholomorphic in
the points of $\mathbf{C}^{n}=F\backslash H_{F}$. Let $H^{\prime}$ be the
hyperplane $\mathbf{P}^{n}\backslash\mathbf{C}^{n}.$ $\mu_{F}^{-1}$ is
holomorphic in a general point $\widetilde{P}^{^{\prime}}$ of $H^{^{\prime}}$,
let $\widetilde{P}\in H_{F}$ be the image of $P^{\prime}$. $(\mu_{F}%
^{-1})_{\ast}$ $\widehat{\chi}^{\prime}$ coincides in $\widetilde{P}$ with
$\widehat{\chi}_{0}$. Thus we get a holomorphic vector field $\widehat{\chi
}^{^{\prime}}$ on $F$. $\widehat{\chi}^{^{\prime}}|\mathbf{C}^{n}$ is the
Euler vector field $\widehat{\chi}$\ on $\mathbf{C}^{n}$ which is induced by
an Euler vector field on $\mathbf{P}^{n}$. We call $\chi^{^{\prime}}$ and any
scalar $(\neq0)$-multiple\ an Euler vector field on $F$. Thus we get Euler
vector fields on $F$, which in the points of $\mathbf{C}^{n}(z_{1},...,z_{n})$
$=F\backslash H_{F}=\mathbf{P}^{n}\backslash H^{^{\prime}}$can be described by
$\sum(z_{i}-a_{i})\cdot\frac{\partial}{\partial(z_{i}-a_{i})}$ with
$(a_{1},...,a_{n})\in\mathbf{C}^{n}(z_{1},...,z_{n})$. The coordinates $z_{1}%
$,..., $z_{n}$ of $\mathbf{C}^{n}=F\backslash H_{F}$come from $\mathbf{C}%
^{n}=$ $\mathbf{P}^{n}\backslash H^{^{\prime}}$\newline\newline The formula of
\ Baker-Campbell-Hausdorff implies: If $F$ is additionally hermitian
symmetric, then translations $\mathbf{C}^{n}\rightarrow\mathbf{C}^{n}$,
$(z_{1},...,z_{n})\rightarrow(z_{1}+a_{1},...,z_{n}+a_{n})$ are induced by
biholomorphic maps $F\rightarrow F$. Otherwise we only claim, that such a
translation stems from a bimeromorphic map $F\rightarrow F$; neverleness the
construction of\ Euler vector fields on $F$ delivers holomorphic vector fields
on $F$, whether $F$ is of the hermitian type or not. The construction given
above is: $\newline$\newline Start with an Euler vector field $\widehat{\chi}$
on $\mathbf{C}^{n}$ resp. with its extension $\chi^{^{\prime}}$ on
$\mathbf{P}^{n}$, extend $\widehat{\chi}$ to a holomorphic vector field
$\widehat{\chi}_{0}^{^{\prime}}$ on $F$ (this is possible), called together
with its scalar multiples $\neq0$ an Euler vector field on $F$. \newline%
\newline Instead of \ vector fields of type $\sum(z_{i}-a_{i})\cdot
\frac{\partial}{\partial(z_{i}-a_{i})}$ on $\mathbf{C}^{n}=F\backslash H_{F}$
we can (for our purposes) work with holomorphic vector \ fields $\sum
\widetilde{m}_{i}\cdot(z_{i}-a_{i})\cdot\frac{\partial}{\partial(z_{i}-a_{i}%
)}$ with say positive integers $\widetilde{m}_{i}$. But in the following we
will call only vector fields on $F$ Euler vector fields, if they can on
$\mathbf{C}^{n}=F\backslash H_{F}$ be described by (multiples of) $\sum
(z_{i}-a_{i})\cdot\frac{\partial}{\partial(z_{i}-a_{i})}$. For our purposes
there is no reason to consider more general types, although the vector fields
with the additional factors $\widetilde{m}_{i}$ could do the job as
well.\newline\newline One can replace $\mathit{N}_{1}$ by the space generated
by $e_{-\alpha(1)},...,e_{-\alpha(n)}$, this was the choice of \cite{Goto}.
Our choice of $\mathit{N}_{1}$ has the notational advantage, that positive
generating line bundles correspond to positive simple roots, a connection
which is established by the theorem of Borel- Weil on p.114 of \cite{Akhiezer}%
. Multiplication of $\ $Euler vector field $\chi^{^{\prime}}$\ by a scalar
$\neq0$ does not change the corresponding \ system of orbital curves.
\newline\newline If $F$ is a quadric or grassmannian of dimension $n$, the
existence of Euler vector fields and "Bruhat cells" $\mathbf{C}^{n}\subset
F$\ can be shown rather easily by computation; see \cite{Hwang1} if $F=$ a
smooth quadric. \newline\newline In the following we suppose $\alpha
_{k}=\alpha_{1}$.\newline\newline

\subsection{The projection map $\mu_{F}:F\rightarrow\mathbf{P}^{n}$ is defined
by sections in the generating line bundle of $F$.\newline}

Now let us start the proof of the statement (*) in 3.1.c above on $\mu_{F}$.
We suppose $F$ to be a flag manifold with $b_{2}=1$ and continue to use
otherwise the notations of 3.1. Choose an affine $(z_{1},...,z_{n}%
)-$coordinate system on ($\mathit{N}_{1}=$) $\mathfrak{N}_{1}=\mathbf{C}^{n},$
such that\ on $\mathfrak{N}_{1}$\ the Euler vector field $\chi^{^{\prime}}$ on
$F$ with center $0\in\mathfrak{N}_{1}$ (see 3.1) can (up to a scalar $\neq0$
multiple) be written in the form
\[
\sum_{i=1}^{n}z_{i}\cdot\frac{\partial}{\partial z_{i}}%
\]
\ We will continue to identify often $\mathit{N}_{1}$ and $\mathfrak{N}_{1}$.
Below$\ \chi^{^{\prime}}|\mathbf{C}^{n}$ is extendible to a holomorphic vector
field on $\mathbf{P}^{n}=\overline{\mathbf{C}^{n}\text{ }}$. Compactify
($\mathit{N}_{1}=$) $\mathfrak{N}_{1}=\exp(\mathit{N}_{1})=\mathbf{C}^{n}$ to
$\mathbf{P}^{n},$ such that $1,$ $z_{1},...,z_{n}$ extend to sections
$s_{0}^{^{\prime}},...,s_{n}^{^{\prime}}$ $\in H^{0}(\mathbf{P}^{n}%
,\mathcal{D}_{\mathbf{P}^{n}})$ in the generating line bundle $\mathcal{D}%
_{\mathbf{P}^{n}}$ of $\mathbf{P}^{n}.$ On ($\mathit{N}_{1}=$) $\mathfrak{N}%
_{1}=\mathbf{C}^{n}$ $\subset F$ the $1,$ $z_{1},...z_{n}$ extend to elements
$\widetilde{s}_{0},...,\widetilde{s}_{n}\in H^{0}(F,\widetilde{d}%
\cdot\mathcal{D}_{F})$. The bimeromorphic map $\mu_{F}:F\rightarrow
\mathbf{P}^{n}$ is given by $(\widetilde{s}_{0}:...:\widetilde{s}_{n}%
)=(s_{0}^{^{\prime}}:...:s_{n}^{^{\prime}})$; let $\widetilde{d}$ be chosen
minimal. We have to show $\widetilde{d}=1$. \newline\newline$H_{F}%
=F\backslash\mathfrak{N}_{1}$ is the complement of a real $2n$-cell, therefore
$H_{F}$ is irreducible and generates $H_{2n-2}(F,\mathbf{Z)}$ $=H^{2}%
(F,\mathbf{Z)=Z}$. \newline\newline We continue to use the\ notations
explained above and will not always distinguish between $N_{1}$ and
$\mathfrak{N}_{1}$. We refer to the theorems of Borel, Weil, Bott discussed in
\cite{Akhiezer}, esp. to the theorem on p. 114, theorem (a) on p. 123, and
furthermore to p. 121-123. The closure $l$\ of \ the image of $(c\cdot
e_{\alpha_{k}}|c\in\mathbf{C})$ in $F$ (the orbital curve through $0$ in
direction $e_{\alpha_{k}}$)\ is a minimal rational curve $l$, generating
$H_{2}(F,\mathbf{Z)=Z}$. Because: $\omega_{k}$ delivers a homomorphism
$\mathbf{C\cdot(}[e_{\alpha_{k}},e_{-\alpha_{k}}])\rightarrow\mathbf{C}$ with
$h_{\alpha_{k}}\rightarrow\omega_{k}(h_{\alpha_{k}})=1$, where $[e_{\alpha
_{k}},e_{-\alpha_{k}}]=h_{\alpha_{k}}$, $\omega_{k}$ is the differential
$\overset{\cdot}{\chi}$ of a multiplicative\ character $\chi\in X(T)$ with $T$
a suitable maximal torus of $Aut_{0}(F)$, which consequently delivers (compare
\cite{Akhiezer}, p. 121 ff. or for instance \cite{Fulton-Harris}, p. 393)
\ the positive generating line bundle $\mathcal{D}_{F}$\ on $F$ and\ a
$\mathbf{P}^{N}-$embedding of $F$ via $H^{0}(F,\mathcal{D}_{F})$, where
$N+1=\dim H^{0}(F,\mathcal{D}_{F}),$ and giving rise to a degree one embedding
of the line $l$ in $F$, as the restriction of $\chi$ to the (maximal) torus of
$Aut_{0}(l)$ delivers on the Lie algebra level just $\overset{\cdot}{\chi
}=\omega_{k}$. Note that $\chi$ defines the transition functions of the line
bundle $\mathcal{D}_{F}$\ on $Aut_{0}(F)/\mathfrak{P,}$the restriction of
these to $l$ gives the induced line bundle on $l$; because of the special
choice of $\overset{\cdot}{\chi}=\omega_{k}$ the induced line bundle on $l$
delivers a degree one embedding according to the above mentioned theorems of
Borel, Weil, Bott. q.e.d$\mathfrak{.}$ - Here $\chi$ is a character; in other
parts of this paper $\chi$ will represent a vector field;because of the
context (hopefully) there will not arise a mixup, if the same symbol
represents different objects at different places.\newline\newline%
\emph{Remark.} \emph{Especially we just proved resp. sketched a proof of the
existence of minimal rational curves }$l$\emph{ on }$F$ \emph{which are
orbital curves of Euler vector fields. }This is the same proof as in
\cite{Kollar2}, p. 249.\ According to the theorem of R.Bott and its
consequences (see chapter 4 in \cite{Akhiezer}, esp. p.123, theorem (a))
$H^{0}(F,\mathcal{D}_{F})$ is the irreducible representation $V^{\omega_{k}}%
$\ of $Aut_{0}(F)$ with highest weight $\omega_{k}$. \newline\newline We can
produce the following situation, see below: $l$ meets $H_{F}$ analytically
just once and this in a general point $P_{F}$ such that $\mu_{F}$ is
holomorphic in $P_{F}$; the by $\mu_{F}$ induced meromorphic map $l\rightarrow
l^{^{\prime}}$ is biholomorphic; it can be seen by looking at the special
construction of $l$, that $l$ is an orbital curve of an Euler vector field
$\chi^{^{\prime}}$ on $F$ with two different simple zeroes of $\chi^{^{\prime
}}|l$, $P_{F}\in H_{F}$ is one of these zeroes. $\mu_{F}(P_{F})$ is a simple
zero of $(\mu_{F})_{\ast}\chi^{^{\prime}}|l^{^{\prime}}$\ \ i.e. the proper
inverse image $l^{^{\prime}}$ of $l$\ in $\mathbf{P}^{n}$ is a line too (this
time a line in $\mathbf{P}^{n}$), cutting $H^{^{\prime}}$ just once in
$\mu_{F}(P_{F})$. Because $l$ and $l^{^{\prime}}$ are orbital curves of (via
$(\mu_{F})_{\ast}$) corresponding\ Euler vector fields on open sets of$\ F$
and $\mathbf{P}^{n}$; this will imply, that the bimeromorphic map $\mu
_{F}:F\rightarrow\mathbf{P}^{n}$ is given by $n+1$ elements $\widetilde{s}%
_{0},...,\widetilde{s}_{n}\in H^{0}(F,\mathcal{D}_{F}).$\ - Every line on $F$
not lying on $H_{F}$\ is an orbital curve of an Euler vector field. - There is
a subgroup of $Aut_{0}(F)$ and a dense $H_{F}$- Zariski open subset of $H_{F}%
$, such that $F\backslash H_{F}$ and this subset of $H_{F}$ are homogeneous
with respect to this subgroup, see for instance the paper \cite{BGG} on
Schubert cells of homogeneous manifolds. \newline\newline A more precise
description of $H_{F}=F\backslash\mathbf{C}^{n}$ will be given below by the
Bruhat decomposition$.$ \newline\newline\emph{We can arrange matters such that
the orbital curve }$l$\emph{ above\ of an Euler vector field cuts }$H_{F}%
$\emph{ just once, and this in a general\ point }$P_{F}$\emph{ outside the
common zero set }$S_{F}$\emph{\ of }$\widetilde{s}_{0},...,\widetilde{s}_{n}%
$\emph{. For the moment let us accept this to be the case without proof. This
will readily imply: Euler vector fields on }$F$\emph{ correspond (via (}%
$\mu_{F})_{\ast})$\emph{\ to vector fields on} $\mathbf{P}^{n}$\emph{;
}$l^{^{\prime}}$\emph{ as introduced above is a }$P^{n}$\emph{-line.\newline%
\newline}We have $H^{^{\prime}}=\mathbf{P}^{n}\backslash\mathbf{C}^{n}$
$=(s_{0}^{^{\prime}}=0)$, $l^{^{\prime}}=\mu_{F}[l]$ is a curve of degree one
in $\mathbf{P}^{n}$, meeting $H^{^{\prime}}$ just once, as $l$ cuts $H_{F}$
just once and this in a general point outside $S_{F}$, we can choose $l$ and
consequently $l^{^{\prime}}$\ as coordinate lines $\mu_{F}:F\rightarrow
\mathbf{P}^{n}$ is given in general points of $F$ by $(s_{0}^{^{\prime}%
}:...:s_{n}^{^{\prime}})=(\widetilde{s}_{0}:...:\widetilde{s}_{n})$ with
$\widetilde{s}_{0},...,\widetilde{s}_{n}\in H^{0}(F,\mathcal{D}_{F})$. The set
of indeterminancy $S_{F}$\ of $\mu_{F}$ is in $F$ at least of codimension $2$.
$(\mu_{F}^{\ast}s_{i}^{^{\prime}}=0)$ represents the positive generating
element in $H_{2\cdot n-2}(F,\mathbf{Z)=Z}$, $i=0,...,n$. \newline\newline
Consequence: Let $\widetilde{l}$ be an arbitrary line in $F$ not lying in
$H_{F}.$Then $\mu_{F}[\widetilde{l}]$ is a line in $\mathbf{P}^{n}.-$%
\newline\newline What is left, is the proof that we can arrange matters such
that $l$ is an orbital curve of the Euler vector field $\chi^{^{\prime}}$,
which\ cuts $H_{F}$ just once and this\ properly in a general\ point ouside
the common zero set of $\widetilde{s}_{0},...,\widetilde{s}_{n}$. We employ
properties of the Bruhat cell decomposition: \newline\newline As we apply
\cite{BGG}, prop. 5, let us recall the notion of the length $l(\omega)$ of an
element $\omega$ of the Weyl group of the (semi)simple linear algebraic group
$\mathfrak{G}$, look up for instance as a start \cite{Akhiezer}, p. 113 and
the remark on p. 121. $\omega$ can be writen as a composition of reflections
$\sigma_{\beta}$ corresponding to the (chosen basis of) simple roots $\beta$.
The minimal number of such composition factors (counted with their number of
occurrence) is called the length $l(\omega)$ of $\omega$. $l(\omega)$ is
exactly the number of positive roots mapped by $\omega$ onto negative
roots.\newline\newline$\mathbf{C}^{n}\subset F$ ($=\mathfrak{G}/\mathfrak{P}$,
where $\mathfrak{P}$ is the isotropy group of $P\in F$) is a Bruhat cell of
complex\ dimension $n.$There is a unique element $w_{0}$\ of the Weyl
group\ of $\mathfrak{G},$ mapping all positive roots onto negative ones; we
use the same notations for the elements of the reflection group and the
corresponding biholomorphic maps $F\rightarrow F$. Let $\Pi$ be a base of the
positive roots, let $w_{0.\alpha_{k}}$ be the element of maximal length in the
subgroup of the Weyl group generated by the reflections $\sigma_{\beta}$,
$\beta\in\Pi$ a simple root $\neq\alpha_{k}$. $\widetilde{w}_{0}:=w_{0}\circ
w_{0,\alpha_{k}}$ has according to \cite{Akhiezer}, p. 122,123, length
$l(\widetilde{w}_{0})=\ n=\dim F$, because we have $\widetilde{w}_{0}%
((\Delta_{\alpha_{k}}^{^{\prime}})^{+})\subset\Delta^{+}$ and $\widetilde
{w}_{0}(\Delta_{\alpha_{k}}^{^{+}})\subset\Delta^{-}$, where $\Delta^{+}$ is
the set of positive roots $\sum_{\beta\in\Pi}k_{\beta}\cdot\beta$\ with
respect to $\Pi$ and where $(\Delta_{\alpha_{k}}^{^{\prime}})^{+}$ is the set
of positive roots $\sum_{\beta\in\Pi}k_{\beta}\cdot\beta$ with $k_{\alpha_{k}%
}=0$. According to \cite{BGG}, prop. 5.1 $\widetilde{w}_{0}=w_{0}\circ
w_{0,\alpha_{k}}$ defines a (complex) $n-$ cell in $F,$ namely the
(open)\ orbit $F\backslash H_{F}=\exp(N_{1})\circ\widetilde{w}_{0}%
(P)=\mathbf{C}^{n}$. By construction $\alpha_{k}$ is mapped by $\widetilde
{w}_{0}$ onto $-\alpha_{k}$; this reasoning implies because of \cite{BGG},
cor. 2.3 furthermore $l(\widetilde{w}_{0})>l(\sigma_{\alpha_{k}}%
\circ\widetilde{w}_{0})$, which means, that the element $\ \sigma_{\alpha_{k}%
}\circ\widetilde{w}_{0}$ has length $l(\sigma_{\alpha_{k}}\circ\widetilde
{w}_{0})=\ n-1$;\ according to prop. 5.1 of \cite{BGG} $\sigma_{\alpha_{k}%
}\circ\widetilde{w}_{0}$ has the (complex) $(n-1)-$cell $\exp(N_{1}%
)\mathfrak{\circ}\sigma_{\alpha_{k}}\circ\widetilde{w}_{0}(P)$ in $H_{F}$ as
an orbit; this cell generates $H_{2n-2}(F,\mathbf{Z)=Z}$. $\sigma_{\alpha_{k}%
}\circ\widetilde{w}_{0}:F\rightarrow F$ maps $\mathbf{C}^{n}$
biholomorphically onto $\mathbf{C}^{n}$ and $H_{F}$ biholomorphically onto
$H_{F}$. We look at $0:=\widetilde{w}_{0}(P)$ as the zero point of
$\mathbf{C}^{n}=\exp(N)\circ\widetilde{w}_{0}(P)$. As above let $l$ be the
line in direction $e_{\alpha_{k}}$. $\sigma_{\alpha_{k}}$ (reflection in the
hyperplane orthogonal to $\alpha_{k}$) will be identified with
$Ad(e^{ad(e_{\alpha_{k}})}\cdot e^{-ad(e_{-\alpha_{k}})}\cdot e^{ad(e_{\alpha
_{k}})})$, mapping $l$ onto itself, interchanging $0$ and $0_{F}=:l\cap H_{F}%
$. $0_{F}=\sigma_{\alpha_{k}}\circ\widetilde{w}_{0}(P)$ is a general point of
$H_{F}$, \ i.e. a point,\ which lies outside a certain proper analytic subset
of $H_{F}$,\ $0_{F}$ can be mapped into any point of $H_{F}$ outside this
proper analytic set by a biholomorphic map of $F$, which maps $H_{F}$ onto
$H_{F}$.\newline\newline Instead of considering (\cite{Goto})\ the embedding
$F\rightarrow G(r,n)$ we might presuppose \cite{Goto}'s results and\ work with
the embedding $F\rightarrow\mathbf{P}^{N}$ given by the positive generating
very ample line bundle on $F$. If this is the case, $l$ has the wanted
properties, as can be seen later.. \newline\newline We already mentioned: If
$F$ is an irreducible, hermitian, symmetric, compact complex manifold,
translations $\mathbf{C}^{n}\rightarrow\mathbf{C}^{n}$ are induced by
biholomorphisms $F\rightarrow F.$ This follows easily from the
Campbell-Hausdorff formula, as $[e_{\alpha(i)},e_{\alpha(j)}]=0$ for $1\leq
i,j\leq n$. \newline\newline Simple special cases of such $F$, $H_{F}$ and
$\mu_{F}$: For $F=\mathbf{Q}_{n}$ $\subset\mathbf{P}^{n+1}$one can employ the
stereographic projection $\mu_{F}:\mathbf{Q}_{n}\rightarrow\mathbf{P}^{n}$
from a point on $\mathbf{Q}^{n};$here $H_{F}$ is the tangent hyperplane
$\subset$ $\mathbf{P}^{n+1}$ to $\mathbf{Q}_{n}$ in this point. For $F$ $=$ a
grassmannian, one can study the Pl\"{u}cker embedding of $F$ in order to
construct the projection $\mu_{F}$:$F\rightarrow\mathbf{P}^{n}$ and $H_{F}%
$\newline\newline\emph{Remark.} Euler vector fields on the considered spaces
$F$ depend on $F$ and $H_{F}$. We continue to stick to the last remark in 1.2.c.

\subsection{The rank concept.}

In order to understand the bimeromorphic map $\mu_{F}$ the rank concept is
useful. Especially it will become clearer, why in general orbital curves of
Euler vector fields\ on flag manifolds (with $b_{2}=1$) are not minimal
rational curves; only some of them are. - We continue to use the notation in
the sense above. Suppose that $F\neq\mathbf{P}^{n}$ is a flag manifold with
$b_{2}=1$. \newline\newline If $l^{^{\prime\prime}}$ is a general line in
$\mathbf{P}^{n},$ i.e. a line\ not meeting the set of indeterminancy of
$\mu_{F}^{-1}$,\ the degree of $\mu_{F}^{-1}\left[  l^{^{\prime\prime}%
}\right]  $ ($=\mu_{F}^{-1}(l^{^{\prime\prime}})$ in this case)\ in $F$ is
$>1$. - This degree is called the \emph{rank} of the flag manifold $F$ (with
$b_{2}=1$); the rank is the minimal degree (in $F$) of a rational curve in
generic tangent direction through a given point; naturally such a rational
curve is not a minimal rational curve in $F$, if $F\neq\mathbf{P}^{n}$. For
instance smooth quadrics of dimension $\geq3$ or grassmannians $G(2,2+q)$
(space of $2-$dimensional linear subspaces of $\mathbf{C}^{2+q}$) with
$q\geq2$ are of rank $2$. \newline\newline We employ: Let $l\subset F$ be a
line, an irreducible curve of degree one meeting $H_{F}$ in one point; then
the proper transform $l^{^{\prime}}:=\mu_{F}\left[  l\right]  $ $\subset
\mathbf{P}^{n}$ is a line too. \emph{Proof:} We use 3.2, 3.3. Suppose that
$l^{^{\prime}}$ is of degree $\widetilde{d}$ $>1;$ let $\mathcal{D}^{^{\prime
}}$ be the hyperplane bundle of $\mathbf{P}^{n}$; the zero set of a general
$s^{^{\prime}}\in\Gamma(\mathbf{P}^{n},\mathcal{D}^{^{\prime}}),$ a general
linear combination of the basis $s_{0}^{^{\prime}},...,s_{n}^{^{\prime}}$,
cuts $l^{^{\prime}}$ outside $H^{^{\prime}}$ in $\widetilde{d}$ \ points; the
zero set of the corresponding $s\in\Gamma\left(  F,\mathcal{D}_{F}\right)  $
cuts $l$ in $\widetilde{d}$ points, contrary to $\deg l=1.$ \emph{q.e.d}. - If
$F\neq\mathbf{P}^{n},$ the inverse is not true, i.e. the proper transform of a
$\mathbf{P}^{n}$-line is not necessarily a line in $F$..\newline\newline If
$F$ is a flag manifold with $b_{2}=1$ and with $rank$ $1$, then $F$ is a
$\mathbf{P}^{n}$. If $F$ is an irreducible hermitian symmetric space\ of
compact type, this rank definition is a reformulation of the ordinary one for
irreducible hermitian symmetric spaces, according to J.-M. Hwang, Geometry of
minimal rational curves on Fano manifolds, manuscript, School on vanishing
theorems and effective results in algebraic geometry, Triest 2000, subsection
1.4.5; apply 3.1. c. esp.the\ statement (*) in 3.1.c., and corresponding
proofs, explanations in 3.2, 3.3 of the present paper. But this differential
geometric characterization useful for symmetric spaces is not used in the
present paper.\ A list with irreducible hermitian symmetric spaces and their
ranks can be found in \cite{Hwang-Kebekus}, Chap. 6, or in \cite{G-Struct};
interesting in this context is the characterization of the rank notion by
\cite{Hwang-Kebekus}, cor. 6.6, compare the following subchapter 3.5. -\ A del
Pezzo manifold of dimension $\geq3$ and degree $5$ is never of $rank$ $1$

\subsection{On chains of lines on flag manifolds.}

We describe a procedure to be found for instance in \cite{Hwang-Kebekus},
\cite{Kebekus2}, esp. consult \cite{Nadel}.p. 685 We continue to suppose that
$F$ is a homogeneous rational manifold with Picard number 1 of dimension $n$.
An irreducible (reduced)\ curve $C$\ in $F$ of degree one, i.e. generating
$H_{2}(F,\mathbf{Z)=Z}$, is called a line (in/of $F$). The positive generating
line bundle $\mathcal{D}_{F}$ of $H^{1}(F,\mathcal{O}_{F}^{\ast})=\mathbf{Z}$
is very ample; embed $F$ via $\mathcal{D}_{F}$ in a $\mathbf{P}^{N}$, call the
embedded $F$ still $F;$ then an irreducible curve $C$ in $F$ is a line in $F$,
if it is a line in $\mathbf{P}^{N}$. Thus a line $C$ in $F$ through $P\in F$
is uniquely determined by $P\in C$ and the tangent direction $\in
\mathbf{P}T_{P}(F)$ of $C$ in $P$. \newline\newline The\ Chow space $C_{1}%
(F)$\ of$\ 1$-cycles in $F$ has (at least) one irreducible component, the
members of which are lines and\ cover a dense subset of $F$. Let $K$ be the
normalization of the union of all components of\ $C_{1}(F)$ the members of
which are lines in $F$; $K$ is compact, as we may identify $K$ via Pl\"{u}cker
coordinates\ with an analytic subset of the grassmannian $G(1,\mathbf{P}%
^{N})=G(2,\mathbf{C}^{N+1})$ of lines in $\mathbf{P}^{N}$; have in mind the
embedding of $F$ in $\mathbf{P}^{N}$.\ If in the following we are talking of
general lines, we may\ look at them as members of $K$. For any $P\in F$ we
define $K_{P}:=(C\in K|P\in C)$. The map $K\rightarrow$ $C_{1}(F)$ is
injective. \newline\newline The map $\tau_{P}:K_{P}\rightarrow\mathbf{P}%
T_{P}(F)$, sending $C\in K_{P}$ to its tangent direction $\in\mathbf{P}%
T_{P}(F)$, is named the tangent map and the image$\ C_{P}:=\tau_{P}%
(K_{P})\subset\mathbf{P}T_{P}(F)=\mathbf{P}^{n-1}$.\ The tangent map induces a
biholomorphism between $K_{P}$ and $C_{P}$; we may identify $K_{P}$ and
$C_{P}$; $\mathbf{P}T_{P}(F)$ can be identified with the exceptional divisor
we get by blowing up $F$ in $P$ (monoidally). \newline\newline Let $U\subset
K\times F$ be the normalization of the graph of the family of lines
parametrized by $K$ with the projections $\iota:$ $U\rightarrow F$,
$\pi:U\rightarrow K$. The fibers of $\pi$ are lines in $F$, analytically
isomorphic to $\mathbf{P}^{1}$; $\pi$ is proper and a $\mathbf{P}^{1}$-bundle
map, for homological reasons all fibers of $\pi$ are mapped by $\iota$
biholomorphically onto lines; $\iota$ is proper, the fiber of $\iota$ over
a$\ P\in F$ is mapped by $\pi$ biholomorphically onto $K_{P}$ \ Naturally $U$
is projective as an analytic subset of the projective space $K\times F$, the
set $K_{P}$ is projective for all $P\in F$.\newline\newline For any point
$P\in F$ define inductively $loc^{1}(P):=\underset{C\in K_{P}}{\cup C}$, and
$loc^{k+1}(P):=\underset{\underset{\text{ }Q\in lok^{k}(P)}{\text{ }C\in
K_{Q}}}{\cup C}$. It is $loc^{1}(P)=\iota(\pi^{-1}(K_{P}))$, $loc^{k+1}%
(P)=\iota(\pi^{-1}(\pi(\iota^{-1}(loc^{k}(P))))))$. Naturally$\ loc^{k}%
(P)\subset loc^{k+1}(P)$; this inclusion is proper, iff all components
increase their dimensions by moving from $loc^{k}(P)$ to $loc^{k+1}(P)$ or if
\ that is not the case, only new components are added. If the last is the
case, there are points on $loc^{k}(P)$ where (together with a local
$loc^{k}(P)$-neighborhood) $loc^{k}(P)$ and $loc^{k+1}(P)$\ coincide.
Therefore after a finite number of steps we arrive at an $l$ with
$loc^{l}(P)=loc^{l+1}(P)$. $loc^{l}(P)$ is the analytic set of all points in
$F$, which can be connected with $P$ by a chain of lines $\subset F$. For
reasons of homogenity $loc^{l}(P)$ is smooth. \newline\newline Call 2 points
of $F$ equivalent, if they can be joined by a chain of lines in $F$. All such
equivalent classes $loc^{l}(P)$, $P\in F$, have the same dimension, as $F$ is
homogeneous. \ Because of \cite{Cartan}, Main Theorem,\ one can provide the
set $R$ of equivalent classes with a\ normal complex structure, the quotient
structure is given in a natural way, such that in our special situation$\ R$
is even\ smooth and such that\ the holomorphic quotient map $F\rightarrow R$
is regular, here one applies, besides \cite{Cartan}, the very interesting
statement \cite{KK}, prop. 49 A. 21 on p.210, due to B. Kaup. Naturally we
could have referred just to \cite{Grauert2}. As $F$ is projective, $R$ is
moishezon.- Suppose $\dim R\geq2$. By lifting a hypersurface of $R$ to $F$, we
get a hypersurface on $F$, not meeting a general equivalence class
$loc^{l}(P)$, contradicting the assumption $b_{2}(F)=1$. Thus we have the
\newline\newline\emph{Lemma}. \emph{Any two points of }$\emph{F}$\emph{ can be
connected by a chain of lines in }$F$\emph{ }\newline\newline Although not
needed, note: If $F$ is \ additionally hermitian symmetric then we can connect
two points of $F$ always by a chain of lines of $length\leq rank$ of $F$ , and
there are pairs of points where the minimal length equals the $rank$. Here the
length of a chain of lines is meant to be the number of lines of such a chain
(\cite{Hwang-Kebekus}, chap. 6, cor. 6.6).\emph{\newline}\newline In our
application we only need: \emph{There is a dense open set in }$F$\emph{ such
that any two points of this subset can be connected by a chain of lines in
}$F.$\newline\newline This result is not not new. The Main Theorem of
\cite{Cartan} is an application of Grauert's image sheaf theorem. It is nice
to see, that \cite{Cartan} shows up on p. 209 of \cite{KK} as an exercise.

\subsection{Euler vector fields on del Pezzo manifolds of degree 5 and
dimension $\geq3$.}

It is of interest, whether there are other manifolds $F$ which are globally
rigid. One class of such manifolds is the family of del Pezzo manifolds of
degree $5$ and dimension $n\geq3$. The proofs of theorem 1 and 2 can be
adapted to deformations\ of this class.\ Characteristic properties of
$n-$dimensional, $n\geq3$, del Pezzo manifolds $F_{n}$\ of degree $5$: They
are projective; the Picard group is $\mathbf{Z}$, namely it is $H^{2}%
(F_{n},\mathbf{Z)=H}^{1}(F_{n},\mathcal{O}_{F_{n}}^{\ast})=\mathbf{Z}$; the
$F_{n}$ are Fano manifolds, for which the positive generating line bundle
$\mathcal{D}_{F_{n}}$ and the canonical bundle $\mathcal{K}_{F_{n}}$ are
related by $\mathcal{K}_{F_{n}}+(n-1)\mathcal{D}_{F_{n}}=0,$ $\mathcal{D}%
_{F_{n}}$ is very ample, we have $c_{1}(\mathcal{D}_{F_{n}})^{n}=5$. - These
properties already imply $n\leq6$.\ It is known that del Pezzo manifolds of
degree 5 of dimension $n\geq3$ are globally rigid under k\"{a}hler
deformations ( a consequence of the main result in \cite{Fujita}, see also
\cite{Parsha} or \cite{Kuhlmann6}), for $V_{5}$ this k\"{a}hler condition is
known to be superfluous. - The following spaces $F_{6}:=G(2,5)=G(\mathbf{P}%
^{1},\mathbf{P}^{4})$, $F_{5}$, $F_{4}$, $F_{3}$ ($=V_{5}$) are all del Pezzo
manifolds of degree 5 and dimension $n$ $\geq3$, more do not exist; their
construction shows, that they fulfill the assumptions of theorem 3.
\newline\newline There is a biholomorphic map $F_{6}=G(2,5)\rightarrow F_{6}$,
mapping $\mathit{N}_{1}=\mathfrak{N}_{1}=$ $\mathbf{C}^{n}\left\langle
e_{\alpha(1)},...,e_{\alpha(n)}\right\rangle $ biholomorphically onto
$V=\mathbf{C}\left\langle u,v,w,x,y,z\right\rangle ,$ as we have $F_{6}-$Euler
vector fields at disposal on $\mathbf{C}^{n}\left\langle e_{\alpha
(1)},...,e_{\alpha(n)}\right\rangle $ and on $V.$\newline\newline There is
another way (as above) to produce the\ Euler vector fields on $F_{6}=G(2,5)$,
the $6-$dimensional del Pezzo manifold of degree $5$. According to
\cite{Fujita}, section 7.12 on p. 421, this manifold can be constructed thus:
Consider in $\mathbf{P}^{6}$ with the homogeneous coordinates $p_{0}%
,...,p_{6}$ the hyperplane $D=(p_{6}=0)$ and the $D$-subspace $C=$
$(p_{0}\cdot p_{3}-p_{1}\cdot p_{2}=$ $p_{2}\cdot p_{5}-p_{3}\cdot p_{4}=$
$p_{0}\cdot p_{5}-$ $p_{1}\cdot p_{4}=$ $0)$. Let $\widetilde{\mathbf{P}}$ be
the blow up of $\mathbf{P}^{6}$ along $C$ and let $\widetilde{D}$ be the
induced blow up of $D$ along $C$, i.e. the proper transform of $D$ on
$\widetilde{\mathbf{P}}$. There is a modification map $\widetilde{\mathbf{P}%
}\rightarrow G(2,5)$, which blows down $\widetilde{D}$ to a $\mathbf{P}%
^{2}\subset G(2,5).$ This construction leaves $\mathbf{P}^{6}\backslash D$
untouched. There are sections $s_{0},...,s_{10}\in\Gamma(F_{6},\mathcal{D}%
_{F_{6}})$ in the generating line bundle $\mathcal{D}_{F_{6}}$, which embed
$F_{6}$ in $\mathbf{P}^{10}$, such that $s_{0},...,s_{6}$ define a
bimeromorphic map $G(2,5)\rightarrow\mathbf{P}^{6},$ which maps
$G(2,5)\backslash(s_{0}=0)$ biholomorphically onto $\mathbf{P}^{6}\backslash
D$. By simple computation one shows that $\mathbf{P}^{6}$-Euler vector fields
with centers outside $D$ can be lifted to holomorphic vector fields on
$\widetilde{\mathbf{P}}$, as these vector fields vanish along $D$ and as only
one blow up (where $C$ is blown up) occurs. Their images on $G(2,5)$ are
holomorphic\ Euler vector fields on $G(2,5)$. \newline\newline Let $F_{5}$ be
the zero set of a general section in the (positive) generating line bundle
$\mathcal{D}_{F_{6}}$, i.e. the $5-$dimensional del Pezzo manifold of degree
$5$. We have $H^{2}(F_{5},\mathbf{Z)}$ $\mathbf{=}$ $H^{1}(F_{5}%
,\mathcal{O}_{F_{5}}^{\ast})$ $=$ $\mathbf{Z}$. Let $\mathcal{D}_{F_{5}}$ be
the positive generating line bundle on $F_{5}$. $\Gamma(F_{5},\mathcal{D}%
_{F_{5}})$ embeds $F_{5}$ in $\mathbf{P}^{9}$. Starting with a $\mathbf{P}%
^{5}\backslash D,$ $D$ a hyperplane in $\mathbf{P}^{5}$ and a subspace
$C\subset D$ of dimension $2$ one can construct $F_{5}$ in a similar fashion
as $G(2,5),$ see \cite{Fujita}, section 7.10. There are sections
$s_{0},...,s_{9}\in\Gamma(F,\mathcal{D}_{F_{5}})$ in the generating line
bundle $\mathcal{D}_{F_{5}}$, which embed $F_{5}$ in $\mathbf{P}^{9}$, such
that $s_{0},...,s_{5}$ define a bimeromorphic map $F_{5}\rightarrow
\mathbf{P}^{5},$ which maps $F_{5}\backslash(s_{0}=0)$ biholomorphically onto
$\mathbf{P}^{5}\backslash D$. By simple computation one shows that
$\mathbf{P}^{5}$-Euler vector fields with centers outside $D$ can be lifted to
holomorphic vector fields on $\widetilde{\mathbf{P}}=$the blow up of
$\mathbf{P}^{5}$ along $C$, as these vector fields vanish along $D$;
$\widetilde{\mathbf{P}}$ can be mapped holomorphically and bimeromorphically
onto $F_{5}.$Their images on $F_{5}$ are holomorphic vector fields on $F_{5}$,
which we may call Euler vector fields on $F_{5}$. \newline\newline Let $F_{4}$
be the zero set of a general section in the (positive) generating line bundle
$\mathcal{D}_{F_{4}}$, i.e. the $5-$dimensional del Pezzo manifold of degree
$5$. We have $H^{2}(F_{4},\mathbf{Z)}$ $\mathbf{=}$ $\mathbf{H}^{1}%
(F_{4},\mathcal{O}_{F_{4}}^{\ast})$ $=$ $\mathbf{Z}$. Let $\mathcal{D}_{F_{4}%
}$ be the positive generating line bundle on $F_{4}$. $\Gamma(F_{4}%
,\mathcal{D}_{F4})$ embeds $F_{4}$ in $\mathbf{P}^{8}$. Starting with a
$\mathbf{P}^{4}\backslash D,$ $D$ a hyperplane in $\mathbf{P}^{4}$ and a
certain subspace $C\subset D$ of dimension $1$ one can construct $F_{4}$ in a
similar fashion as $G(2,5),$ see \cite{Fujita}, section 7.8. There are
sections $s_{0},...,s_{8}$ $\in\Gamma(F_{4},\mathcal{D}_{F_{4}})$ in the
generating line bundle $\mathcal{D}_{F_{4}}$, which embed $F_{4}$ in
$\mathbf{P}^{8}$, such that $s_{0},...,s_{4}$ define a bimeromorphic map
$F_{4}\rightarrow\mathbf{P}^{4},$ which maps $F_{4}\backslash(s_{0}=0)$
biholomorphically onto $\mathbf{P}^{4}\backslash D$. By simple computation one
shows that $\mathbf{P}^{4}$-Euler vector fields with centers outside $D$ can
be lifted to holomorphic vector fields on $\widetilde{\mathbf{P}}=$the blow up
of $\mathbf{P}^{4}$ along $C$, as these vector fields vanish along $D$;
$\widetilde{\mathbf{P}}$ can be mapped holomorphically and bimeromorphically
onto $F_{4}.$Their images on $F_{4}$ are holomorphic vector fields on $F_{4}$,
which we may call Euler vector fields on $F_{4}$.\newline\newline These
constructions of $F_{4}$, $F_{5}$, $F_{6}$ are rough descriptions of
\cite{Fujita}, 7.8, 7.10, 7.12. The construction of $F_{6}$ induces the
construction of $F_{5}$ as a subspace of $F_{6}:$ We produce a modification of
$\mathbf{P}^{6}$ along a hyperplane $D$, leaving $\mathbf{P}^{6}\backslash D$
intact. If we take a suitable linear subspace $\mathbf{P}^{5}$ of
$\mathbf{P}^{6}$, then the construction of $F_{6}$ starting with
$\mathbf{P}^{6}$ induces the construction of $F_{5}$ starting with
$\mathbf{P}^{5}\ $leaving\ $\mathbf{P}^{5}\backslash a$ $hyperplane$
unchanged. Therefore, if an orbital curve of an Euler vector field on $F_{6}%
$\ with center $\in F_{5}$ meets this $\mathbf{P}^{5}\backslash a$
$hyperplane$ in a point $\neq$center, then this orbital curve must lie in
$F_{5}$. With other words: Orbital curves of Euler vector fields on $F_{5}$
are orbital curves of Euler vector fields on $F_{6}$. Analogous statements can
be made for the $F_{4}$ as a subspace of \ the $F_{5}$.\newline\newline%
$F_{3}=V_{5}$ can be dealt with using \cite{Fujita}, section 7.5, or
\cite{Furushnaka};\ one can find a hypersurface $H_{F_{5}}$ and sections
$s_{0},...,s_{3}$ in the positive generating (very ample) line bundle, which
define a bimeromorphic map $F_{3}\rightarrow\mathbf{P}^{3},$ which maps
$F_{3}\backslash H_{F}$ biholomorphically onto $\mathbf{C}^{3}=\mathbf{P}%
^{3}\backslash hyperplane$ $of$ $\inf inity$, such that Euler vector fields on
$\mathbf{P}^{3}$ with centers in $\mathbf{C}^{3}$ can be lifted to holomorphic
vector fields on $F_{3}.$ Sketch of proof: Let $\mathbf{Q}_{3}\subset
\mathbf{P}^{4}$ be a smooth quadric of dimension $3$, project $\mathbf{Q}_{3}$
stereographically onto $\mathbf{P}^{3},$ let $D$ be the hypersurface of
$\mathbf{Q}_{3}$ blown down by this projection, which maps $\mathbf{Q}%
_{3}\backslash D$ biholomorphically onto $\mathbf{C}^{3}=\mathbf{P}%
^{3}\backslash hyperplane~of$ $\inf inity$, $D$ is the intersection of
$\mathbf{Q}_{3}$ and a hyperplane of $\mathbf{P}^{4}.$ Take an Euler vector
field on $\mathbf{P}^{3}$ with center in $\mathbf{C}^{3}$, this vector field
can be lifted to a holomorphic vector field on $\mathbf{Q}_{3}$. This lifted
vector field vanishes on $D$ along a Veronese curve $C^{^{\prime}}$of degree
$3$ in the hyperplane inducing $D.$ This vector field can be lifted to a
holomorphic vector field on the blow up $\widetilde{\mathbf{Q}}$\ of
$\mathbf{Q}_{3}$ along $C^{^{\prime}}$, the proper transform $\widetilde{D}$
of $D$ on $\widetilde{\mathbf{Q}}$ can be blown down to a rational curve; thus
$V_{5}=F_{3}$ is constructed, the above constructed vector field can be taken
down to $F_{3}$; this holomorphic vector field is of the wanted type we can
work with and is called an Euler vector field on $F_{3}=V_{5}$. \cite{Fujita},
7.7 describes a construction of $F_{4}$ by modifying $\mathbf{P}^{4}$ along a
hyperplane, which induces on a suitable linear\ subspace $\mathbf{P}^{3}$ a
modification delivering the $V_{5}$-construction described above. Therefore,
if we consider $V_{5}$ as a subspace of $F_{4}$ as described above, orbital
curves of Euler vector fields on $V_{5}$ are orbital curves of Euler vector
fields on $F_{4}$.\newline\newline The space $T$ of lines on $V_{5}$ is a
$\mathbf{P}^{2}$. Let $\lambda:$ $M\rightarrow T$ be the universal family of
lines on $V_{5}$ with the corresponding \ map $\rho:M\rightarrow V_{5}$.
$\rho$ is surjective and finite. Through a general point of $V_{5}$ there pass
3 lines. $SL(2,\mathbf{C})$ is acting on $V_{5}$, such that the following
holds: There are on $V_{5}$ a line $\Sigma$, a hypersurface $H_{5}$
(generating $H^{2}(V_{5},\mathbf{Z})=\mathbf{Z}$ and containing $\Sigma$),
such that $V_{5}\backslash H_{5}$, $H_{5}\backslash\Sigma$, $\Sigma$ are the
orbits of $SL(2,\mathbf{C})$, $SL(2,\mathbf{C})$ has no fixed points. $H_{5}$
is a ruled surface swept out by lines which meet $\Sigma$ ( \cite{Furushnaka},
esp. p. 113, 118; \cite{Muku}). \newline\newline Thus any 2 points of $V_{5}$
can be connected by a chain of lines as any line in $V_{5}$meets $H_{5}$.
\newline\newline Embed $G(2,5)$ in $\mathbf{P}^{9}$. The intersection of
$G(2,5)$ and a general $\mathbf{P}^{9-\iota}$, $i=1,2,3$ is an $F_{6-i}%
$\thinspace; consequently there exists an open, dense set on $F_{6-i}$, such
that any two points of $F_{6-i}$ can be connected by a chain of $V_{5}$'s and
therefore by a chain of lines.\newline\newline\emph{Remark}. It has been known
since long, that each k\"{a}hler manifold $F$ homeomorphic to $V_{5}$ is
analytically isomorphic to $V_{5}$\newline

\section{\textbf{Discussion of the bimeromorphic map }$\mu:X\rightarrow
Y\times P^{n}$\textbf{.}}

\subsection{Construction and some properties\ of the bimeromorphic
maps$\ \mu:X\rightarrow X^{^{\prime}}:=Y\times P^{n},$\textbf{\ }$\widehat
{\mu}:X\rightarrow Y\times F$\textbf{. }}

\textbf{a. The construction of }$\mu$, $\widehat{\mu}$\textbf{.}\ We continue
to use the notations of the introduction and of\ 3.1 and continue to suppose,
that the deformation maps we consider are differentiably trivial, and that the
theorems 1, 2, 3 we want to prove, are false, if nothing else is agreed upon
or obvious because of the context, in order to arrive at a contradiction.
\newline\newline So let $F$ be an $n-$dimensional flag manifold with $b_{2}%
=1$; remember that the anticanonical bundle of $F$ is very ample. - $F$ has
according to the last chapter 3, esp. 3.2, 3.3,\ the following properties:
\newline\newline$F$\emph{ is a projective }$n-$\emph{dimensional
complex\ manifold with }$b_{2}=1$\emph{ satisfying the
following\ assumptions:\newline}\newline\emph{a) }$F$\emph{ is a
Fano\ manifold of dimension }$n\geq3$\emph{ with}\textbf{ }$\mathbf{Z}%
=H^{2}(F,\mathbf{Z})$\emph{ }$=$\emph{ }$H^{1}(X,\mathcal{O}_{X}^{\ast}%
)$\emph{. The positive generating line bundle }$\mathcal{D}_{F}$\emph{ }$\in
H^{1}(X,O_{X}^{\ast})$\emph{, called also the hyperplane bundle, is very
ample}. \newline\newline\emph{b) }$F$ \emph{is a compactification of
}$\mathbf{C}^{n}$\emph{.} \emph{There exists a bimeromorphic map
}$F\rightarrow\mathbf{P}^{n}$, \emph{given by sections in }$\mathcal{D}_{F}%
$\emph{, together with a hypersurface }$H_{F}\subset F$\emph{, such that
}$F\backslash H_{F}$\emph{ is mapped biholomorphically onto the complement of
a hyperplane in }$\mathbf{P}^{n}$\emph{, a }$\mathbf{C}^{n}$\emph{. (Naturally
such an\ }$H_{F}$\emph{ can be proven to be of degree one in }$F.)$%
\newline\newline Let us collect below\ some generalities of the bimeromorphic
maps $\mu:X\rightarrow Y\times\mathbf{P}^{n},$ $\widehat{\mu}:X\rightarrow
\widehat{X}^{^{\prime}}:=Y\times F$ with the graphs $G$ and $\widehat{G},$
which are the concern of the present paper. We consider the following setup:
\newline\newline Let $\tau:X\rightarrow Y$ \ be a deformation map,
differentiably trivial, with $X_{t}=F$ for $t\neq0$. As $F$ has a positive
Chern class $c_{1}(F):=c_{1}(-\mathcal{K}_{F})>0,$ $X_{0}$ has to be
moishezon; see 2.1.c. We now apply 2.2: Let $\mathcal{D}$ be the positive
generating line bundle on $X$ with the properties stated in 2.2. We claim,
possibly after replacing $Y$ by a smaller neighborhood of $0\in Y$, but
without the necessity of a suitable base change, the following\newline%
\newline\emph{Lemma.} \emph{There are a hypersurface }$H\subset X,$\emph{ the
zero set of \ a section in the positive generating line bundle }$D,$\emph{ a
corresponding hyperplane }$H^{\prime}\subset X^{\prime}:=Y\times P^{n},$\emph{
elements }$s_{0},...,s_{n}\in\Gamma(X,D)$ \emph{with }$H=(s_{0}=0),$\emph{
linearly independent along all fibers }$X_{t}$\emph{, such that the
meromorphic map }$\mu:X\rightarrow X^{^{\prime}}=Y\times P^{n}$\emph{, defined
by }$\tau$\emph{ and }$s_{0},...,s_{n}$\emph{, maps }$X\setminus H\setminus
X_{0}$\emph{ biholomorphically onto }$X^{\prime}\setminus H^{\prime}\setminus
X_{0}^{\prime}$\emph{. We may assume that\ }$t$ \emph{vanishes along all
components of (the normalizations of)\ }$G_{0}$ \emph{and }$\widehat{G}_{0}%
$\emph{ with order one; if one feels more secure one can perform a suitable
base change.}. \emph{- }The proof will be given a few lines below.\newline%
\newline Here $\mu(P)=(\tau(P);s_{0}(P):$ $...:s_{n}(P))=$ $(\tau
(P);s_{0}^{^{\prime}}(\mu(P)):...:s_{n}^{^{\prime}}(\mu(P)))$ for general
$P\in X$.\ Let $G$ be the graph of $\mu$ with the projections $\varphi
:G\rightarrow X,$ $\psi:G\rightarrow X^{^{\prime}}$, let $X_{0}^{\ast}$ be the
component of $G_{0},$ which is a modification of $X_{0}$, and let $G_{0}%
^{\ast}$ be the component of $G_{0},$ which is a proper modification of
$X_{0}^{^{\prime}}$; let $\mathbf{P}^{n}$ be provided with a homogeneous
$(s_{0}^{^{\prime}},...,s_{n}^{^{\prime}})-$coordinate system. We will denote
$X_{0}^{\prime\prime}:=\mu\left[  X_{0}\right]  =\psi(X_{0}^{\ast}).$ Let $S$
be the set of indeterminancy of $\mu;$ because of 2.2.b we have $\dim
S_{t}\leq n-2$ for all $t.$ Let $\mathcal{D}^{\prime}$ be the positive
generating line bundle on \ $X^{^{\prime}}=Y\times\mathbf{P}^{n}$; we may
identify $s_{0}^{^{\prime}},...,s_{n}^{^{\prime}}$ with elements $\in$
$\Gamma(X^{^{\prime}},\mathcal{D}^{^{\prime}}\mathcal{)}$; it is $\mu^{\ast
}[(s_{i}^{^{\prime}}=0)]=(s_{i}=0)$;$\ \mu^{\ast}s_{i}^{^{\prime}}$ vanishes
outside $S$\ with order one along $(s_{i}=0)$, $0\leq i\leq n$, because of
2.2.b. We will identify $\mu^{\ast}s_{i}^{^{\prime}}$ and $s_{i}$, and $H$
with $(s_{0}=0),$ $H^{^{\prime}}$ with $(s_{0}^{^{\prime}}=0)$\ \newline%
\newline We can suppose that the set $C=(s_{0}\neq0,s_{1}=..s_{n}=0)$ defines
a general\ section $Y\rightarrow C\subset X$ over $Y$, and that the family of
tangent spaces of $X$ along the curve $C$\ can be identified with
$Y\times\mathbf{C}^{n};$ by construction $\mu$ is on $X\backslash H$ a
holomorphic map into $Y\times\mathbf{C}^{n}\subset Y\times\mathbf{P}^{n}$,
mapping the section over $Y$ onto $Y\times(0)$.\newline\newline Let $N$ be the
embedding dimension of $F;$ there are $s_{n+1},...,s_{N}\in\Gamma
(X,\mathcal{D)},$ such that $\tau$ and $s_{0},...,s_{n},...,s_{N}$ are
linearly independant along $X_{0}$\ (we may even suppose that the $s_{i}$ are
linearly independent along all $X_{t}$)\ and\ define a bimeromorphic map
$\widehat{\mu}:X\rightarrow\widehat{X}^{^{\prime}}:=Y\times F\subset
Y\times\mathbf{P}^{N}$ (, where $\mathbf{P}^{N}$ is supposed to be provided
with a homogeneous $(s_{0}^{^{\prime}},...,s_{n}^{^{\prime}},...,s_{N}%
^{^{\prime}})$ -coordinate system), biholomorphic outside $X_{0},$ with the
following properties: $\mu$ equals the composition of $\widehat{\mu}$ and the
projection $\widehat{X}^{^{\prime}}\rightarrow X^{^{\prime}};$ the projection
$\widehat{X}_{0}^{^{\prime\prime}}:=\widehat{\mu}\left[  X_{0}\right]
\rightarrow X_{0}^{^{\prime\prime}}$ is biholomorphic over $X_{0}%
^{^{\prime\prime}}\backslash H^{^{\prime}};$ the set of indeterminancy
$\widehat{S}$ of $\widehat{\mu}$ lies in $X_{0}$ and is here at least of
codimension $2$; a general fiber of $\widehat{\mu}_{0}:=\widehat{\mu}|X_{0}$
is a general fiber of $\mu_{0}$. Some more notations: Let $\widehat{G}$ be the
graph of $\widehat{\mu}$ with the projections $\widehat{\varphi}:\widehat
{G}\rightarrow X,$ $\widehat{\psi}:\widehat{G}\rightarrow\widehat{X}%
^{^{\prime}},$ let $\widehat{X}_{0}^{\ast}$ be the component of $\ \widehat
{G}_{0}$ which is mapped by $\widehat{\varphi}|\widehat{X}_{0}^{\ast}$
bimeromorphically onto $X_{0},$ let $\widehat{G}_{0}^{\ast}$ be the component
of $\widehat{G}_{0},$ which lies bimeromorphically over $\widehat{X}%
_{0}^{^{\prime}}:=(0)\times F$. - $\widehat{\varphi}(\widehat{G}_{0}^{\ast})$
and $\varphi(G_{0}^{\ast})$ coincide; we denoted $S^{\ast}:=\varphi
(G_{0}^{\ast})=$ $\widehat{\varphi}(\widehat{G}_{0}^{\ast})$. \newline\newline
Let $fib$ be a fiber of $\mu_{0}$; $X_{0}^{\ast}$ is the graph of $\mu_{0}$;
we can consider $fib$ as a fiber of $\psi|X_{0}^{\ast}$:$X_{0}^{\ast
}\rightarrow X_{0}^{^{\prime\prime}}$. Property b) listed\ above at the
beginning of this subchapter 4.1.a implies: If $\psi(fib)$ is a general point
of $X_{0}^{^{\prime\prime}}\backslash H_{F}$, then $fib\,\ $is also a
$\widehat{\mu}_{0}$-fiber. Have in mind that $X_{0}^{^{\prime\prime}}$ cannot
lie in the linear subspace $H_{F}$. Thus for general points $P\in X_{0}$ the
$\mu_{0}$- and the $\widehat{\mu}_{0}$- fiber through $P$ coincide.\newline%
\newline\emph{Proof of the lemma}: We are going to use 3.2, 3.3 and Barlet
resp. Chow spaces; compare \cite{Barlet}, esp. th\'{e}or\`{e}me 6, p. 109,
\cite{Fujiki2},\ \cite{Shafarevich}, p. 65ff., \cite{Timmerscheidt}, p. 268
and the reasoning on p. 271, 272, subsection e..By possibly\ shrinking $Y$ to
a smaller neighborhood of $0$\ we can arrange the following situation: There
exists a component $\widetilde{G}$ of the cycle space $C(X\times(Y\times
F)/Y)$ with the by $\tau$ induced proper\ map $\pi:\widetilde{G}\rightarrow
Y$, such that for uncountably many $t\neq0$ some points of$\ \widetilde{G}%
_{t}$ represent graphs of biholomorphic maps $X_{t}\rightarrow(t)\times F$,
thus we get $\pi(\widetilde{G})=Y$, because the set of irreducible components
of $\widetilde{G}$\ \ is countable. Consequently firstly all points in a dense
open subset\ of $\widetilde{G}_{t}$ are graphs of bimeromorphic maps
$X_{t}\rightarrow(t)\times F$, which secondly stem from biholomorphic maps.
There is a section $\gamma$ over $Y$ through a suitable point of
$\widetilde{G}_{0}$, such that for all $t\neq0$ the points of $\gamma_{t}$
represent graphs of biholomorphic maps $X_{t}\rightarrow(t)\times F$ , at
least after possibly\ shrinking $Y$ to a smaller neighborhood of $0$; at this
point it is not necessary to perform a suitable base change. This delivers a
bimeromorphic map $X\rightarrow Y\times F$, which maps $X\backslash X_{0}$
biholomorphically and fiber preservingly\ onto $(Y\backslash(0))\times F$.
This proves our assertions made above. q.e.d. \newline\newline\emph{Remark.
}There is no need to employ Chow spaces for some special manifolds, for
instance for $F=\mathbf{P}^{n}$ \ With the help of Grauert's image sheaf
theorem, esp. \cite{Grauert1}, p. 61, Satz 2 one can produce $s_{0},...,s_{n}$
$\in$ $H^{0}(X,D)$\ which are linearly independent along $X_{0}$ and generate
outside a discrete set of points of $Y$ the vector spaces $H^{0}(X_{t},D_{t})$
for $t\in Y$. For our purposes we may suppose in this case, that
$s_{0},...,s_{n}$ generate for all $t\neq0$ the $(n+1)-$dimensional vector
space $H^{0}(X_{t},D_{t})$.\newline\newline\textbf{b. It is }$\dim
X_{0}^{^{\prime\prime}}\geq2$: In our proof we may suppose $\dim
X_{0}^{^{\prime\prime}}\geq2$. Suppose $\dim X_{0}^{\prime\prime}=1.$ We may
assume that \thinspace$(s_{0}^{^{\prime}}=0)\cap X_{0}^{\prime\prime}%
$\thinspace\thinspace consists of $d$\thinspace$\geq2$\ simple points.
$\varphi(\psi^{-1}((s_{0}^{^{\prime}}=0)\cap X_{0}^{\prime\prime}%
))=\,(s_{0}=0)\cap X_{0}$ consists of at least $d$ homological equivalent
hypersurfaces, as $n-2\geq\dim S\cap X_{0}$, contrary to subsection 2.2.b of
the preceding chapter 2. Therefore $\dim X_{0}^{\prime\prime}\geq2$ and
\thinspace$n\geq3$. \newline\newline As $s_{0},...,s_{n}$ are linearly
independent along $X_{0}$, $X_{0}^{^{\prime\prime}}$ cannot lie in a proper,
linear subspace of $\mathbf{P}^{n}$, esp. the degree $d:=\deg X_{0}%
^{\prime\prime}$ in $\mathbf{P}^{n}$ is $>1$. \newline\newline\textbf{c. Why
the proof of }$X_{0}^{^{\prime\prime}}=\mu\left[  X_{0}\right]  =(0)\times
\mathbf{P}^{n}$\textbf{\ is important. }From 2.3, applied to $\widehat{\mu
}:X\rightarrow Y\times F$, we can deduce: $\widehat{\mu}$ is biholomorphic,
iff $\widehat{X}_{0}^{^{\prime\prime}}=\widehat{\mu}\left[  X_{0}\right]
=(0)\times F$, i.e. iff $X_{0}^{^{\prime\prime}}=\mu\left[  X_{0}\right]
=(0)\times\mathbf{P}^{n}$. Suppose that this is not the case, i.e.:\newline%
\newline\emph{Let us suppose }$X_{0}^{^{\prime\prime}}\neq(0)\times
\mathbf{P}^{n}$\emph{\ or what is the same }$\widehat{X}_{0}^{^{\prime\prime}%
}\neq(0)\times F$\emph{\ for the rest of this paper, unless the contrary is
explicitely stated.}\newline\newline\emph{Remark}\textbf{.} One may wonder why
at all\ we consider $\mu:X\rightarrow Y\times\mathbf{P}^{n}$ and not only
$\widehat{\mu}:X\rightarrow\widehat{X}^{^{\prime}}$. The reason is: Our aim
will be the proof of \ $X_{0}^{^{\prime\prime}}=(0)\times\mathbf{P}^{n}$,
resp. of $\widehat{X}_{0}^{^{\prime\prime}}=(0)\times F.$ For the proof we
employ Euler vector fields on $Y\times F,$ which can easily be characterized
by certain\ holomorphic vector fields $\chi^{^{\prime}}$\ on $Y\times
\mathbf{P}^{n}$; Euler vector fields on $F$ are described with the help of the
coordinate functions on $F\backslash H_{F}=\mathbf{C}^{n}\subset\mathbf{P}%
^{n}$ and become thus computable.\ For this purpose we have to investigate
also$\ \mu$ and not only $\widehat{\mu}$.\newline\newline\textbf{d. On a map
}$f^{^{\prime}}\rightarrow$\textbf{\ }$f\in H^{0}(X,\widetilde{d}\cdot
D)$\textbf{\ for certain }$f^{\prime}\in H^{0}(X^{\prime},\widetilde{d}\cdot
D^{\prime}\mathbf{)}$\textbf{. } $\mathbf{\alpha}$\textbf{.} Suppose
$3\leq\widetilde{n}\leq n$. If below we apply Bertini's theorem\ we use
tacitly that the meromorphic map $X_{0}\rightarrow\mathbf{P}^{\widetilde{n}}$
defined by $s_{0},...,s_{\widetilde{n}}$ has an image of dimension $\geq2$.
\newline\newline Take an $f^{\prime}\in H^{0}(X^{\prime},\widetilde{d}%
\cdot\mathcal{D}^{\prime}$) with $\widetilde{d}>0$, not vanishing in all
points of the $X_{0}^{^{\prime}}$, i.e. not being\ divisible by $t,$
$f^{\prime}=$ $\sum a_{\nu_{0}...\nu_{\widetilde{n}}}\cdot s_{0}^{^{\prime}%
\nu_{0}}\cdot\cdot s_{\widetilde{n}}^{^{\prime}\nu_{\widetilde{n}}}$, with on
$Y$ holomorphic coefficients $a_{\nu_{0}...\nu_{\widetilde{n}}}$. We say that
$f^{^{\prime}}$ depends on $s_{0}^{^{\prime}},...,s_{\widetilde{n}}^{^{\prime
}}$ \ Suppose that $\mu^{\ast}f^{^{\prime}}:=$ $\sum a_{\nu_{0}...\nu
_{\widetilde{n}}}\cdot s_{0}^{\nu_{0}}\cdot\cdot s_{\widetilde{n}}%
^{\nu_{\widetilde{n}}}$ vanishes along $X_{0}$ with order $\sigma
^{^{\prime\prime}}$. We may associate with $f^{^{\prime}}$ the element
$f:=\mu^{\ast}f^{`}/t^{\sigma^{^{\prime\prime}}}\in H^{0}(X,\widetilde{d}%
\cdot\mathcal{D}$). We say that $f$ depends on $s_{0},...,s_{\widetilde{n}}$.
\newline\newline$(f=0)$ is the $X-$closure of $\varphi\circ\psi^{-1}%
((f^{^{\prime}}=0))\backslash X_{0}^{^{\prime}}$, as $\dim S_{t}\leq n-2$ for
all t, and we have $(f=0)=\mu^{\ast}[(f^{^{\prime}}=0)]=\varphi(\psi
^{-1}[(f^{^{\prime}}=0])$. $(f=0)$ is of degree $\widetilde{d}$ $\ $in $X$ and
$(f^{^{\prime}}=0)$ is of degree $\widetilde{d}$ in $X^{^{\prime}}$, if
$f,f^{^{\prime}}$ do not have factors with multiplicity $>1$. - \newline%
\newline The $f^{^{\prime}}$-image $f$ $\in H^{0}(X,\widetilde{d}%
\cdot\mathcal{D})$ is over $Y\backslash(0)$ a linear combination of
$\widetilde{d}$-fold products of the $s_{0},...,s_{\widetilde{n}}$\ with on
$Y\backslash(0)$ holomorphic coefficients and is not divisible by $t.$
\newline\newline Vice versa. Suppose that $f$ $\in H^{0}(X,\widetilde{d}%
\cdot\mathcal{D})$ is over $Y\backslash(0)$ a linear combination of
$\widetilde{d}$-fold products of the $s_{0},...,s_{\widetilde{n}}$\ with on
$Y\backslash(0)$ holomorphic coefficients and is not divisible by $t.$ Then
$f$ is the image of an $f^{\prime}\in H^{0}(X^{\prime},\widetilde{d}%
\cdot\mathcal{D}^{\prime}$), not being\ divisible by $t,$ $f^{\prime}=$ $\sum
a_{\nu_{0}...\nu_{\widetilde{n}}}\cdot s_{0}^{^{\prime}\nu_{0}}\cdot\cdot
s_{\widetilde{n}}^{^{\prime}\nu_{\widetilde{n}}}$, with on $Y$ holomorphic
coefficients $a_{\nu_{0}...\nu_{\widetilde{n}}}$. \newline\newline\emph{We
call }$f$\emph{\ and }$f^{^{\prime}}$\emph{\ associated.} \newline%
\newline\emph{Remark}\textbf{. }The map defined by the association
$f^{^{\prime}}\rightarrow f$ described above is for $\widetilde{d}\geq\deg
X_{0}^{^{\prime\prime}}$\ not\ additive, as by assumption$\ X_{0}%
^{^{\prime\prime}}\neq X_{0}^{^{\prime}}$. \newline\newline$\mathbf{\beta.}$
Suppose that $f$ and $f^{^{\prime}}$ are associated, and that $f$ does not
have components with a multiplicity $\geq2$; this is for instance the case if
$f$ is a general $f\in H^{0}(X,\widetilde{d}\cdot\mathcal{D})$ depending on
$s_{0},...,s_{\widetilde{n}}$, because of Bertini's theorem, in this case
($f=0)\cap X_{0}$ is even irreducible, $f|X_{0}$ vanishes along \ $(f=0)\cap
X_{0}$ with order one. \newline\newline If \ $F\neq\mathbf{P}^{n}$, a general
$f\in H^{0}(X,\widetilde{d}\cdot\mathcal{D}$) is never the \textquotedblright
image\textquotedblright\ of such an $f^{\prime}\in H^{0}(X^{\prime}%
,\widetilde{d}\cdot\mathcal{D}^{\prime}$), as such an $f$ cannot be described
as a linear combination of products only of the $s_{0},...,s_{\widetilde{n}}$
with on $Y$ $\backslash(0)$ holomorphic coefficients. \newline\newline Let
$E^{^{\prime}}$ be a general hypersuface in $X^{^{\prime}},$ not containing
any $X_{t}^{^{\prime}};$ if $X_{0}^{^{\prime\prime}}$ $\subset E^{^{\prime}},$
$\mu^{-1}(E^{^{\prime}})$ contains $X_{0};$ the degrees of $E^{^{\prime}}$ in
$X^{^{\prime}}$ and of $E:=\overline{\mu^{-1}(E^{^{\prime}})\backslash X_{0}}$
in $X$ coincide.\newline\newline Take an $f^{^{\prime}}=\sum a_{\nu_{0}%
...\nu_{n}}\cdot s_{0}^{^{\prime}\nu_{0}}\cdot\cdot s_{n}^{^{\prime}\nu_{n}}$
$\in H^{0}(X^{^{\prime}},\widetilde{d}\cdot\mathcal{D}^{^{\prime}})$ as above,
not being divisible by $t$. For $f=\sum a_{\nu_{0}...\nu_{n}}\cdot s_{0}%
^{\nu_{0}}\cdot\cdot s_{n}^{\nu_{n}}/t^{\sigma^{^{\prime\prime}}}$ as above
the inverse image $\varphi^{\ast}f$ vanishes along $G_{0}^{\ast}$ with a
certain order $\widetilde{\sigma}$, we have (up to a factor without zeroes)
$f^{\prime}=(\mu^{-1})^{\ast}f/t^{\widetilde{\sigma}}.$ \newline%
\newline$\mathbf{\gamma}$\textbf{.}\emph{\ If }$\widetilde{d}$ \emph{is large
and }$f^{^{\prime}}$\emph{\ is suitably chosen, depending on }$s_{0}%
^{^{\prime}},...,s_{n}^{^{\prime}}$\emph{, then according to 2.1.c the zeroset
(f=0) of }$f$\emph{\ is cutting a general fiber of }$\mu\mid X_{0}\setminus
S$\emph{\ transversly in general points P, which implies }$X_{0}%
^{^{\prime\prime}}\subset(f^{\prime}=0)$\emph{, because there is an }%
$X-$\emph{neighborhood }$U$\emph{\ of }$P$\emph{\ such that }$(f=0)\cap
U\subset$ $\psi^{-1}[(f^{^{\prime}}=0)]$\emph{.} $f^{^{\prime}}$\emph{\ is
suitably chosen, if }$f$\emph{\ is general among those elements of }%
$H^{0}(X^{^{\prime}},\widetilde{d}\cdot D^{^{\prime}})$\emph{, which depend on
}$s_{0},...,s_{n}$\emph{. \newline}\newline But vice versa$\ X_{0}%
^{^{\prime\prime}}\subset(f^{\prime}=0)$ does not necessarily imply that the
zeroset of $f$ is cutting a general fiber of $\mu_{0}=\mu|X_{0}$ properly in
general points. \newline\newline We continue to suppose\ that $f^{^{\prime}%
}\in\Gamma\left(  X^{^{\prime}},\widetilde{d}\mathcal{D}^{^{\prime}}\right)  $
is not divisible by $t$ , and that $f\in\Gamma\left(  X,\widetilde{d}%
\cdot\mathcal{D}\right)  $is associated to $f^{^{\prime}}$. We have (let
$s^{^{\prime}}\in\Gamma\left(  X^{^{\prime}},\mathfrak{D}^{^{\prime}}\right)
\,$ be general)
\[
\frac{f}{s^{\widetilde{d}}}\circ\mu^{-1}=\frac{t^{\sigma^{\prime}}\cdot
f^{\prime}}{s^{\prime\widetilde{d}}\cdot t^{\widetilde{d}\cdot\sigma}}%
\]
with $\sigma^{\prime}\,,\,\sigma\in\mathbf{N,}$ $s^{\prime}\in\Gamma\left(
X^{\prime},\mathcal{D}^{\prime}\right)  ,$ modulo nonzero factors assumed to
be $1;$ here $\sigma^{\prime}$ is the vanishing order of $\varphi^{\ast}f$
along $G_{0}^{\ast}$, $\sigma$ the vanishing order of $\varphi^{\ast}s$ along
$G_{0}^{\ast}.$ We denote by $\sigma^{\prime\prime}$ the vanishing order of
$\psi^{\ast}f^{\prime}$ along $X_{0}^{\ast}.$ We have $\sigma^{\prime}%
+\sigma^{\prime\prime}=\widetilde{d}\cdot\sigma$ because of $\left(  \frac
{f}{s^{\widetilde{d}}}\circ\mu^{-1}\right)  \circ\mu=\frac{f}{s^{\widetilde
{d}}};$ and according to chapter 2 for large $\widetilde{d}$ we can arrange
matters such that $f$ is general among those elements in $H^{0}(X,\widetilde
{d}\cdot\mathcal{D})$ which depend on $s_{0},...,s_{n}$, i.e. such that
$\sigma^{^{\prime\prime}}$ is maximal (and certainly $>0$) and $\sigma
^{^{\prime}}$ minimal for given $\widetilde{d}$. \newline\newline%
$\sigma^{\prime}$ and $\sigma^{\prime\prime}$ depend only on $\widetilde{d}$.
\newline\newline$\mathbf{\delta}$. Analogously one can extend these
considerations to $\widehat{\mu}:X\rightarrow Y\times F\subset Y\times
\mathbf{P}^{N}$. We denoted by $\widehat{\mathcal{D}}^{^{\prime}}$\ be the
positive generating line bundle on $Y\times F$. We can associate to each
$f^{^{\prime}}=\sum a_{\nu_{0}...\nu_{N}}\cdot s_{0}^{^{\prime}\nu_{0}}%
\cdot\cdot s_{N}^{^{\prime}\nu_{N}}$ $\in H^{0}(\widehat{X}^{^{\prime}%
},\widetilde{d}\cdot\widehat{\mathcal{D}}^{^{\prime}})$, not vanishing in all
points $\widehat{X}_{0}^{^{\prime}}$, i.e. not being divisible by $t$, the
element $f=\mu^{\ast}f^{^{\prime}}/t^{\sigma^{^{\prime\prime}}}=\sum
a_{\nu_{0}...\nu_{N}}\cdot s_{0}^{\nu_{0}}\cdot\cdot s_{N}^{\nu_{N}}%
/t^{\sigma^{^{\prime\prime}}}\in$ $H^{0}(X,\widetilde{d}\cdot\mathcal{D}%
)$\ with $\sigma^{^{\prime\prime}}$= vanishing order of $\widehat{\psi}^{\ast
}f^{^{\prime}}$ along $\widehat{X}_{0}^{\ast}$. Each $f\in H^{0}%
(X,\widetilde{d}\cdot\mathcal{D})$, not vanishing in all points of $X_{0}$, is
of this type $f=\mu^{\ast}f^{^{\prime}}/t^{\sigma^{^{\prime\prime}}}$. The
inverse image $\widehat{\varphi}^{\ast}f$ vanishes along $\widehat{G}%
_{0}^{\ast}$ with a certain order $\widetilde{\sigma},$ we have (up to a
factor without zeroes) $f^{\prime}=(\widehat{\mu}^{-1})^{\ast}f/t^{\widetilde
{\sigma}}.$ We have $\widehat{\mu}[(f=0)]=$ $\widehat{\psi}(\widehat{\varphi
}^{-1}([(f=0)]))$ $=$ $(f^{^{\prime}}=0),(\widehat{\mu}^{-1})[(f^{^{\prime}%
}=0)]=(f=0)$. \newline\newline Let $\pi:Y\times F\rightarrow Y\times
\mathbf{P}^{n}=X^{^{\prime}}$ be the projection given by $(a_{0}%
:....:a_{N})\rightarrow(a_{0}:...a_{n})$ for general $(a_{0}:....:a_{N})\in
F$,\ and take an $f^{^{\prime}}=\sum a_{\nu_{0}...\nu_{n}}\cdot s_{0}%
^{^{\prime}\nu_{0}}\cdot\cdot s_{n}^{^{\prime}\nu_{n}}$ $\in H^{0}%
(X^{^{\prime}},\widetilde{d}\cdot\mathcal{D}^{^{\prime}})$ as above, not
vanishing in all points of $X_{0}$, then the associated $f=\sum a_{\nu
_{0}...\nu_{n}}\cdot s_{0}^{\nu_{0}}\cdot\cdot s_{n}^{\nu_{n}}/t^{\sigma
^{^{\prime\prime}}}$ equals $\widehat{\mu}^{\ast}(\pi^{\ast}f^{^{\prime}%
})/t^{\sigma^{^{\prime\prime}}}$; studying the relationship between
$f^{^{\prime}}$ and $f$ means studying the relation between $\pi^{\ast
}f^{^{\prime}}/t^{\sigma^{^{\prime\prime}}}$ and $f$. \newline\newline%
\textbf{e. } \textbf{On vanishing orders. }Vanishing orders are defined on
normalizations. $t$ vanishes with order one along $\widehat{G}_{0}^{\ast}$,
$G_{0}^{\ast}$, $\widehat{X}_{0}^{\ast}$, $X_{0}^{\ast}$. If there are more
components of $\widehat{G}_{0}$, $G_{0}$, we are not sure about the vanishing
orders of $t$. But by a suitable base change we can arrange that $t$ vanishes
with order one along all components (of the normalizations) of $G_{0}$,
$\widehat{G}_{0}$. We will assume in the following,\ that $t$ vanishes with
order one along all components (of the normalizations) of $G_{0}$,
$\widehat{G}_{0}$. \newline\newline$\emph{Claim}$\emph{:} \emph{Let }%
$s\in\Gamma(X,\mathcal{D})$ \emph{be general, let }$\sigma$\emph{ be the
vanishing order of }$\widehat{\varphi}^{\ast}s$\emph{ along }$\widehat{G}%
_{0}^{\ast}$\emph{. Then }$\widehat{\varphi}^{\ast}s$\emph{ vanishes along all
components }$\widehat{E}^{\ast}\neq\widehat{X}_{0}^{\ast}$\emph{ of }%
$\widehat{G}_{0}$\emph{\ with the same order }$\sigma$\emph{.} \newline%
\newline Because: Let $\widehat{s}^{^{\prime}}\in\Gamma(\widehat{X}^{^{\prime
}},\widehat{\mathcal{D}}^{^{\prime}})$ be associated to $s$. Consider points
$\widehat{P}_{1}\in\widehat{G}_{0}^{\ast}$, $\widehat{P}_{2}\in$ $\widehat
{E}^{\ast}$ with the same $\widehat{\psi}$-image $\widehat{P}^{^{\prime}}%
\in\widehat{X}_{0}^{^{\prime\prime}}$. Let $\widehat{C}_{i}\subset\widehat{G}$
be in $\widehat{P}_{i}$ irreducible curves lying\ biholomorphically\ over a
neighborhood of $0\in Y$, which we may assume to be $Y$ , $i=1$, $2$, we are
talking of sections over $Y$.\ According to the preceding (compare subsection
$\gamma$ above)\ we may identify $\widehat{\psi}^{\ast}\widehat{s}^{^{\prime}%
}|\widehat{C}_{i}$ with $(\widehat{\varphi}^{\ast}s)/t^{\sigma}|\widehat
{C}_{i}$. This gives that $\widehat{s}(\widehat{P}^{^{\prime}})\neq0$ is
equivalent to $(\widehat{\varphi}^{\ast}s)/t^{\sigma}|\widehat{C}_{i}$ is
holomorphic and $\neq0$ in $\widehat{P}_{i}$, $i=1$, $2$, which gives the
assertion. \emph{Remark}: De facto this claim means, that we can reason as if
$\widehat{G}_{0}$ decomposes only into $\widehat{G}_{0}^{\ast}$ and
$\widehat{X}_{0}^{\ast}$. - In the same fashion one proves:\newline%
\newline$\emph{Claim}$\emph{:} \emph{Let } $f$ $\in H^{0}(X,\widetilde{d}%
\cdot\mathcal{D})$\emph{ be general, let }$\widetilde{\sigma}$\emph{ be the
vanishing order of }$\widehat{\varphi}^{\ast}f$\emph{ along }$\widehat{G}%
_{0}^{\ast}$\emph{. Then }$\widehat{\varphi}^{\ast}f$\emph{ vanishes along all
components }$\widehat{E}^{\ast}\neq\widehat{X}_{0}^{\ast}$\emph{ of }%
$\widehat{G}_{0}$\emph{\ with the same order }$\widetilde{\sigma}$%
\emph{.}\newline\newline Let $f^{\prime}\in H^{0}(X^{\prime},\widetilde
{d}\cdot\mathcal{D}^{\prime}$) be associated to $f$. The connection between
$\widetilde{\sigma}$ and the vanishing order $\sigma^{^{\prime\prime}}$ of
$\widehat{\mu}^{\ast}f^{^{\prime}}$ along $X_{0}$ is described in subsection d
above.\newline\newline\emph{Remark. If } $f$ $\in H^{0}(X,\widetilde{d}%
\cdot\mathcal{D})$\emph{ is general, the associated }$f^{\prime}\in
H^{0}(X^{\prime},\widetilde{d}\cdot\mathcal{D}^{\prime}$) \emph{need not be
general. }\newline\newline\textbf{f. On }$\widehat{G}_{0}^{\ast}$,
$G_{0}^{\ast}$: $\widehat{G}_{0}^{\ast}$\emph{\ meets }$\widehat{X}_{0}^{\ast
}$\emph{\ in codimension one, i.e. }$\dim\widehat{G}_{0}^{\ast}\cap\widehat
{X}_{0}^{\ast}=n-1$\emph{.} \newline\newline Let $\widehat{S}$ be the common
zero set of $s_{0},...,s_{N}$. We have $\widehat{S}\subset X_{0}$. Let
$\sigma$ have the same meaning as in e. above. For our purposes here we
suppose (if one wishes this for the sake of notational\ simplicity)$\ \sigma
=1$. $\widehat{\varphi}^{\ast}(\frac{t^{\sigma}}{s})$, restricted to
$\widehat{G}_{0}^{\ast}$, is a meromorphic section in $\widehat{\varphi}%
^{\ast}(\mathcal{D}^{-1})|\widehat{G}_{0}^{\ast}$, with a proper\ zero set and
a proper\ pole set, as $\frac{t^{\sigma}}{s}$ is not locally constant on
$\widehat{G}_{0}^{\ast}$. According to e. above $\widehat{\varphi}^{\ast
}(\frac{t^{\sigma}}{s})$ (naturally we identify in such a situation $s$ with a
local function) vanishes for general $s\in H^{0}(X,\mathcal{D)}$ outside
$\widehat{\psi}^{-1}[(s^{^{\prime}}=0)]$\ exactly along the graph$\ \widehat
{X}_{0}^{\ast}$ of $\mu_{0}$ (here $s$ and $s^{^{\prime}}\in H^{0}%
(X^{^{\prime}},\mathcal{D}^{^{\prime}}\mathcal{)}$ are associated), if we
replace $\widehat{G}$ by its normalization; this implies the claim for
$\widehat{G}$ itself; this normalization does not change anything on
$\widehat{G}$ in general points of $\widehat{G}_{0}^{\ast}$: \newline%
\newline$\widehat{G}$ is normal, even a manifold in general points of
$\widehat{G}_{0}^{\ast}$; $\widehat{G}_{0}^{\ast}$ is a modification of
$(0)\times F$.\ - Let $Q$ be a general point $\in S^{\ast}$,$\ U$ a small
$X$-neighborhood of $Q$; consider the meromorphic map $\widetilde{\mu
}:\widehat{U}:=\widehat{\varphi}^{-1}(U)\rightarrow$ $Y\times\mathbf{P}^{N+1}$
(with graph $\widetilde{U}$), defined by $\widetilde{P}\rightarrow
(\tau(\widehat{\varphi}(\widetilde{P}));s_{0}(\widehat{\varphi}(\widetilde
{P})):.$ $..$ $:s_{N}(\widehat{\varphi}(\widetilde{P})):$ $t^{\sigma}%
(\widehat{\varphi}(\widetilde{P})))$ for general $\widetilde{P}\in\widehat{U}%
$. The restriction of $\widetilde{\mu}$ to $\widehat{U}\cap$ $\widehat{X}%
_{0}^{\ast}$ is given by the holomorphic map $(s_{0}(\widehat{\varphi
}(\widetilde{P})):...:s_{N}(\widehat{\varphi}(\widetilde{P})):$ $0)$\ for
general $\widetilde{P}\in\ \widehat{U}\cap$ $\widehat{X}_{0}^{\ast}$, the
graph of which is biholomorphic to $\widehat{U}\cap$ $\widehat{X}_{0}^{\ast}$.
$\widetilde{U}$ lies between $\widehat{U}$ and its normalization $\widehat
{U}^{n}$. The projection $\widetilde{\pi}:\widetilde{U}\rightarrow\widehat{U}$
is holomorphic, bimeromorphic, discrete, biholomorphic over a neighborhood of
$\widehat{U}\cap$ $\widehat{X}_{0}^{\ast}$. General fibers of $\widehat{G}%
_{0}^{\ast}\rightarrow S^{\ast}$ are cut by the $\widetilde{\pi}-$images
of$\ (\emph{(}\widehat{\varphi}\circ\widetilde{\pi})^{\ast}(\frac{t^{\sigma}%
}{s})=0)$ properly (this is a consequence of e.), esp.\ this intersection is
not empty. Consequently $\dim\widehat{G}_{0}^{\ast}\cap\widehat{X}_{0}^{\ast
}=n-1.$- Remark: For our purposes we may suppose $\widehat{G}$ to be normal.
\newline\newline Observe $\widehat{\psi}(\widehat{G}_{0}^{\ast}\cap\widehat
{X}_{0}^{\ast})=\widehat{X}_{0}^{^{\prime\prime}}$\emph{, }$\widehat{\varphi
}(\widehat{G}_{0}^{\ast})=$\emph{\ }$\widehat{\varphi}(\widehat{G}_{0}^{\ast
}\cap\widehat{X}_{0}^{\ast})=:S^{\ast}$\emph{, consequently }$\psi(G_{0}%
^{\ast}\cap X_{0}^{\ast})=X_{0}^{^{\prime\prime}}$\emph{, }$\varphi
(G_{0}^{\ast}\cap X_{0}^{\ast})=$ $\varphi(G_{0}^{\ast})$ $=S^{\ast}$%
.\newline\newline\emph{Remark.} Let $fib$ be a (component of a) general fiber
of $\widehat{X}_{0}^{\ast}\rightarrow$ $\widehat{X}_{0}^{^{\prime\prime}}$,
let $\widehat{\varphi}:\widehat{X}_{0}^{\ast}\rightarrow X_{0}$ be the
modification map introduced above, let $s\in H^{0}(X,\mathcal{D)}$ be
general.\ The procedure of Siu, see for instance \cite{Siu2}, delivers
$fib=\mathbf{P}^{m}$, $fib\cap(\widehat{\varphi}^{\ast}s=0)$ is analytically a
hyperplane in $\mathbf{P}^{m}$ and equals $fib\cap\widehat{G}_{0}^{\ast}$. But
we want to introduce Siu's non trivial\ procedure as late as possible and will
employ Siu's set up only in the proof of theorem 2. Naturally Siu's procedure
can be employed not before a discussion of holomorphic liftings of Euler
vector fields has taken place as in chap. 5 and chap. 7 below..

\subsection{Applying 2.1.c.}

We continue to use the notations above and discuss here\ first\ the
hypersurface case $\dim X_{0}^{^{\prime\prime}}=n-1$. The general fibers of
$X_{0}^{\ast}\rightarrow X_{0}^{^{\prime\prime}}$ are of dimension $m=1$, i.e.
are\ curves. Suppose that the ideal of functions vanishing on $X_{0}%
^{^{\prime\prime}}$ is generated on $Y\times(F\backslash H_{F})=Y\times
\mathbf{C}^{n}$\ by $t$ and $f_{d}^{^{\prime}}\in\Gamma(X^{^{\prime}},d\cdot
D^{^{\prime}})$ naturally with $d>1$.\newline\newline According to 2.1.d or to
Hironaka's Chow lemma (There are proper\ modifications of $G$ resp
$\widehat{G}$ lying in a $Y\times P^{\widetilde{N}}$, \cite{Hironaka})\ for
every large $\widetilde{d}\in N$ any general $f_{\widetilde{d}}\in
\Gamma(X,\widetilde{d}\cdot D)$, which depends outside $(s_{0}=0)$ only on
$z_{1}=\frac{s_{1}}{s_{s_{0}}}$,..., $z_{n}=\frac{s_{n}}{s_{0}}$, has the
following property: $\psi^{-1}[(f_{\widetilde{d}}^{^{\prime}}=0)]$\ cuts all
irreducible components of general fibers of $X_{0}^{\ast}\rightarrow
X_{0}^{^{\prime\prime}}$ in at least one general point $P^{\ast}$
transversally and coincides in a $G-$neighborhood of this point with
$(f_{\widetilde{d}}=0)$; this assertion includes, that for a suitable choice
of the $f_{\widetilde{d}}$ the proper inverse image $\psi^{-1}[(f_{\widetilde
{d}}^{^{\prime}}=0)]$ cuts a chosen general\ (component of a) general fiber in
at least one point transversally;\ here $f_{\widetilde{d}}^{^{\prime}}%
\in\Gamma(X^{^{\prime}},\widetilde{d}\cdot D^{^{\prime}})$ is associated to
$f_{\widetilde{d}}$, which can be considered as a function in $z_{1}%
=\frac{s_{1}}{s_{s_{0}}}$,..., $z_{n}=\frac{s_{n}}{s_{0}}$\newline\newline
Take such a general point $P^{\ast}\in(f_{\widetilde{d}}=0)\cap X_{0}^{\ast},$
then $P^{^{\prime}}=\psi(P^{\ast})$ can be supposed to be a smooth point $\in
X_{0}^{^{\prime\prime}}\backslash H_{\mathbf{P}^{n}}$\ of (the reduced)
$\ X_{0}^{^{\prime\prime}}$. There are a $G-$neighborhood $U^{\ast}$ of
$P^{\ast}$ and an $X^{^{\prime}}-$neighborhood $U^{^{\prime}}$ of
$P^{^{\prime}},$ such that $\psi^{-1}[(f_{\widetilde{d}}^{^{\prime}}=0)]\cap
U^{\ast}$ $=$ $(f_{\widetilde{d}}=0)\cap U^{\ast}$, that the map
$(f_{\widetilde{d}}=0)\cap U^{\ast}\rightarrow U^{^{\prime}}$ is defined,
proper,\ nondegenerate, and maps $(f_{\widetilde{d}}=0)\cap U^{\ast}$
biholomorphically onto a reduced, irreducible,\ analytic, smooth\ hypersurface
analytically described by $(f_{\delta}^{^{\prime}}=0)$\ in $U^{\prime}$ with
$(f_{\delta}^{^{\prime}}=0)\cap X_{0}^{^{\prime}}=X_{0}^{^{\prime\prime}}\cap
U^{^{\prime}}$. $(f_{\delta}^{^{\prime}}=0)$ is in $P^{^{\prime}}$ a local
component of $(f_{\widetilde{d}}^{^{\prime}}=0)$ and coincides set
theoretically in $U^{^{\prime}}$\ with the closure of $(\psi((f_{\widetilde
{d}}=0)\cap(U^{\ast}\backslash X_{0}^{\ast})))$. \newline\newline We let
$\sigma^{^{\prime\prime}}$ be the vanishing order of $\psi^{\ast}%
f_{\widetilde{d}}^{^{\prime}}$ along $X_{0}^{\ast}$; then $\frac
{f_{\widetilde{d}}^{^{\prime}}}{t^{\sigma^{^{\prime\prime}}}}$\ is a local
holomorphic parameter on $X_{0}^{\ast}$ in $P^{\ast}$ and coincides here up to
a factor $\neq0$ with $f_{\widetilde{d}}$. This vanishing order $\sigma
^{^{\prime\prime}}$is maximal among the elements of $\Gamma(X,\widetilde
{d}\cdot D)$ depending outside $(s_{0}=0)$\ only on $z_{1}=\frac{s_{1}%
}{s_{s_{0}}}$,..., $z_{n}=\frac{s_{n}}{s_{0}}$\ and is\ common to all general
pairs $P^{\ast}$ of $X_{0}^{\ast}$\ and $f_{\widetilde{d}}\in\Gamma
(X,\widetilde{d}\cdot D)$ with fixed $\widetilde{d}$, as $\mu$ and
$\widehat{\mu}$ are bimeromorphic. \newline\newline Naturally one cannot
expect that $X_{0}^{^{\prime\prime}}$ is\ in a neighborhood of $P^{^{\prime}}$
analytically described by $t=f_{\widetilde{d}}^{^{\prime}}=0$. But
$f_{\widetilde{d}}^{^{\prime}}$ has in $P^{^{\prime}}$a local holomorphic
irreducible factor $f_{\delta}^{^{\prime}}$ with the following properties: In
a neighborhood of $P^{^{\prime}}$ the ideal sheaf of $X_{0}^{^{\prime\prime}}$
is generated by $t$ and $f_{\delta}^{^{\prime}}$; let $\sigma^{^{\prime
\prime\prime}}$ be the vanishing order of $\psi^{\ast}$ $f_{\delta}^{^{\prime
}}$ along $X_{0}^{\ast}$; then $\frac{\psi^{\ast}f_{\delta}^{^{\prime}}%
}{t^{\sigma^{^{^{\prime\prime\prime}}}}}$ is a holomorphic parameter in
$P^{\ast}$. $\frac{\psi^{\ast}f_{\delta}^{^{\prime}}}{t^{\sigma^{^{^{\prime
\prime\prime}}}}}$ and \ $\frac{\psi^{\ast}f_{\widetilde{d}}^{^{\prime}}%
}{t^{\sigma^{^{\prime\prime}}}}$ differ\ in (a neighborhood of) $P^{\ast}$ by
a holomorphic factor $\neq0$ in $P^{\ast}$. In a suitable $G$-neighborhood of
$P^{\ast}$ the proper transforms $\psi^{-1}[(f_{\delta}^{^{\prime}}=0)]$ and
$\psi^{-1}[(f_{\widetilde{d}}^{^{\prime}}=0)]$ coincide; both sets can in a
$G$- neighborhood of $P^{\ast}$ analytically be described by $\frac{\psi
^{\ast}f_{\widetilde{d}}^{^{\prime}}}{t^{\sigma^{^{\prime\prime}}}}=0$ and by
$\frac{\psi^{\ast}f_{\delta}^{^{\prime}}}{t^{\sigma^{^{^{\prime\prime\prime}}%
}}}=0$. - Small continuous movements of $P^{\ast}$along $X_{0}^{\ast}$ and
$f_{\widetilde{d}}$ in $\Gamma(X,\widetilde{d}\cdot D)$\ will not change the
vanishing orders $\sigma^{^{\prime}}$, $\sigma^{^{\prime\prime}}$ and
$\sigma^{^{\prime\prime\prime}}$, apply the preceding subsection
4.1.d.$\gamma$. This vanishing order $\sigma^{^{\prime\prime}}$is maximal
among the elements of $\Gamma(X,\widetilde{d}\cdot D)$ depending outside
$(s_{0}=0)$ only on $z_{1}=\frac{s_{1}}{s_{s_{0}}}$,..., $z_{n}=\frac{s_{n}%
}{s_{0}}$. Later it will become clear that in the present case$\ \sigma
^{^{\prime\prime\prime}}$is the vanishing order $k$ introduced above in 1.2.c.
The geometric picture will show towards the end of this paper (esp. in the
hypersurface case) the at first improbably looking equality $\sigma
^{^{\prime\prime\prime}}=1$.\ In order to reduce the number of symbols and
notations (a little bit) we may without loss in the following assume
$f_{\widetilde{d}}^{^{\prime}}=f_{\widetilde{\delta}}^{^{\prime}}$ and
$\sigma^{^{\prime\prime}}=\sigma^{^{\prime\prime\prime}}=1$, if no confusion
arises and if we are not led to unjustified applications of these assumptions.
But one must be aware that these conventions mean a simplification.\newline%
\newline It can be seen that these considerations depending only formally on
the hypersurface case $m=1$ can be extended to the non hypersurface case. Let
$P^{\ast}\in X_{0}^{\ast}$ be general, $P^{^{\prime}}=\psi(P^{\ast})$ is a
general point of (the reduced) $X_{0}^{^{\prime\prime}}$, especially a smooth
point. The case $n=4$, $m=2$, $\dim X_{0}^{^{\prime\prime}}=2$ can be
considered as already the general case.From 2.1.c it follows, that there exist
for large $\widetilde{d}$ general homogeneous polynomials $f_{1\widetilde{d}}$
, $f_{2\widetilde{d}}$ $\in\Gamma(X,\widetilde{d}\cdot D)$ with the following
properties: These polynomials depend only on $s_{0}$, ..., $s_{n}$. The zero
set$\ (f_{1\widetilde{d}}$ $=$ $f_{2\widetilde{d}}=0)$ cuts the $\mu_{0}%
$-fiber through $P^{\ast}$ in $P^{\ast}$ transversely. Let $f_{1\widetilde{d}%
}$ $^{^{\prime}}$, $f_{2\widetilde{d}}^{^{\prime}}$\ be associated to
$f_{1\widetilde{d}}$ , $f_{2\widetilde{d}}$. Then there are a $G-$neighborhood
$U^{\ast}$ of $P^{\ast}$ and an $X^{^{\prime}}-$neighborhood $U^{^{\prime}}$
of $P^{^{\prime}},$ such that in $U^{\ast}$ the analytic sets $(f_{1\widetilde
{d}}=f_{2\widetilde{d}}=0)$ and $\psi^{-1}[(f_{1\widetilde{d}}^{^{\prime}%
}=f_{2\widetilde{d}}^{^{\prime}}=0)]$\ coincide and such that\ the map
$(f_{1\widetilde{d}}=f_{2\widetilde{d}}=0)\cap U^{\ast}\rightarrow
U^{^{\prime}}$ is defined, proper, nondegenerate, and maps $(f_{1\widetilde
{d}}=f_{2\widetilde{d}}=0)\cap U^{\ast}$ biholomorphically onto a (reduced)
irreducible, smooth\ analytic set $(f_{1\delta}^{^{\prime}}=f_{2\delta
}^{^{\prime}}=0)$\ in $U^{\prime}$ with $(f_{1\delta}^{^{\prime}}=f_{2\delta
}^{^{\prime}}=0)\cap U^{^{\prime}}\cap X_{0}^{^{\prime}}\ =X_{0}%
^{^{\prime\prime}}\cap U^{^{\prime}}$. $\psi^{-1}[(f_{1\widetilde{d}%
}^{^{\prime}}=f_{2\widetilde{d}}^{^{\prime}}=0)]$ coincides with
$(f_{1\widetilde{d}}=f_{2\widetilde{d}}=0)$ in the $G$-neighborhood $U^{\ast}$
of $P^{\ast}$. We can assume that $\psi^{\ast}f_{1\widetilde{d}}^{^{\prime}}$
and $\psi^{\ast}f_{2\widetilde{d}}^{^{\prime\prime}}$ vanish along
$X_{0}^{\ast}$ with the same order $\sigma^{^{\prime\prime}}$and that
$\frac{\psi^{\ast}f_{1\widetilde{d}}^{^{\prime}}}{t^{\sigma^{^{\prime\prime}}%
}}$, $\frac{\psi^{\ast}f_{2\widetilde{d}}^{^{\prime}}}{t^{\sigma
^{^{\prime\prime}}}}$ are local parameters on $U^{\ast}$ in a neighborhood of
$P^{\ast}$. This\ number $\sigma^{^{\prime\prime}}$ is common to all such
general pairs $P^{\ast}$ and $(f_{1\widetilde{d}}^{^{\prime}}$,
$f_{2\widetilde{d}}^{^{\prime}})$ with fixed large degree $\widetilde{d}$.
$(f_{1\delta}^{^{\prime}}=f_{2\delta}^{^{\prime}}=0)$ is in $P^{^{\prime}}$ a
local component of $(f_{1\widetilde{d}}^{^{\prime}}=f_{2\widetilde{d}%
}^{^{\prime}}=0)$: \newline\newline One cannot expect that $X_{0}%
^{^{\prime\prime}}$ is\ in a neighborhood of $P^{^{\prime}}$ analytically
described by $t=f_{1\widetilde{d}}^{^{\prime}}=f_{2\widetilde{d}}^{^{\prime}%
}=0$. - Suppose that $P^{^{\prime}}$ is a general point of \ $X_{0}%
^{^{\prime\prime}}$. But then we may suppose that $P^{^{\prime}}$ is a\ smooth
point of $X_{0}^{^{\prime\prime}}$ and that\ the in $P^{^{\prime}}$
holomorphic functions $f_{i\delta}^{^{\prime}}$, $i=1$,$2$, have the following
property: In a neighborhood of $P^{^{\prime}}$ the ideal sheaf of
$X_{0}^{^{\prime\prime}}$ is generated by $t$ and the $f_{i\delta}^{^{\prime}%
}$, here the $f_{i\delta}^{^{\prime}}$ are chosen with maximal vanishing order
$\sigma^{^{\prime\prime\prime}}$ of $\psi^{\ast}f_{i\delta}^{^{\prime}}$ along
$X_{0}^{\ast}$. Naturally $\sigma^{^{\prime\prime\prime}}$ $\leq
\sigma^{^{\prime\prime}}$ and $\sigma^{^{\prime\prime\prime}}$ does not depend
on $i$ . The $\frac{\psi^{\ast}f_{i\delta}^{^{\prime}}}{t^{\sigma
^{^{^{\prime\prime\prime}}}}}$ are holomorphic parameters in $P^{\ast}$. \ In
a $G-$neighborhood $U^{\ast}$ of $P^{\ast}$ the proper transforms $\psi
^{-1}[(f_{1\delta}^{^{\prime}}=f_{2\delta}^{^{\prime}}=0)]$ and $\psi
^{-1}[(f_{1\widetilde{d}}^{^{\prime}}=f_{2\widetilde{d}}^{^{\prime}}=0)]$
coincide; both sets can in $U^{\ast}$ analytically be described by $\frac
{\psi^{\ast}f_{1\delta}^{^{\prime}}}{t^{\sigma^{^{^{\prime\prime\prime}}}}%
}=\frac{\psi^{\ast}f_{2\delta}^{^{\prime}}}{t^{\sigma^{^{^{\prime\prime\prime
}}}}}=0$, by $\frac{\psi^{\ast}f_{1\widetilde{d}}^{^{\prime}}}{t^{\sigma
^{^{\prime\prime}}}}=\frac{\psi^{\ast}f_{2\widetilde{d}}^{^{\prime}}%
}{t^{\sigma^{^{\prime\prime}}}}=0$ and by$(f_{1\widetilde{d}}$ $=$
$f_{2\widetilde{d}}=0)$. It will turn out that in the present situation
$\sigma^{^{^{\prime\prime\prime}}}$\ coincides with the vanishing order $k$
talked about in 1.2.c. \newline\newline In the above we used: Because of
\cite{Hironaka} we may suppose that $\widehat{G}$ is dominated by a
bimeromorphic subspace of a $Y\times\mathbf{P}^{\widetilde{N}}$, for our
present\ purposes we may even assume $\widehat{G}\subset Y\times\mathbf{P}%
^{N}$ with $N=\widetilde{N}$, if we want to reduce the number of
notations.\ Small continuous movements of $P^{\ast}$along $X_{0}^{\ast}$ and
of $f_{1\widetilde{d}}$, $f_{2\widetilde{d}}$ in $\Gamma(X,\widetilde{d}\cdot
D)$ will not change the vanishing orders $\sigma^{^{\prime}}$, $\sigma
^{^{\prime\prime}}$ and $\sigma^{^{\prime\prime\prime}}$. The $f_{i\delta
}^{^{\prime}}$ can be considered as local factors of the $f_{i\widetilde{d}%
}^{^{\prime}}$ in general points of $X_{0}^{^{\prime\prime}}$. We have (degree
of a general $\mu_{0}$-fiber in $(0)\times\mathbf{P}^{\widetilde{N}}$%
)$\cdot\sigma^{^{\prime\prime\prime}}=\sigma^{^{\prime\prime}}$. If we do not
like the reasoning with $N=\widetilde{N}$, $\widehat{G}\subset Y\times
\mathbf{P}^{N}$ (which only describes the procedure) we have to replace
$X_{0}^{\ast}$ by its proper inverse image in the above mentioned
bimeromorphic subspace of $Y\times\mathbf{P}^{\widetilde{N}}$. \newline%
\newline For reasons of notational simplicity we will often suppose in a
situation like this $\sigma^{^{\prime}}=\sigma^{^{\prime\prime}}%
=\sigma^{\prime\prime\prime}=1$, $f_{1\widetilde{d}}^{^{\prime}}=f_{1\delta
}^{^{\prime}}$, $f_{2\widetilde{d}}^{^{\prime}}=f_{2\delta}^{^{\prime}}$, will
abstain from unjustified applications of these identifications, confusion by
this unlikely. \newline

\section{\textbf{Euler vector fields and deformations (mainly) in the
hypersurface case }$m=1$\textbf{.}}

\subsection{Introductory remarks on lifting Euler vector fields
from\textbf{\ }$X^{^{\prime}}$\ to\textbf{\ }$X$.}

\textbf{a. Generalities. }We continue to use the set up of the preceding
chapter, especially continue to suppose $X_{0}^{^{\prime\prime}}\neq
X_{0}^{^{\prime}}$, i.e. $X_{0}\neq F$, unless the contrary is explicitely
stated. General fibers of $\mu_{0}:X_{0}\rightarrow X_{0}^{^{\prime\prime}}$
resp. of $\widehat{\mu}_{0}:X_{0}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$
have dimension $m:=n-\dim X_{0}^{^{\prime\prime}}$ and are supposed (for
notational reasons) to be irreducible; our reasoning will not depend on the
irreducibility of fibers. We will show that there are $m$ additive, commuting
holomorphic vector fields $\chi_{1},...,\chi_{m}$ on $X_{0},$ such that any
irreducible component of a general $\mu_{0}$-fiber is the closure of a
maximal\ integral manifold defined by this collection of these vector fields,
i.e. an orbital variety of $\chi_{1},...,\chi_{m}.$ \newline\newline With the
exception of the proof of the lemma below and related statements (see the
remark of subchapter 5.1.d\ at the end of this subchapter 5.1) the
considerations of 5.1 hold for arbitrary flag manifolds with Picard number 1
and del Pezzo manifolds of dimension $\geq3$ and\ degree \ $5$. In general it
should be clear (from the context), which of the following assertions,
procedures and definitions hold for (or can immediately extended to) arbitrary
flag manifolds (with Picard number one) or del Pezzo manifolds of dimension
$\geq3$ and\ degree $5$\ \newline\newline Employing the results of chapter 3
we can produce holomorphic vector fields on $Y\times F$, $Y\times
\mathbf{P}^{n}$, $X$ thus: We start with a\textbf{ }section $\sigma
:Y\rightarrow X$ through a general point of $X_{0}$, denote $C:=\sigma(Y)$,
the induced section (the following symbol represents at other places a
vanishing order, because of the different contexts there will be no mix up)
$\sigma^{^{\prime}}:=\mu\circ\sigma$ with $C^{^{\prime}}:=\mu\lbrack C]$, the
induced section $\sigma^{\ast}:=\varphi^{-1}\circ\sigma$ with $C^{\ast
}:=\varphi^{-1}[C]$, the induced section $\widehat{\sigma}^{^{\prime}%
}:=\widehat{\mu}\circ\sigma\ $\ with $\widehat{C}^{^{\prime}}:=$ $\widehat
{\mu}[C]$, $\widehat{C}^{\ast}:=$ the image of the corresponding section
$\widehat{\varphi}^{-1}\circ\sigma$\ on $\widehat{G}$, and furthermore we
start with\ Euler vector fields $\chi^{^{\prime}}=\chi^{^{\prime}}%
(\sigma^{^{\prime}})$ on $X^{^{\prime}}=Y\times\mathbf{P}^{n}$ which on all
$X_{t}^{^{\prime}}$ have center in $C_{t}^{^{\prime}}$, such that the
following is true:\ Lifted to $\widehat{X}^{^{\prime}}:=Y\times F$ we get a
holomorphic vector field $\widehat{\chi}^{^{\prime}}=\widehat{\chi}^{^{\prime
}}(\widehat{\sigma}^{^{\prime}})$,\ lifted to $X$ we get a meromorphic vector
field $\widetilde{\chi}=\widetilde{\chi}(\sigma)$, which has a\emph{\ proper
}pole, of positive order $k>0,$ along $X_{0}$, as will be shown;
$\widetilde{\chi}$ is the unique meromorphic extension of $(\widehat{\mu}%
^{-1}|(\widehat{X}^{^{\prime}}\backslash\widehat{X}_{0}^{^{\prime}}))_{\ast
}\widehat{\chi}^{^{\prime}}$ to $X$; the existence of this meromorphic
extension can be proved as on p. 259 in \cite{Siu2};$\ $in subsection 5.2 and
7.b of the present paper\ another computational\ approach is chosen, which
gives some more information; $\chi=$ $t^{k}\cdot\widetilde{\chi}$ (we write
instead of $\chi$ also $\chi(\sigma)$), with $k>0$, restricted to $X_{0},$ is
not identically zero along $X_{0}$ and is holomorphic on $X$; we call $\chi$
an \emph{Euler vector field with center} $C$, the corresponding holomorphic
vector field on $\widehat{G}$ to be defined next an \emph{Euler vector
field\ with center} $\widehat{C}^{\ast}$, etc. In the same fashion as
above\ we can lift $\widehat{\chi}^{^{\prime}}$ to a meromorphic vector field
on $\widehat{G}$, which after multiplication with $t^{k}$ becomes a
holomorphic vector field $\widehat{\chi}^{\ast}$ (also named $\widehat{\chi
}_{\widehat{G}}$)\ on $\widehat{G}$,\ and is $\neq0$ in general points of
$\widehat{X}_{0}^{\ast}$; it will do no harm assuming $\widehat{G}$ to be
normal. -Similarly we can lift $\chi$ to a meromorphic vector field $\chi_{G}$
on $G$, which is holomorphic and $\neq0$ in general points of $X_{0}^{\ast}$.
It is clear that the choice of $\widehat{\chi}^{^{\prime}}$ resp.
$\chi^{^{\prime}}$ determines the above mentioned holomorphic and meromorphic
liftings uniquely. Euler vector fields depend (up to a multiplication with a
factor without zeros) on the choice of the centers and the suitable\ choice of
the so "hyperplanes" $H_{F}$ in $F$ with $F\backslash H_{F}=\mathbf{C}^{n}$.
\newline\newline If nothing else is agreed upon, we may suppose (as mentioned
above) that the\textbf{ }section $\sigma:Y\rightarrow X$ is general;
especially we may suppose that $\sigma(0)\notin S$ (= set of indeterminancy of
$\mu)$\ is a general point of $X_{0}$ and that $\mu\circ\sigma(0)$ is a
general point of $X_{0}^{^{^{\prime\prime}}}$. For continuity reasons
($\sigma(0)$ is the limit of centers $\sigma(t)$, $t\neq0$) the vector field
$\chi=\chi(\sigma)$ has a zero in $\sigma(0)$.\newline\newline If we can
assume this generality assumption, we can guarantee, in the hypersurface case,
that the holomorphic liftings $\chi_{0}$ on $X_{0}$, which we construct,
\emph{have divisorial zeroes}. \newline\newline\emph{We need that general
orbital curves of }$\chi_{0}=\chi|X_{0}$\emph{\ lie in }$\mu_{0}%
-$\emph{fibers}. It is for this purpose that \ we also study the
meromorphic\ map $\mu:X\rightarrow X^{^{\prime}}=Y\times\mathbf{P}^{n}$ and
the Euler vector fields $\chi^{^{\prime}}$ on $X^{^{\prime}}$ in order to be
sure that we construct $\chi$'s with the wanted properties. Here we use in an
essential way that $X_{0}^{^{\prime\prime}}=\mu\lbrack X_{0}]$, which is a
proper analytic subset of $(0)\times\mathbf{P}^{m}$,\ does not lie in a proper
linear subspace of $(0)\times\mathbf{P}^{m}$, this means especially $d=\deg
X_{0}^{^{\prime\prime}}>1$.\newline\newline One should have in mind, that
$X_{0}\neq F$, i.e. $X_{0}^{^{\prime\prime}}\neq X_{0}^{^{\prime}}$, is
equivalent to the assertion, that the $\widetilde{\chi}$ are not holomorphic
along $X_{0}$. The statement, that the $\widetilde{\chi}$ are holomorphic, is
therefore equivalent to the statement that $X_{0}=F.$ This equivalence was
proved by Mabuchi in case $F$ is a\emph{ compact} $i$%
\emph{rreducible,hermitian symmetric} \emph{space }(see \cite{Hwang1}, p.
327); for $F=$ a smooth quadric $\mathbf{Q}_{n}$, $n\geq3$ a simpler proof is
given in \cite{Hwang1}. The assertion, that the meromorphic vector
fields$\ \widetilde{\chi}$ have proper poles along $X_{0}$ is equivalent to
the assertion that general orbital curves of the $\chi_{0}$ lie in $\mu_{0}%
-$fibers. \newline\newline\textbf{b. Lifting Euler vector fields, a first
approach in the hypersurface case. }First\textbf{\ }we study special liftings
of\emph{\ Euler vector fields, }living on $n-$dimensional flag manifolds $F$
with $b_{2}=1$. \newline\newline\emph{In this subsection 5.1 the proofs are
carried out only in the hypersurface case} - In this case we can employ that
$X_{0}^{^{\prime\prime}}$ $\neq X_{0}^{^{\prime}}=\mathbf{P}^{n}$ is not a
linear subspace of $X_{0}^{^{\prime}}$, in the more general case we can only
apply, that $X_{0}^{^{\prime\prime}}$ resp. $\widehat{X}_{0}^{^{\prime\prime}%
}$ is not contained in a proper linear subspace of $\mathbf{P}^{n}$ resp.
$\mathbf{P}^{N}$, i.e. is not contained in a hyperplane. . \newline Remember
the notion of an Euler vector field on the projective space $\mathbf{P}^{n}$
with \textquotedblright center\textquotedblright\ $a=(a_{1},..,a_{n})$\ resp.
on $Y\times\mathbf{P}^{n}$ with \textquotedblright center\textquotedblright%
$\ (t,a(t)=(t,(a_{1}(t),...,a_{n}(t)))$. - Let $H^{\prime}=(s_{o}^{\prime}=0)$
be suitably chosen$,$ $z_{i}=s_{i}^{\prime}/s_{0}^{\prime}$ for
$i=1,...,n,Y\times\mathbf{C}^{n}=Y\times\mathbf{P}^{n}\backslash H^{\prime}.$
$t,\,z_{1},...,z_{n}$ is a holomorphic coordinate system on $Y\times
\mathbf{C}^{n}$. We denote the Euler vector field
\[
\sum_{i=1}^{n}(z_{i}-a_{i})\cdot\frac{\partial}{\partial(z_{i}-a_{i})}%
\]
and the extension of this vector field to a holomorphic vector field on
$Y\times\mathbf{P}^{n}$ by $\chi^{\prime}$. The orbital curves of
$\chi^{^{\prime}}$ through $a$ are exactly the $\mathbf{P}^{n}-$lines through
$a$. Let $\sigma_{1}:Y\rightarrow X$ \ be a section, denote $\sigma
_{1}^{^{\prime}}:=\mu\circ\sigma_{1}$, $\widehat{\sigma}_{1}^{^{\prime}%
}:=\widehat{\mu}\circ\sigma_{1}$. We can introduce a coordinate system, such
that $\sigma_{1}^{^{\prime}}(Y)=Y\times a$. \ Let $\chi^{^{\prime}}%
=\chi^{^{\prime}}(\sigma_{1})$ be the Euler\ vector field on $X^{^{\prime}}%
$\ with zeroes (centers) along $\sigma_{1}^{^{\prime}}(Y)$, where naturally
$\chi_{t}^{^{\prime}}$ shall have an isolated zero in $\sigma_{1}^{^{\prime}%
}(Y)_{t}$, let $\widehat{\chi}^{^{\prime}}=\widehat{\chi}^{^{\prime}}%
(\sigma_{1})$ be the Euler\ vector field on $\widehat{X}^{^{\prime}}$\ with
zeroes (center) along $\widehat{\sigma}_{1}^{^{\prime}}(Y)$, corresponding to
$\chi^{^{\prime}}$ and which\ is constructed by extending $\chi^{^{\prime}%
}|Y\times\mathbf{C}^{n}$ to a holomorphic vector field on $Y\times F$.
\newline\newline\emph{On }$\widehat{G}$\emph{\ \ there is a meromorphic vector
field }$\widetilde{\chi}_{\widehat{G}}$\emph{\ with }$\widehat{\psi}_{\ast
}(\widetilde{\chi}_{\widehat{G}})=\widehat{\chi}^{^{\prime}}$\emph{. On }%
$G$\emph{\ there is a meromorphic vector field }$\widetilde{\chi}_{G}%
$\emph{\ with }$\psi_{\ast}\widetilde{\chi}_{G}=\chi^{^{\prime}}$\emph{.}
\emph{\newline\newline}Proof:\textbf{\ }\ Let $a=0$. See \cite{Siu2}, p. 259.
We only prove the existence of $\widetilde{\chi}_{\widehat{G}}$. \ For our
purposes we may suppose $\widehat{G}$ to be normal, consequently general
points of irreducible components of $\widehat{G}_{0}$ are manifold points of
$\widehat{G}.$ Becau$%
%TCIMACRO{\unit{s}}%
%BeginExpansion
\operatorname{s}%
%EndExpansion
$e of Riemann$^{\prime}$s removing singularity theorem it is enough to show
the existence of $\widetilde{\chi}_{\widehat{G}}$ in the manifold points of
$\widehat{G}$ along $\widehat{G}_{0}$, the parameter $t$ is one of the
manifold parameters. Let $w_{1},...,w_{n},t$ be local holomorphic parameters
in a such a manifold point $\in\widehat{G}_{0},$ let $z_{v}=f_{\nu}\left(
w_{1},...,w_{n},t\right)  $ for $v=1,...,n.$ For $n\geq v,\mu\geq1$ we have
$\delta_{\nu,\mu}=\frac{\partial z_{v}}{\partial z_{\mu}}=\sum\frac{\partial
f_{v}}{\partial w_{\lambda}}\cdot\frac{\partial w_{\lambda}}{\partial z_{\mu}%
}.$ We may interpret this as a system of equations for the unknown functions
$\frac{\partial w_{\lambda}}{\partial z_{\mu}}$ and solving this system of
equations we get $\widetilde{\chi}_{\widehat{G}}$ by identifying $\sum z_{\mu
}\cdot\frac{\partial}{\partial z_{\mu}}=\sum f_{\mu}\cdot\frac{\partial
w_{\lambda}}{\partial z_{\mu}}\cdot\frac{\partial}{\partial w_{\lambda}}.$
This finishes the proof. - This result and proof holds whether we aare in the
hypertsurface case or not.\newline\newline By $X_{0}^{\ast}$ we denoted the
component of $G_{0}$ which is mapped by $\varphi$ bimeromorphically onto
$X_{0}.$ Recall that we assume $\mu_{0}=\mu|X_{0}:X_{0}\rightarrow
X_{0}^{^{\prime\prime}}$ not to be bimeromorphic, i. e. (in the present
context) that $\psi$ maps $X_{0}^{\ast}$ onto the proper subspace
$X_{0}^{\prime\prime}$ of $X_{0}^{^{\prime}}$; compare 2.3. By $\widehat
{X}_{0}^{\ast}$ we denoted the component of $\widehat{G}_{0}$ which is mapped
by $\widehat{\varphi}$ bimeromorphically onto $X_{0}$; $\widehat{\psi}$ maps
$\widehat{X}_{0}^{\ast}$ onto the proper subspace $\widehat{X}_{0}%
^{\prime\prime}$ of $\widehat{X}_{0}^{^{\prime}}$; again compare 2.3. The
first\ proof of the next lemma only works, if $\ X_{0}^{^{\prime\prime}}$is
a\ proper linear subspace of $X_{0}^{^{\prime}}=(0)\times\mathbf{P}^{n}$,
therefore certainly in the hypersurface case and in the case $n=3\ $;
naturally the lemma itself is true for arbitrary flag manifolds $F$ with
Picard number $1$ and del Pezzo manifolds of degree 5 and dimension $\geq3$.
\newline\newline\emph{Lemma. The vector field }$\widetilde{\chi}_{G}$ \emph{is
nowhere holomorphic on }$X_{0}^{\ast};$ $X_{0}^{\ast}$ \emph{belongs to the
pole set of} $\widetilde{\chi}_{G}.$\emph{The vector field }$\widetilde{\chi
}_{\widehat{G}}$ \emph{is not holomorphic in any point of} $\widehat{X}%
_{0}^{\ast};$ $\widehat{X}_{0}^{\ast}$ \emph{belongs to the pole set of}
$\widetilde{\chi}_{\widehat{G}}$. \emph{Here we suppose, that the center of
}$\chi_{0}^{^{\prime}}$\emph{ is a general point of }$X_{0}^{^{\prime\prime}}%
$\emph{. }\newline\newline(First) proof:\textbf{ }It suffices to discuss
$\widetilde{\chi}_{\widehat{G}}$.\textbf{\ }In the following subsections we
compute $\widetilde{\chi}_{\widehat{G}}$ explicitely in general points of
$\widehat{X}_{0}^{\ast}$ resp. $\widetilde{\chi}_{G}$ in general points of
\ $X_{0}^{\ast}$. These calculations deliver (among other facts) our
assertion. But first let us consider here\ another proof, which is more
conceptual. \newline\newline We use: $X_{0}^{^{\prime\prime}}$ is a\ linear
subspace of $X_{0}^{^{\prime}}=(0)\times\mathbf{P}^{n}$, iff there is a
nonempty $X_{0}^{^{\prime\prime}}$-\ open subset $W$\ of $X_{0}^{^{\prime
\prime}}$, such that each point of this subset $W$ can serve as the vertex of
a cone which equals $X_{0}^{^{\prime\prime}}$, i. e. any $\mathbf{P}^{n}$-line
connecting 2 points of this subset of $X_{0}^{^{\prime\prime\prime}}$ lies on
$X_{0}^{^{\prime\prime\prime}}$; for a proof show, that, given such a $W$,
$W\backslash($\emph{a} \emph{suitable} $\mathbf{P}^{n}-$\emph{hyperplane}$)$
is a non empty subset of a linear $\mathbf{P}^{n}$-subspace, i.e. show that
$X_{0}^{^{\prime\prime}}$ is a proper linear subspace of $\mathbf{P}^{n}$,
which cannot be. This means:\ As $X_{0}^{^{\prime\prime}}$ is not a
proper\ linear subspace of $X_{0}^{^{\prime}}=(0)\times\mathbf{P}^{n}$, there
is an open, dense set of points $(P^{^{^{\prime\prime}}},...)$ on
$X_{0}^{^{\prime\prime}}$ such that $X_{0}^{^{\prime\prime}}$ is not a cone
with vertex in one of these points $P^{^{^{\prime\prime}}}\in$%
\ $(P^{^{^{\prime\prime}}},...)$, i. e. a line through $P^{^{^{\prime\prime}}%
}$ and a general point of $X_{0}^{^{\prime\prime}}$ meets $X_{0}%
^{^{\prime\prime}}$ pointwise.\newline\newline It suffices to discuss
meromorphic vector fields $\widetilde{\chi}_{G}$ (coming from an Euler vector
field $\chi^{^{\prime}}$ on $X^{^{\prime}}$with center $C^{^{\prime}}$ as
described above) in general points $P^{\ast}$ of $X_{0}^{\ast}$, lying over
points$\ P^{^{\prime\prime}}=\psi\left(  P^{\ast}\right)  $, such that
$X_{0}^{^{\prime\prime}}$ is not a cone with vertex $P^{^{\prime\prime}}$. We
may suppose $\chi^{\prime}\left(  P^{^{\prime\prime}}\right)  \neq0$ as the
set of suitable $P^{^{\prime\prime}}$ lies in a dense open subset of
$X_{0}^{^{\prime\prime}}$. Assume that $\widetilde{\chi}_{G}$ is holomorphic
in these points $P^{\ast}.$ Because of $\psi_{\ast}\left(  \widetilde{\chi
}_{G}\right)  =\chi^{\prime}$ and $\chi^{\prime}\left(  P^{^{\prime\prime}%
}\right)  \neq0$ we have $\widetilde{\chi}_{G}\left(  P^{\ast}\right)  \neq0$.
$\widetilde{\chi}:=\varphi_{\ast}\left(  \widetilde{\chi}_{G}\right)  $ is
holomorphic and $\neq0$ in general points of $X_{0}$. We can arrange matters
that the $\mathbf{P}^{n}$-line $l^{\prime}\subset X_{0}^{\prime}$ through
$P^{\prime}=C^{\prime}\cap X_{0}^{\prime\prime}$ and $P^{\prime\prime}$ meets
$X_{0}^{\prime\prime}$ only pointwise. $l^{\prime}$ is the closure of an
integral curve of $\chi^{\prime}$. The orbital curve $l^{\ast}$ of
$\widetilde{\chi}_{G}$ through $P^{\ast}$ must necessarily lie on a fiber of
$\psi,$ because otherwise $\psi(l^{\ast})$ has to be an orbital curve $\neq
l^{^{\prime}}$ through $P^{^{\prime\prime}},$ a $\mathbf{P}^{n}$-line lying in
$X_{0}^{^{\prime\prime}}$. But this implies that $\chi^{\prime}=\psi_{\ast
}\left(  \widetilde{\chi}_{G}\right)  $ has a zero in $P^{^{\prime\prime}},$
which should not be.\newline\newline With other words: Assume $X_{0}$ $\neq
F$. Suppose that for general\ holomorphic Euler\ vectorfields $\widehat{\chi
}^{\prime}$ on $Y\times F$ the (meromorphic) liftings $\widetilde{\chi}$,
$\widetilde{\chi}_{\widehat{G}}$ to $X$ resp. $\widehat{G}$\ are holomorphic;
here we suppose that the center $P^{^{\prime}}$of $\widehat{\chi}_{0}^{\prime
}$ is a general point on $\widehat{X}_{0}^{^{\prime\prime}}$; identify this
point with the corresponding point on $X_{0}^{^{\prime\prime}}\subset
(0)\times\mathbf{P}^{n}$. Let $\chi^{^{\prime}}$ be the corresponding Euler
vector field on $X^{^{\prime}}=Y\times\mathbf{P}^{n}$. The holomorphy of
$\widetilde{\chi}_{\widehat{G}}$\ implies $\widehat{\psi}_{\ast}%
\widetilde{\chi}_{\widehat{G}}=\widehat{\chi}^{\prime}$ and that general
orbital curves of $\widetilde{\chi}_{\widehat{G}}$ on $\widehat{X}_{0}^{\ast}$
are mapped by $\widehat{\psi}$ onto orbital curves in $\widehat{X}%
_{0}^{^{\prime\prime}}$, that general orbital curves of $\widetilde{\chi}_{G}$
on $X_{0}^{\ast}$ are mapped by $\psi$ onto orbital curves of$\ \chi
^{^{\prime}}$ in $X_{0}^{^{\prime\prime}}$. As general orbital curves of
$\chi^{^{\prime}}$ are $\mathbf{P}^{n}$-lines through $P^{^{\prime}}$,
$X_{0}^{^{\prime\prime}}$ is a cone with vertex $P^{^{\prime}}$.
\newline\newline The set of these $P^{^{\prime}}$contains a nonempty,
$X_{0}^{^{\prime\prime}}$-\ open subset, consequently $X_{0}^{^{\prime\prime}%
}$\ is a proper linear subspace of $X_{0}^{^{\prime}}$, which cannot be by
construction of $\mu$. - This finishes the proof of the lemma. q.e.d.\newline%
\newline In 1.2.c and 5.1.a we denoted by $k$ the pole order of $\varphi
_{\ast}\left(  \widetilde{\chi}_{G}\right)  $ along $X_{0}.$ $\chi:=$
$t^{k}\cdot\varphi_{\ast}\left(  \widetilde{\chi}_{G}\right)  $ is holomorphic
on $X_{0}$ and $\neq0$ in a general point of $X_{0}.$ Let us call $\chi$ the
\emph{holomorphic lifting }of $\chi^{^{\prime}}$ resp $\widehat{\chi
}^{^{\prime}}$\ to $X.$ Below we will see that $t^{k}\cdot\widetilde{\chi
}_{\widehat{G}}$ is holomorphic on $\widehat{G}$. \ - We have to point out
that the proof above only gives a pole order $k$, which possibly depends on
the chosen $\chi^{^{\prime}}$ \newline\newline A better description of the
pole order $k$ will be given in the following. It will even turn out later
that $k=1$; this equality (as will be seen) is closely connected with the
realization that $\widehat{\varphi}^{\ast}s$ vanishes along $\widehat{G}%
_{0}^{\ast}$ with order one for general $s\in H^{0}(X,\mathcal{D\dot{)}}$.
Compare 5.3.\ - The same arguments as used in the proof above give$:$%
\newline\newline\emph{The orbital curves (closures of integral curves) of
}$\chi$\emph{\ through general points of }$X_{0}$\emph{\ lie in fibers of
}$\mu_{0}.$\emph{\ \newline}\newline This statement about orbital curves is
most important. Here we exploited the nonlinearity of $X_{0}^{^{\prime\prime}%
},$ implying that there does not exist a nonempty open set in $X_{0}%
^{^{\prime\prime}}$ such that any point of this set can serve as a vertex of a
cone which equals $X_{0}^{^{\prime\prime}}$, we used further that the orbital
curves\ $l^{^{\prime}}$of an Euler vector field on $\mathbf{P}^{n}$ through
the center $P^{^{\prime}}$ are all the $\mathbf{P}^{n}-$lines through
$P^{^{\prime}}$. \newline\newline\emph{Remark} \ The proof above does not
necessarily\ work, if only $X_{0}^{^{\prime\prime}}$ lies in a proper\emph{
linear} subspace of $(0)\times\mathbf{P}^{n}$. -\ For instance one can produce
a bimeromorphic map $\xi:$ $Y\times\mathbf{P}^{n}\rightarrow Y\times
\mathbf{P}^{n},$ biholomorphic outside $(0)\times\mathbf{P}^{n},$ where
$\xi\left[  (0)\times\mathbf{P}^{n}\right]  $ is a proper linear subspace
$(0)\times\mathbf{P}^{n}$ and where the assertion above on integral curves of
certain Euler vector fields being contained in fibers is wrong, i.e. where the
extension $\widetilde{\chi}$ is holomorphic, where no multiplication with
$t^{k}$ is needed and where consequently correponding orbital curves $\subset
X_{0}$ are not all contained in fibers of $\mu_{0}$; these Euler vector fields
have centers lying in $\xi\lbrack(0)\times\mathbf{P}^{n}].$ - As an example of
this sort consider the bimeromorphic map $\xi:Y\times\mathbf{P}^{2}%
\rightarrow$ $Y\times\mathbf{P}^{2}$, given by $(t;p_{0}:p_{1}:p_{2}%
)\rightarrow(t;t\cdot p_{0}:t\cdot p_{1}:p_{2})$ in general points, where
$(t;p_{0}:p_{1}:p_{2})\in(0)\times\mathbf{P}^{2}$, with $\xi\lbrack
(0)\times\mathbf{P}^{2}]=(0;0:0:1)$ and $(0)\times(p_{2}=0)$ as set of
indeterminancy. Let the image space $Y\times\mathbf{P}^{2}$ carry the
homogenous coordinates $q_{0}:q_{1}:q_{2}$; let $\chi_{1}^{^{\prime}}$ be the
Euler vector field given on the subspace $q_{2}\neq0$ (put $q_{2}=1$ and work
with $t,q_{0},q_{1}$ as parameters on this subspace) by $q_{0}\frac{\partial
}{\partial q_{0}}+q_{1}\frac{\partial}{\partial q_{1}}$; as described above we
can lift$\ \chi_{1}^{^{\prime}}$ to a meromorphic vector field $\widetilde
{\chi}_{1}$ on the preimage space$\ Y\times\mathbf{P}^{2}=Y\times
\mathbf{P}^{2}(p_{0},p_{1},p_{2})$; elementary computation shows that
$\widetilde{\chi}_{1}$ is holomorphic on $Y\times\mathbf{P}^{2}(p_{0}%
,p_{1},p_{2})$ and is given on the subspace $p_{2}\neq0$ (put $p_{2}=1$) by
$p_{0}\frac{\partial}{\partial p_{0}}+p_{1}\frac{\partial}{\partial p_{1}}$.-
If we replace $q_{0}\frac{\partial}{\partial q_{0}}+q_{1}\frac{\partial
}{\partial q_{1}}$ by $(q_{0}-a)\frac{\partial}{\partial q_{0}}+(q_{1}%
-b)\frac{\partial}{\partial q_{1}}$ with $(a,b)\neq(0,0)$ and lift the new
Euler vector field to $Y\times\mathbf{P}^{2}(p_{0},p_{1},p_{2})$, we get a
meromorphic vector field with a proper pole along $(0)\times\mathbf{P}%
^{2}(p_{0},p_{1},p_{2})$\newline\newline\textbf{c. Generalities on holomorphic
liftings}. For our purposes we may suppose (in most cases) $\widehat{G}$ \ to
be normal (thus we can apply Riemann's extension theorem)\ and that $t$
vanishes along all components of $\widehat{G}_{0}$ with order one, which can
be arranged by a base change. We will show:\newline\newline$t^{k}%
\cdot\widetilde{\chi}_{\widehat{G}}$ is holomorphic on $\widehat{G}$.
\newline\newline Suppose that\ there is a component $\widehat{K}$ of
$\widehat{G}_{0}$ which is a pole set of order $k^{\ast}>0$ of $t^{k}%
\cdot\widetilde{\chi}_{\widehat{G}};$ take a $\widehat{K}$ with a maximal
$k^{\ast}>0;$ naturally $\widehat{K}\neq X_{0}^{\ast}.$ Then $t^{k+k^{\ast}%
}\cdot\widetilde{\chi}_{\widehat{G}}$ is holomorphic on $\widehat{G}$ and not
zero in any general point of $\widehat{K}.$ There are orbits of dimension $>0$
of $t^{k+k^{\ast}}\cdot\widetilde{\chi}_{\widehat{G}}$ on $\widehat{K},$ which
are mapped onto orbits of dimension $>0$ of $t^{k+k^{\ast}}\cdot
\widetilde{\chi}$ on $X_{0}.$ But $t^{k+k^{\ast}}\cdot\widetilde{\chi}$ has
zeroes in all points of $X_{0}.$The only orbits on $X_{0}$ of $t^{k+k^{\ast}%
}\cdot\widetilde{\chi}$ are points. - Thus $t^{k}\cdot\widetilde{\chi
}_{\widehat{G}}$ is holomorphic on $\widehat{G}$ and $\widehat{G}_{0}^{\ast}$
belongs to the zero set of this vector field.\newline\newline Let
$\widehat{\chi}^{^{\prime}}$ be the holomorphic vector field on $\widehat
{X}^{^{\prime}}=Y\times F$ corresponding to $\chi^{^{\prime}}$.We call
$\chi=t^{k}\cdot\mu^{\ast}\chi^{^{\prime}}$ the \emph{holomorphic lifting} of
$\chi^{^{\prime}}$ resp. $\widehat{\chi}^{^{\prime}}$ to $X$ and call
$\chi_{\widehat{G}}$\emph{\ :=} $t^{k}\cdot\widetilde{\chi}_{\widehat{G}}$ the
\emph{holomorphic lifting }of $\chi^{^{\prime}}$ resp. $\widehat{\chi
}^{^{\prime}}$\ to $\widehat{G}.$ We have $\widehat{\varphi}_{\ast}%
\chi_{\widehat{G}}=\chi.$ Instead of $\chi_{\widehat{G}}$ , $\chi_{\widehat
{G}}|\widehat{X}_{0}^{\ast}$we will also write $\chi^{\ast}$, $\chi_{0}^{\ast
}$. \newline\newline If $\widehat{G}$ is not normal, let $\nu:$ $_{n}%
\widehat{G}\rightarrow\widehat{G}$ be the normalization map, let $\chi
_{_{n}\widehat{G}}$ be the holomorphic lifting of $\chi^{^{\prime}}$ to
$_{\nu}\widehat{G}$. Then $\chi_{\widehat{G}}:=\nu_{\ast}\chi_{_{n}\widehat
{G}}$ is a holomorphic vector field on $\widehat{G}$, the holomorphic lifting
of $\chi^{^{\prime}}$ resp. $\widehat{\chi}^{^{\prime}}$\ to $\widehat{G}%
$.\newline\newline\textbf{d. }\emph{Remark}. We already have \ at disposal
nearly all of the main means to show theorem 1 (where$X_{0}$ is supposed to be
k\"{a}hler). Let us give a (slightly oversimplified) sketch, for more details
see chap. 6. These general\ meromorphic vector fields $\widetilde{\chi}$
(living on $X$) and $\widetilde{\chi}_{\widehat{G}}$ (living on $\widehat{G}%
$)\ are not holomorphic, let $\chi_{\widehat{G}}$on $\widehat{G}$\ be the
holomorphic lifting of $\widehat{\chi}^{\prime}$, $\chi_{\widehat{G}}$ is zero
on $\widehat{G}_{0}^{\ast}$. - Let $\chi=\widehat{\varphi}_{\ast}%
\chi_{\widehat{G}}$ be the holomorphic lifting of such a vector field
$\widehat{\chi}^{\prime}$ to $X$, $\chi$ is $0$ in all points of $S^{\ast
}=\widehat{\varphi}(\widehat{G}_{0}^{\ast})$, because $\chi_{\widehat{G}}$ is
zero on $\widehat{G}_{0}^{\ast}$, each point of $S^{\ast}$ belongs to at least
2 different orbital curves of $\chi_{0}$, general orbital curves of $\chi_{0}%
$meet $S^{\ast}$. They lie in $\mu_{0}$-fibers. Suppose $\dim S^{\ast}>0$. Let
the line $l^{^{\prime}}\subset F$ be suitably chosen. For $t\neq0$ in $Y$\ the
curve $\widehat{\varphi}(\widehat{\psi}^{-1}[Y\times l^{\prime}])_{t}$
generates $H_{2}(X,\mathbf{Z)=Z}$, for reasons of continuity $\widehat
{\varphi}(\widehat{\psi}^{-1}[Y\times l^{\prime}])_{0}$ should generate
$H_{2}(X,\mathbf{Z)=}H\mathbf{_{2}(}X_{0}\mathbf{,\mathbf{Z)=}Z}$ too.
Therefore on the one hand $\widehat{\varphi}(\widehat{\psi}^{-1}[Y\times
l^{\prime}])_{0}$ has to be irreducible, as $X_{0}$ is k\"{a}hler. But on the
other hand we can arrange because of $\dim S^{\ast}>0$ and because of
subsection 3.5, that $\widehat{\varphi}(\widehat{\psi}^{-1}[Y\times l^{\prime
}])_{0}$ has at least 2 components, compare the reasoning in the following
subsection 6.2.e. Consequently $S^{\ast}$ must be a point. Thus we are in the
hypersurface case, which can be handled as described at the end of this
chapter 5.

\subsection{Lifting Euler vector fields in the case $\dim X_{0}^{^{\prime
\prime}}=n-1$, a calculatory approach.}

\textbf{a. Computing lifted Euler vector fields for }$F$\textbf{ = a flag
manifold with }$b_{2}=1$. $\alpha.$\textbf{\ }Now to a different approach of
the above, which is of a calculatory nature. In this chapter 5 we discuss
mainly\ the "hypersurface case" $\dim X_{0}^{^{\prime\prime}}=n-1,$ the
general case $2\leq\dim X_{0}^{^{\prime\prime}}\leq n-1$ is handled in the
next chapter 6. - Look up the notations and the simplifications in 4.2, made
for notational reasons. General fibers of $\widehat{\psi}|\widehat{X}%
_{0}^{\ast}$, $\psi|X_{0}^{\ast}$ (i.e. general $\mu_{0}$-fibers) are supposed
to be irreducible, so we talk of fibers where we should talk of their
components. We point out that sometimes we will identify homogeneous
polynomials locally with inhomogeneous polynomials. We use the notations and
conventions\ of 4.2, and also the simplifications made in 4.2, where the
$f_{\delta}^{^{\prime}}$ are sometimes mixed up with the $f_{\widetilde{d}%
}^{^{\prime}}$ \ etc.\newline\newline Let $P^{^{\prime}}\in X_{0}%
^{^{\prime\prime}},$ $P^{\ast}\in X_{0}^{\ast}$ with $\psi(P^{\ast
})=P^{^{\prime}}$ be general\textbf{ }(especially these points are manifold
points), let $z_{2},...,z_{n}$ be regular parameters in an $X_{0}%
^{^{\prime\prime}}-$neighborhood of $P^{^{\prime}},$ which are centered in
$P^{^{\prime}}$. Have in mind the simplifications we are making in the
following, for reasons of simple notations; according to 4.2 there is a
general homogeneous polynomial $f_{\widetilde{d}}^{^{\prime}}$ of possibly
high degree $\widetilde{d}$\ depending only on $s_{0}^{^{\prime}}%
,...,s_{n}^{^{\prime}}$\ on $X_{t}$ for $t\neq0$ with the following
properties: $\psi^{-1}[(f_{\widetilde{d}}^{^{\prime}}=0)]$ cuts the fiber of
$X_{0}^{\ast}\rightarrow X_{0}^{^{\prime\prime}}$ through $P^{\ast}$ in a
neighborhood of $P^{\ast}$\ (analytically) just\ once; $\frac{\psi^{\ast
}f_{\widetilde{d}}^{^{\prime}}}{t^{\sigma^{^{\prime\prime}}}},z_{2},...,z_{n}$
are on $X_{0}^{\ast}$ regular manifold parameters in $P^{\ast},$ centered in
$P^{\ast};$ here we denote by $\sigma^{^{\prime\prime}}$ the vanishing order
of $\psi^{\ast}f_{\widetilde{d}}^{^{\prime}}$ along $X_{0}^{\ast}$ and think
of the $z_{i}$ as $\frac{s_{i}^{^{\prime}}}{s_{0}^{^{\prime}}}$.
$f_{\widetilde{d}}^{^{\prime}}$ possesses in $P^{^{\prime}}$ a local
irreducibel holomorphic factor $f_{\delta}^{^{\prime}}$ with vanishing order
$\sigma^{^{\prime\prime\prime}}$ of $\psi^{\ast}f_{\delta}^{^{\prime}}$ along
$X_{0}^{\ast}$, such that $\frac{\psi^{\ast}f_{\delta}^{^{\prime}}}%
{t^{\sigma^{^{\prime\prime\prime}}}}$ is in $P^{\ast}$ a local parameter and
that $\frac{\psi^{\ast}f_{\delta}^{^{\prime}}}{t^{\sigma^{^{\prime\prime
\prime}}}}$\ and $\frac{\psi^{\ast}f_{\widetilde{d}}^{^{\prime}}}%
{t^{\sigma^{^{\prime\prime}}}}$ differ in a $G$-neighborhood of $P^{\ast}$ by
a factor which is $\neq0$ in $P^{\ast}$. \newline\newline Let $f^{^{\prime}%
}:=f_{\delta}^{^{\prime}}$. For computing purposes suppose that $(s_{1}%
^{^{\prime}}=0)$ is tangent to $X_{0}^{^{\prime\prime}}$ in the general point
$P^{^{\prime}}=\psi(P^{\ast})$ of $X_{0}^{^{\prime\prime}}$, and that
$s_{0}^{^{\prime}}(P^{\prime})\neq0,$ but $s_{i}^{^{\prime}}(P^{\prime})=0$
for $i\geq1.$ We denoted $z_{1}=\frac{s_{1}^{^{\prime}}}{s_{0}^{^{\prime}}%
},z_{2}=\frac{s_{2}^{^{\prime}}}{s_{0}^{^{\prime}}},...$.. Consider
$\psi^{\ast}$ $(z_{1}\cdot$ $\frac{\partial}{\partial z_{1}}+z_{2}\cdot
\frac{\partial}{\partial z_{2}}+...).$ In the general point $P^{\ast}$ over
$P^{^{\prime}}$ we have as parameters $t,$ $\widetilde{f}=$ $\frac
{f^{^{\prime}}}{t^{\sigma^{^{\prime\prime\prime}}}},z_{2},...$with
$\sigma^{^{\prime\prime\prime}}=1$. The lifted vector field has a pole of
order $k$ along $X_{0}^{\ast},$ where $k$ equals $\sigma^{^{\prime\prime
\prime}}$(for notational reasons we suppose $\sigma^{^{\prime\prime}}=1$ and
$f^{^{\prime}}=f_{d}^{^{\prime}}$)$,$ as the computations give, in the
following we may assume because of 4.2 that $k$ does not depend on the special
choice of $\chi^{^{\prime}}$ as $k=1$, compare 5.3. We \ can solve $f^{\prime
}=0$ for $z_{1},$ $z_{1}=f^{\prime}+h$ with $h$ not identically zero, as
$f^{^{\prime}}$ cannot be linear, because otherwise $X_{0}^{^{\prime\prime}}$
would lie in a proper linear subspace of $(0)\times\mathbf{P}^{n}\ $The
holomorphic lifting $\chi$ is in general points over $P^{^{\prime}}$ of \ the
form \thinspace$factor\cdot\frac{\partial}{\partial\widetilde{f}},$ where
$\ $this time $factor$ has a $1-$co$\dim$ensional zero set on $X_{0}$; this
function $factor$\ should not be confused with the function $factor$ \ a few
lines above. For the verification of these announcements work in general
points of $X_{0}^{\ast}$ resp. $X_{0},$ where $t,\widetilde{f}=\frac
{f^{^{\prime}}}{t^{^{\sigma^{^{\prime\prime\prime}}}}\text{ }}$ are local
parameters. The holomorphic lifting is here of the form ($z_{1}\cdot f_{z_{1}%
}^{^{\prime}}+z_{2}\cdot f_{z_{2}}^{^{\prime}}+..)\frac{\partial}%
{\partial\widetilde{f}}+t\cdot\ast.$ The factor $\chi^{^{\prime}}(f^{^{\prime
}})$ of $\frac{\partial}{\partial\widetilde{f}}$, not vanishing identically
along $X_{0}^{\ast}$ resp. $X_{0}$ (this has to be justified; see below)$,$
\emph{depends only on parameters defined in an} $X^{\prime}-$%
\emph{neighborhood of} $P^{^{\prime}};$\emph{\ the coordinates }%
$z_{2},...,z_{n}$\emph{\ stay constant along an orbital curve; consequently
orbital curves have to lie in fibers. \newline}\newline\emph{Let us record
that in the hypersurface case the holomorphic liftings }$\chi_{0}%
$\emph{\ constructed above\ have\ divisorial zeros over (}$\chi^{^{\prime}%
}(f^{^{\prime}})=0$\emph{)}$\cap X_{0}^{^{\prime}}$, \emph{which in
general\ do not lie completely over }$X_{0}^{^{\prime\prime}}\cap H^{^{\prime
}},$ \emph{to be more precise, which are lifted }$X_{0}^{^{\prime}}%
-$\emph{hypersurfaces, which do not contain a component of \ }$X_{0}%
^{^{\prime\prime}}\cap H^{^{\prime}}.$\newline\newline What counts in the
preceding is: 1) $P^{^{\prime}}$ is a smooth point of $X_{0}^{^{\prime\prime}%
}$. 2) $P^{^{\prime}}$ belongs to the zero set of $\chi^{^{\prime}}(f_{\delta
}^{^{\prime}})$.\ 3) $(\chi^{^{\prime}}(f_{\delta}^{^{\prime}})=0)$ cuts
$X_{0}^{^{\prime\prime}}$ properly. - One can guarantee 2) by choosing
$P^{^{\prime}}$ as the center of $\chi^{^{\prime}}$, as was done above.
Property 3) still has to be discussed. \newline\newline$\chi_{0}^{^{\prime}%
}f^{^{\prime}}=0$ in all points of a non empty $X_{0}^{^{\prime\prime}}$-open
set of$\ X_{0}^{^{\prime\prime}}$, means the same as, this open set
$\subset\ X_{0}^{^{\prime\prime}}$ is covered by orbital curves of $\chi
_{0}^{,\prime},$ because $f^{^{\prime}}$ generates $\operatorname{mod}t$ the
functions vanishing along $X_{0}^{^{\prime\prime}}.$ Let us continue with this
proof of the statement on orbital curves lying in fibers of $\mu_{0}.$ We are
still in the hypersurface case. \newline\newline\emph{Remark. }Let us already
point out, that (here in the hypersuface case)$\ \sigma^{^{\prime\prime\prime
}}$ equals the pole order $k$ \ discussed in 5.1.c. This will follow from
subsection\ 5.2.a.$\beta$\ below. Compare 5.3 why $k=1$.\newline\newline In
situations like this very often we write $f_{\widetilde{d}}^{^{\prime}}$ ,
$f^{^{\prime\prime}},...$instead of $\psi^{\ast}f_{\widetilde{d}}^{^{\prime}}%
$, $\psi^{\ast}f^{^{\prime}}$,.... For the\ sake of \ simple notations we will
often assume $d=\widetilde{d}$, $f^{^{\prime}}=$ $f_{\widetilde{d}}^{^{\prime
}}=f_{d}^{^{\prime}}$, as already remarked. Furthermore we will often suppose
for notational simplifying reasons $\sigma^{^{\prime\prime}}=\sigma
^{^{\prime\prime\prime}}=1$. From the context it should become clear whether
$f^{^{\prime}}$ can be looked at as $f_{\widetilde{d}}^{^{\prime}}$ or as
$f_{\delta}^{^{\prime}}$.\ Naturally we will refrain from drawing unjustified
conclusions from these simplifying assumptions. \newline\newline$\beta$. We
continue to suppose $\sigma^{^{\prime\prime}}=1,$we denoted $f^{^{\prime}%
}:=f_{\delta}^{^{\prime}};$ and we continue to suppose $(z_{1}=0)$ to be
tangent to $X_{0}^{^{\prime\prime}}$ in the general point $P^{^{\prime}}$ with
$z_{1}=0,\,$ $...$, $z_{n}=0$ in $P^{^{\prime}}$, esp. not all partial
derivatives of \ $f^{^{\prime}}$ in $P^{^{\prime}}$ should vanish, let
$\chi^{^{\prime}}$ be the Euler vector field $\chi^{^{\prime}}$ = $z_{1}%
\frac{\partial}{\partial z_{1}}+...+z_{n}\frac{\partial}{\partial z_{n}}$ on
$X^{^{\prime}}.$ In a neighborhood of $P^{^{\prime}}$ we can write
$\chi^{^{\prime}}=g_{1}\frac{\partial}{\partial f^{^{\prime}}}+z_{2}%
\frac{\partial}{\partial z_{2}}+...+z_{n}\frac{\partial}{\partial z_{n}}.$
Here $g_{1}$ can be described as $z_{1}\frac{\partial f^{^{\prime}}%
(z_{1},...z_{n})}{\partial z_{1}}$ $+...+z_{n}\frac{\partial f^{^{\prime}}%
}{z_{n}}$. In a neighborhood of a general point over $P^{\ast}$ on
$X_{0}^{\ast}$ we may look at $t,\frac{1}{t}\cdot\psi^{\ast}(f^{^{\prime}%
})=\widetilde{f},$ as part of a parametersystem centered in this point. $Mod$
$t$ the lifted vectorfield, multiplied by $t$, can be written as $\psi^{\ast
}(g_{1})\cdot\frac{\partial}{\partial\widetilde{f}}$ in a neighborhood of this
point. \newline\newline Now take a second general vector field $\chi
_{2}^{^{\prime}}=(z_{1}-a_{1})\frac{\partial}{\partial z_{1}}+...+(z_{n}%
-a_{n})\frac{\partial}{\partial z_{n}}$ $=\overline{h}_{1}\frac{\partial
}{\partial f^{^{\prime}}}+$ $(z_{2}-a_{2})\frac{\partial}{\partial z_{2}%
}+...+(z_{n}-a_{n})\frac{\partial}{\partial z_{n}}$ \thinspace with
$\overline{h}_{1}=(z_{1}-a_{1)}\frac{\partial f^{^{\prime}}(z_{1},...,z_{n}%
)}{\partial z_{1}}+$ $...+(z_{n}-a_{n})\frac{\partial f^{^{\prime}}}{\partial
z_{n}},$ where $Q^{^{\prime}}=(a_{1},...,a_{n})\in X_{0}^{^{\prime\prime}}$ is
general. Suppose that $\chi^{^{\prime}}(f^{^{\prime}})=$ $\chi_{2}^{^{\prime}%
}(f^{^{\prime}})=0$ on an open nonempty set of $X_{0}^{^{\prime\prime}}.$ As
$Q^{^{\prime}}\in X_{0}^{\prime\prime}$ is general, one gathers (vary
$Q^{^{\prime}}\in X_{0}^{^{\prime\prime}},$ fix $(z_{1},...,z_{n})\in
X_{0}^{^{\prime\prime}})$ \ that $X_{0}^{^{\prime\prime}}\subset$ $(a_{1}%
\frac{\partial f}{\partial z_{1}}^{^{\prime}}(z_{1},...,z_{n})+...$
$+a_{n}\frac{\partial f^{^{\prime}}}{\partial z_{n}}^{^{\prime}}%
(z_{1},...,z_{n})=0)$, i.e. that $X_{0}^{^{\prime\prime}}$ lies in the normal
vector space of the fixed vector $(\frac{\partial f}{\partial z_{1}}%
^{^{\prime}}(z_{1},...,z_{n}),...,\frac{\partial f^{^{\prime}}}{\partial
z_{n}}^{^{\prime}}(z_{1},...,z_{n}))$, which is linear. But $X_{0}%
^{^{\prime\prime}}$ cannot lie in a proper linear subspace of $(0)\times
\mathbf{P}^{n}.$ Consequently: \newline\newline If the center $\in$
$X_{0}^{^{\prime\prime}}$ of $\chi_{0}^{^{\prime}}$ is general, $\chi
^{^{\prime}}(f^{^{\prime}})$ does not vanish in all points of\ a dense subset
of $X_{0}^{^{\prime\prime}}$, i.e. does not vanish identically along
$X_{0}^{^{\prime\prime}}.$ - As a\ small movement of the $a_{i}$ doesn't
change everywhere inequalities, we can also state: If the center $\in$
$X_{0}^{^{\prime}}$ of $\chi_{0}^{^{\prime}}$ is general, $\chi^{^{\prime}%
}(f^{^{\prime}})$ does not vanish in all points of\ an open dense subset of
$X_{0}^{^{^{\prime}}}.$ If the center of $\chi^{^{\prime}}$is a general point
of $X_{0}^{^{\prime\prime}}$, the center is a zero of $\chi^{^{\prime}%
}(f^{^{\prime}})$. With other words: Suppose that for all or at least for
somewhere on $X_{0}^{^{\prime\prime}}$ dense sets of points $(z_{1}%
,...,z_{n})\in X_{0}^{^{\prime\prime}}$, $(a_{1},...,a_{n})\in X_{0}%
^{^{\prime\prime}}$ $(a_{1}\frac{\partial f}{\partial z_{1}}^{^{\prime}}%
(z_{1},...,z_{n})+...$ $+a_{n}\frac{\partial f^{^{\prime}}}{\partial z_{n}%
}^{^{\prime}}(z_{1},...,z_{n})=0)$. Let $(a_{1},...,a_{n})\in X_{0}%
^{^{\prime\prime}}$ be general. Then for all $(z_{1},...,z_{n})\in
X_{0}^{^{\prime\prime}}$ we have $a_{1}\frac{\partial f}{\partial z_{1}%
}^{^{\prime}}(z_{1},...,z_{n})+...$ $+a_{n}\frac{\partial f^{^{\prime}}%
}{\partial z_{n}}^{^{\prime}}(z_{1},...,z_{n})=0)$, which is not
possible.\newline\newline$\gamma$. \emph{Remark}. Now let $\chi^{^{\prime}}$
be an Euler vector field on $Y\times F$, where $\chi_{0}^{^{\prime}}$ has a
(general)\ center on $(0)\times\mathbf{C}^{n}$ outside $X_{0}^{^{\prime\prime
}}$. As in $\alpha.$ above, we can produce a holomorphic lifting $\chi_{0}$ of
$\chi^{^{\prime}}$, which has on $X_{0}$\ the zero set of the lifted $\chi
_{0}^{^{\prime}}(f_{d}^{^{\prime}})$ as zero divisor. This time the orbital
curves of $\chi_{0}^{^{\prime}}$ through the center meet $X_{0}^{^{\prime
\prime}}$ pointwise, thus $X_{0}^{^{\prime\prime}}\nsubseteq(\chi
_{0}^{^{\prime}}(f_{d}^{^{\prime}})=0)$. But this will not suffice to
guarantee divisorial zeroes. Therefore we dealt in the preceding with Euler
vector fields, whose centers on $X_{0}^{^{\prime}}$\ belonged to
$X_{0}^{^{\prime\prime}}$. - This statement also holds, if $F$ is an arbitrary
flag manifold with $b_{2}=1$\ or an arbitrary at least 3-dimensional\ del
Pezzo manifold of degree 5, which are discussed below.\newline\newline%
$\delta.$\emph{Important is: }All vector fields $\chi_{0}$\ of this type on
$X_{0}$ commute with each other (\cite{Siu2}, \S 10)\emph{.} Let $\gamma
_{0}^{\ast(1)}$ be a general orbital curve of $\chi_{0}$; $\gamma_{0}%
^{\ast(1)}$ lies in a general $\mu_{0}$-fiber and meets $S^{\ast}$ set
theoretically in just one point, this point is set theoretically the only zero
of $\chi_{0}|\gamma_{0}^{\ast(1)}$, analytically this point is a double zero
of $\chi_{0}|\gamma_{0}^{\ast(1)}$; $\chi_{0}|\gamma_{0}^{\ast(1)}$ is\emph{
is additive. - }This statement holds for arbitrary rational\ flag manifolds
with $b_{2}=1$ and in all cases $m\geq1$, and not only in the hypersurface
case as will be seen below. Compare \emph{\cite{Hwang1}, 2.5, prop. 6}. In the
hypersurface case $m=1$\ one gets: Let $(\widetilde{s}=0)$ be the zero divisor
of $\chi_{0}$ on $X_{0}$. $\frac{1}{\widetilde{s}}\chi_{0}$ must have a zero
in the point $\gamma_{0}^{\ast(1)}\cap S^{\ast}$ as here different orbital
curves of $\chi_{0}$ meet. This implies easily that $(\widetilde{s}=0)$ and
$\gamma_{0}^{\ast(1)}$ are in $X_{0}$ of degree one. \ $\frac{1}{\widetilde
{s}}\chi_{0}$ must have a zero in the point $\gamma_{0}^{\ast(1)}\cap S^{\ast
}$.\newline\emph{\newline}From that what was said in the preceding it is
clear, that our results also do not depend on the special simplifying
assumptions (i.e. irreducibility of general $\mu_{0}$-fibers, $f^{^{\prime}%
}=f_{\delta}^{^{\prime}}=f_{d}^{^{\prime}}=f_{\widetilde{d}}^{`^{\prime}},$
$\sigma^{^{\prime\prime}}=1)$, we made at the beginning of these computations.
Remark. The$\ f^{^{\prime}}$,... are not meant to be derivatives.
\newline\newline\textbf{b. Some remarks on general }$\mu_{0}$-\textbf{fibers}.
Let us\ drop the assumption that general $\mu_{0}-$fibers are irreducible.
Then what has been proved is, that the components of general $\mu_{0}$-fibers
can be looked at (for our purposes) as orbital curves of additive holomorphic
vector fields $\chi_{0}$, holomorphic liftings of Euler vector fields, and
thus are $\mathbf{P}^{1}$'s; compare subsection a.$\varepsilon$ above.\newline%
\newline Our computations above give that the holomorphic liftings $\chi_{0}$
on $X_{0}$ have divisorial zeros $D$. Lemma 1 resp. its proof in \cite{Hwang1}
implies that $D$ is of degree one in $X_{0}$ and that the components of
general fibers of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}%
^{^{\prime\prime}}$ are smooth minimal rational curves $\gamma_{0}^{\ast(1)}%
$\ in $X_{0}$, i.e. of degree one in $X_{0}$. These $\gamma_{0}^{\ast(1)}$ can
be looked at as orbital curves of additive vector fields $\widehat{\chi}%
_{0}^{\ast}$, which restricted to $\gamma_{0}^{\ast(1)}$\ have a double zero
in $\gamma_{0}^{\ast(1)}\cap\widehat{G}_{0}^{\ast}$ and no zero else. Let
$s\in\Gamma(X,\mathcal{D)}$ be general, denote $\widehat{\varphi}^{\ast}s$
also by $s$. $\frac{1}{s^{2}}\cdot\widehat{\chi}_{0}^{\ast}$ has no zeroes in
general points of $\widehat{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$,
therefore different components of general fibers of $\widehat{X}_{0}^{\ast
}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ cannot meet along $\widehat
{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$. But different components of such
general fibers can only meet in $\widehat{X}_{0}^{\ast}\cap\widehat{G}%
_{0}^{\ast}$. Therefore general fibers of $\widehat{X}_{0}^{\ast}%
\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are irreducible; over outside a
proper analytic subset of $\widehat{X}_{0}^{^{\prime\prime}}$ the map
$\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ is a
$\mathbf{P}^{1}$-bundle map.

\subsection{On the nondeformability of $F=\mathbf{P}^{n}$, (mainly) the
hypersurface case.}

\textbf{a. General remarks on deformations of }$\mathbf{P}^{n}$. Let
$F=\mathbf{P}^{n}$, suppose $X_{0}\neq\mathbf{P}^{n}$\ i.e. $X_{0}%
^{^{\prime\prime}}\neq(0)\times\mathbf{P}^{n}$. Choose a curve $C$ through a
general point of $X_{0}$, lying biholomorphically over a neighborhood of $0\in
Y$, which we suppose to be $Y$, let $C^{^{\prime}}:=\mu(C)$. With other words:
$C$ and $C^{^{\prime}}$ are sections in $X$ and $X^{^{\prime}}$ over $Y.$\ We
may suppose $C^{^{\prime}}=Y\times(0)\subset Y\times\mathbf{C}^{m}$
$=Y\times\mathbf{P}^{m}\backslash H^{^{\prime}}$ with $H^{^{\prime}}%
=(s_{0}^{^{\prime}}=0\dot{)}$. Let $\chi^{^{\prime}}$ be the Euler vector
field on $X^{^{\prime}}=Y\times\mathbf{P}^{n}$ with centers along
$C^{^{\prime}}$. $\chi^{^{\prime}}$ has $H^{^{\prime}}$ as divisorial zero.
Then the holomorphic lifting $\chi$ of $\chi^{^{\prime}}$ to $X$ has $\mu
^{-1}[H^{^{\prime}}]=(s_{0}=0)$ as a divisorial zero. Consequently, according
to \cite{Hwang1}, lemma 1, the attractor set $S^{\ast}$ is a point, all
generic orbital curves of $\chi_{0}$ are rational and of degree $1$ in $X_{0}%
$. Consequently we are in the hypersurface case $\dim X_{0}^{^{\prime\prime}%
}=n-1$. Subsection 1.4 of \cite{Hwang1} delivers the nondeformability of
$\mathbf{P}^{n}$. \newline\newline\textbf{b. Remarks in the hypersurface
case}: $\alpha.$ According to \cite{Hwang1}, lemma 1, we are in the
hypersurface case. The computations in 5.2 show the existence of an additional
divisorial zero of $\chi_{0}$ through $C\cap X_{0}$, i.e. the zero set of
$\chi_{0}$ contains a divisor of degree $\geq2$. Part is the divisor
$(s_{0}=0)\cap X_{0}$, which is in $X_{0}$ of degree one. Let $\widetilde
{D}_{0}$ be the other divisorial zeros of $\chi_{0}$. Because of $b_{2}=1$
there exists a meromorphic function $g$ on $X_{0}$ having $\widetilde{D}_{0}$
as pole divisor and a multiple of $(s_{0}=0)\cap X_{0}$ as zero divisor. The
holomorphic vector field $\eta:=g\cdot\chi_{0}$ has $(s_{0}=0)^{j}\cap X_{0}$
with $j\geq2$ as divisorial zero. $\eta$ and $\chi_{0}$ have the same general
orbital curves.The only possibility is $j=2$. The restriction $\eta|\gamma
_{0}^{\ast(1)}$ of $\eta$ to a general orbital curve $\gamma_{0}^{\ast(1)}$
has a double zero in $S^{\ast}.$We can consider $\frac{1}{s_{0}^{2}}\cdot\eta$
as a $\mathcal{D}_{0}^{-2}$ valued holomorphic vector field on $X_{0}$, having
no zero in $S^{\ast}$, an impossibility, as there are different generic
orbital curves of $\eta$\ which meet in points of $S^{\ast}$. \newline%
\newline$\beta$. The preceding part of this chapter 5 gives: Drop the
assumption $F=\mathbf{P}^{n}$. If such holomorphic liftings $\chi_{0}$ have
divisorial zeros (if these exist they are necessarily of degree 1), we are in
the case $F=\mathbf{P}^{n}$ and are done according to \cite{Hwang1}, theorem
1. In the hypersurface case such holomorphic liftings $\chi_{0}$ have
divisorial zeros, necessarily of degree one. If $\dim F=3$, we are in the
hypersurface case. In the following we can suppose $\dim X_{0}^{^{\prime
\prime}}\geq2$, $F\neq\mathbf{P}^{n}$, if nothing else is explicitly agreed
upon; the last assumption has a bearing, if the adjunction formula is applied,
because then we can make use of $c_{1}(F)<n+1.-$These assertions follow (as
already mentioned) from \cite{Siu2},\ (10.3), case 1, \ p.278.

\subsection{\textbf{Further comments on the proof, that }$P^{n}$\textbf{ is
(globally) nondeformable.}}

\textbf{a. On Siu's proof. } Let us illustrate how Siu's proof fits into our
set up for $F=\mathbf{P}^{n}$, where we study the \ bimeromorphic map
$\mu:X\rightarrow X^{^{\prime}}=Y\times\mathbf{P}^{n}$ with $X_{t}%
=\mathbf{P}^{n}$ for $t\neq0$, i.e. where $\mu$ is biholomorphic outside
$X_{0}$.\ Partly we are repeating ourselves.\newline\newline We denoted the
graph of $\mu$ by $G$. Suppose that $X_{0}$ is not biholomorphic to
$\mathbf{P}^{n}$, i.e. that the fiber of $G$ over $0\in Y$ \ is reducibel
(compare 2.3). This fiber$\ G_{0}$ has at least two different\ components
$G_{0}^{\ast}$ and $X_{0}^{\ast}$, where $G_{0}^{\ast}$ is a proper
modification of $X_{0}^{^{\prime}}$ and where $X_{0}^{\ast}$ is a proper
modification of $X_{0}$, the attractor set $S^{\ast}$equals $\varphi
(G_{0}^{\ast})$, $\ \mu$ and $\mu_{0}$ are indeterminate in each point
of$\ S^{\ast}$. Let $\chi^{^{\prime}}$ be a general Euler vector field on
$X^{^{\prime}}$ the center of which on $X_{0}^{^{\prime}}$ lies on
$X_{0}^{^{\prime\prime}}=\mu\lbrack X_{0}]$; $\chi^{^{\prime}}$ on
$X^{^{\prime}}$ has a divisorial zero $(s^{^{\prime}}=0)$ of degree one, i.e.
where $s^{^{\prime}}\in H^{0}(X^{^{\prime}},\mathcal{D}^{^{\prime}})$, because
of $X^{^{\prime}}=Y\times\mathbf{P}^{n}$;\ here $\mathcal{D}^{^{\prime}}$ is
the line bundle defined by the $\mathbf{P}^{n}$-hyperplanes; we may identify
$s^{^{\prime}}$ with a homogeneous polynomial of degree one. Let $s$ be the
corresponding associated\ section $\in H^{0}(X,\mathcal{D)}$, we may identify
without loss of generality $s^{^{\prime}}$ with $s_{0}^{^{\prime}}$ and $s$
with $s_{0}$. Let $\chi$ be the holomorphic lifting of $\chi^{^{\prime}}$ to
$X$ and let $\chi^{\ast}$ be the holomorphic lifting of $\chi$ to $G$,
$\chi^{\ast}$ vanishes along $G_{0}^{\ast}$. \newline\newline Let $\gamma
_{0}^{\ast(1)}$\ be a general orbital curve of $\chi_{0}$. $\gamma_{0}%
^{\ast(1)}$ meets $S^{\ast}$, we may suppose in one point, which must be a
double zero point of $\chi_{0}$, as $\chi_{0}$%
%TCIMACRO{\TEXTsymbol{\vert}}%
%BeginExpansion
$\vert$%
%EndExpansion
$\gamma_{0}^{\ast(1)}$ is additive. Different orbital curves of $\chi_{0}$
meet in $\gamma_{0}^{\ast(1)}\cap S^{\ast}$. Consequently the holomorphic line
bundle valued vector field $\frac{1}{s}\chi_{0}$ has a zero in $\gamma
_{0}^{\ast(1)}\cap S^{\ast}$ and $(s=0)$ meets $\gamma_{0}^{\ast(1)}$ in
$\gamma_{0}^{\ast(1)}\cap S^{\ast}$ analytically just once. $\frac{1}%
{\varphi^{\ast}s}\chi^{\ast}$ is a line bundle valued holomorphic vector field
on $G$. \newline\newline First a few words on the hypersurface case $\dim
X_{0}^{^{\prime\prime}}=n-1$, i.e. the case, where a general $\mu_{0}$-fiber
is of dimension $m=1$: Direct computation shows (see above) that besides
$(s=0)|X_{0}$ the vector field $\chi_{0}$ has still another zero divisor
$(h=0)$, we may suppose that$(h=0)$ meets $\gamma_{0}^{\ast(1)}\cap S^{\ast}$
because of $b_{2}(X_{0})=1$ (there is no torsion), in this intersection
different orbital curves of $\chi_{0}$ meet. One does not get rid of these
other orbital curves through the point $\gamma_{0}^{\ast(1)}\cap S^{\ast}$by
constructing $\frac{1}{h\cdot s}\chi_{0}|$ $\gamma_{0}^{\ast(1)}$ , which is
holomorphic in the point $\gamma_{0}^{\ast(1)}\cap S^{\ast}$ and must have a
zero here because of the other orbital curves going through $\gamma_{0}%
^{\ast(1)}\cap S^{\ast}$. But this implies that on the rational curve
$\gamma_{0}^{\ast(1)}$ the vector field\ $\chi_{0}|$ $\gamma_{0}^{\ast(1)}%
$\ has at least 3 zeros which cannot be as exactly 2 zeros (counted with
multiplicities)\ have to be expected. - This contradiction proves
$X_{0}=\mathbf{P}^{n}$ (in the hypersurface case). \newline\newline Taking now
already\ into account Siu's procedure say of \cite{Siu2}\ resp. the following
chapter 8, one can handle the non hypersurface case $m>1$ thus: Let the
holomorphic lifting $\chi_{0}$ have $(s=0)$ as zero divisor,\ let $fib$ be a
general $\mu_{0}$-fiber. According to chapter 8 resp. Siu $fib$ is a
$\mathbf{P}^{m}$, where $m=\dim fib$, and $S^{\ast}\subset$ $\mathbf{P}^{m}$
is a hyperplane, the infinity hyperplane of $\mathbf{P}^{m}$. We can arrange
that $fib\cap(s=0)=S^{\ast}$ analytically. Consequently $\frac{1}{s}\chi_{0}$
is $\neq0$ in general points of $S^{\ast}.$The decisive property of $X_{0}$ in
this proof is: $\chi_{0}$ has a zero divisor. The additivity of $\chi_{0}$
with a double zero of $\chi_{0}|\gamma_{0}^{\ast(1)}$ in $\gamma_{0}^{\ast
(1)}\cap S^{\ast}$\ follows from the fact, that $\gamma_{0}^{\ast(1)}$ is
rational and that\ different orbital curves of $\chi_{0}$ meet in $\gamma
_{0}^{\ast(1)}\cap S^{\ast}$; the only zero of $\chi_{0}|\gamma_{0}^{\ast(1)}$
is the double zero in $\gamma_{0}^{\ast(1)}\cap S^{\ast}$.\newline\newline
There exist on $X_{0}$ holomorphic liftings $\chi_{0}^{(1)}:=\chi_{0}$,...,
$\chi_{0}^{(m)}$\ with the following property: $\chi_{0}^{(1)}\wedge
...\wedge\chi_{0}^{(m)}$ can on $X$ \ be written as $s^{m}\cdot K$, where $K$
is holomorphic on $X$ and $0$ in all points of $S^{\ast}$. One can show that
$K$ has a zero divisor (\cite{Siu2}, p.278), without restriction $(s=0)$, i.e.
we can write $\chi_{0}^{(1)}\wedge...\wedge\chi_{0}^{(m)}=s^{m+1}\cdot
\widehat{K}$, where $\widehat{K}$ has no zeros, because of $c_{1}%
(fib)=c_{1}(\mathbf{P}^{m})=m+1$. But this cannot be, as different general
$\mu_{o}$-fibers $fib$ meet along $S^{\ast}$. \newline\newline Very satisfying
in the case $F=\mathbf{P}^{n}$ are those non deformability proofs which use
\cite{Hwang1}, lemma 1/theorem1/Cor.1 or related arguments; but compare also
the somehow related result of\ the next subsection d below. On the other hand,
Siu's procedure in, say \cite{Siu2}, seems to be stronger, can be improved and
extended in order to get \ stronger non deformability results (theorem 2 and
3) for $F\neq\mathbf{P}^{n}$. \newline\newline\textbf{b. Another approach to
the global rigidity of }$\mathbf{P}^{n}$. The subsequent\ proof belongs to the
context of 5.1.and can be considered (with more than just cum grano salis) as
a new proof of the $P^{n}$- non deformability. The following situation can be
realized: Let $F=\mathbf{P}^{n}$, $\mathcal{D}^{^{\prime}}$ the positive
generating line bundle on $X^{^{\prime}}=Y\times F\subset Y\times
\mathbf{P}^{N}$, $\mathcal{D}$ the positive generating line bundle on $X$,
$s\in H^{0}(X,\mathcal{D})$ and $s^{^{\prime}}\in H^{0}(X^{^{\prime}%
},\mathcal{D}^{^{\prime}})$ associated elements, $\chi^{^{\prime}}$a general
Euler vector field on $X^{^{\prime}}$ with a center on $X_{0}^{^{\prime}}$ in
$X_{0}^{^{\prime\prime}}$ and\ with $(s^{^{\prime}}=0)$ as divisorial zero,
$\chi$ the holomorphic lifting of $\chi^{^{\prime}}$ to $X$, $\chi^{\ast}$ the
holomorphic lifting of $\chi^{^{\prime}}$ to $G$ (= graph of $\mu:$
$X\rightarrow Y\times F$), $\varphi:G$.$\rightarrow X$, $\psi:G\rightarrow$
$X^{^{\prime}}=Y\times F$ the corresponding projections. $\frac{1}{s}\cdot
\chi$ is a line bundle valued holomorphic vector field on $X$, $\frac
{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$ is a line bundle valued holomorphic
vector field on $G$. \newline\newline Let $\gamma_{0}^{\ast(1)}$ be the
orbital curve of $\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$ through a general
point of a general fiber $fib$ of $X_{0}^{\ast}\overset{\psi}{\rightarrow
}X_{0}^{^{\prime\prime}}$, we can interpret $\gamma_{0}^{\ast(1)}$ also as a
subset of \ $X_{0}$. $\gamma_{0}^{\ast(1)}$ is met by $(s=0)$ pointwise just
once, namely in the point $P^{\ast}\ =G_{0}^{\ast}\cap X_{0}^{\ast}\cap
\gamma_{0}^{\ast(1)}$. Here we use the additivity of $\chi^{\ast}$ along
$\gamma_{0}^{\ast(1)}$. If the intersection multiplicity of $(s=0)$ and
$\gamma_{0}^{\ast(1)}$ in $P^{\ast}$ would be two, $\frac{1}{s}\cdot\chi$
would be $\neq0$ in the point $\varphi(P^{\ast})$. This is not possible as
different orbital curves of $\frac{1}{s}\cdot\chi$ meet in $\varphi(P^{\ast}%
)$. A higher intersction multiplicity in $P^{\ast}$ is not possible.$\ $%
\newline\newline Thus $\varphi^{\ast}s$ vanishes along $G_{0}^{\ast}$ with
order $1$, $P^{\ast}$ is a zero of $\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$
of order one, because $\frac{1}{s}\cdot\chi$ has a zero in $\varphi(P^{\ast}%
)$, as different orbital curves of $\frac{1}{s}\cdot\chi$ meet in
$\varphi(P^{\ast})$, $\ G\ $is smooth in general points of$\ G_{0}^{\ast}$. It
can be seen that the pole order $k$ introduced in 5.1.a equals 1: Otherwise
$\chi^{\ast}$ vanishes along $G_{0}^{\ast}$ with order $\geq2$, $\frac
{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$ vanishes along $G_{0}^{\ast}$ with order
$\geq1$, which is not possible.\newline\newline There are plenty of choices
for $\chi^{^{\prime}}$ and we may assume that $\chi$ $^{^{\prime}}$ is not
zero in general points of $X_{0}^{^{\prime\prime}}$. The reason:
$X_{0}^{^{\prime\prime}}$ is not a linear subspace of $(0)\times\mathbf{P}%
^{n}$ and therefore in general the lines of $(0)\times\mathbf{P}^{n}$ through
the center of $\chi^{^{\prime}}$\ meet $X_{0}^{^{\prime\prime}}$
pointwise\ also in points $\neq$ center, in these points $\chi^{^{\prime}}$ is
$\neq0$, but locally $\chi^{^{\prime}}$ can be identified here with the image
of $\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}|G_{0}^{\ast}$. Thus $\frac
{1}{\varphi^{\ast}s}\cdot\chi^{\ast}|G_{0}^{\ast}$ has no zero over general
points $P^{^{\prime}}$\ of $X_{0}^{^{\prime\prime}}$. But over such points two
orbital curves of $\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$\ meet, one lying
in such a fiber $fib$ over $P^{^{\prime}}$, one lying on $G_{0}^{\ast}$,
$\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$ should have a zero in these
intersection points. Contradiction.- $Remark$: We used $\chi^{^{\prime}%
}(P^{^{\prime}})\neq0$ and\ that the orbital curve of $\chi^{^{\prime}}$
through $P^{^{\prime}}$\ does not lie on $X_{0}^{^{\prime\prime}}$ in order to
be sure, that over a neighborhood of $P^{^{\prime}}$ the orbital curves of
$\frac{1}{\varphi^{\ast}s}\cdot\chi^{\ast}$ through points of$\ G_{0}^{\ast}$
$\neq$ center\ stay in $G_{0}^{\ast}$.

\section{\textbf{On a proof of theorem 1.}}

\subsection{The main purpose of this chapter.}

We continue to use previous notations. Let $\tau:X\rightarrow Y:=(|t|<1)$ be a
deformation map, i.e. a holomorphic, proper, surjective, regular map
$\tau:X\rightarrow Y,$ where $X$ is a (connected) complex manifold, let $\ F$
be a rational homogeneous complex manifold\ with Picard number 1. Let $\tau$
be differentiably trivial\ As the homology groups\ of $F$ are without torsion
(\cite{Borel}), duality and $H^{2}(F,\mathbf{Z})=\mathbf{Z}$ give
$H_{2}(F,\mathbf{Z})=\mathbf{Z}$. Suppose that for $t\neq0$ the fiber
$X_{t}:=\tau^{-1}(t)$ is biholomorphic to $F$ and that $X_{0}$ is k\"{a}hler.
These assumptions allow to prove global rigidity, i.e. theorem 1. It seems
worthwhile to outline first a proof of the simpler\ theorem 1 before
discussing the general case of theorem 2, as in the case of theorem 1 we can
de facto neglect chap. 7. \newline\newline In a first step we have to make
sure (in the case of theorem 1) that general $\mu_{0}$-fibers contain minimal
rational curves, which is not difficult. \newline

\subsection{A discussion \ in case $\mathbf{X}_{0}$ is known to be k\"{a}hler
by assumption, a proof of theorem1.}

\textbf{a.} For our purposes we may\ continue to suppose that general $\mu
_{0}$-\ fibers are irreducibel. We can arrange the following situation
described in \textbf{a, b, c}: \ \newline\newline There are general\ points
$\widehat{P}^{\ast}\in\widehat{G}_{0}^{\ast}$, $\widehat{Q}^{\ast}\in
\widehat{X}_{0}^{\ast}$\ (i.e. general with respect to $\widehat{G}_{0}^{\ast
}$ resp. to $\widehat{X}_{0}^{\ast}$)\ with $\widehat{Q}^{\ast}\notin
\widehat{G}_{0}^{\ast}$, such that $\widehat{P}^{`}:=\widehat{\psi}%
(\widehat{P}^{\ast})\neq\widehat{Q}^{^{\prime}}:=\widehat{\psi}($ $\widehat
{Q}^{\ast})$ and that $\widehat{Q}^{^{\prime}}$ , $\widehat{P}^{`}$can be
joined in $\widehat{X}_{0}^{\prime}=F$ by an $F-$ line $l^{^{\prime}}$ which
meets $\widehat{X}_{0}^{^{\prime\prime}}$ pointwise and such that
$(\widehat{\psi}|\widehat{G}_{0}^{\ast})^{-1}[l^{^{\prime}}]$ is a curve,
which does not lie in a fiber of $\widehat{G}_{0}^{\ast}\rightarrow S^{\ast}%
$\ \newline\newline The existence of such a line $l^{`^{\prime}}$ is a
consequence of 3.5, as $\widehat{\psi}$ induces a bimeromorphic map
$\widehat{G}_{0}^{\ast}\rightarrow$ $\widehat{X}_{0}^{^{\prime}}=(0)\times F$
and as we may assume that this map is biholomorphic in $\widehat{P}^{\ast}$;
so we may apply 3.5, as for practical purposes we are operating on $F$. We may
wiggle $\widehat{Q}^{^{\prime}}$ , $\widehat{P}^{`}$ a little bit suitably, if
we are compelled to. We start with $\widehat{Q}^{^{\prime}}$, choose
subsequently $l^{^{\prime}}$ through $\widehat{Q}^{^{\prime}}$\ such that
$(\widehat{\psi}|\widehat{G}_{0}^{\ast})^{-1}[l^{^{\prime}}]$ does not lie in
a fiber of $\widehat{G}_{0}^{\ast}\rightarrow S^{\ast}$. Then we fix
$\widehat{P}^{\ast}$ etc.\newline\newline We construct a section
$Y\rightarrow$ $\widehat{C}^{\ast}$ through $\widehat{Q}^{\ast}\in\widehat
{X}_{0}^{\ast}$ with $\widehat{C}^{^{\prime}}=\widehat{\psi}(\widehat{C}%
^{\ast})$. Let $L^{^{\prime}}\subset Y\times F$ be the family of the
$L_{t}^{^{\prime}}=$\emph{line in (t)}$\times F$ \emph{through the
point\ }$\widehat{C}_{t}^{^{\prime}}$ \emph{of }$\widehat{C}^{^{\prime}}%
$\emph{over} $t\in Y$\ \emph{in the direction of }$l^{^{\prime}}$\emph{.
}$\widehat{C}_{t}^{^{\prime}}$ and the direction uniquely define
$L_{t}^{^{\prime}}$.\emph{ }Define\emph{ }$L:=$.$\widehat{\varphi}%
(\widehat{\psi}^{-1}[L^{^{\prime}}])$, $\widehat{L}^{\ast}:=\widehat{\psi
}^{-1}[L^{^{\prime}}]$. Here we work with the $\mathbf{C}^{n}$-coordinates of
the Bruhat embedding of $\mathbf{C}^{n}=F\backslash H_{F}$ discussed in
chapter 3, we assume \emph{ }$\widehat{C}_{t}^{^{\prime}}\in\mathbf{C}^{n}$.
It will do no harm to suppose $L^{^{\prime}}=Y\times l^{^{\prime}}$\ The
holomorphic maps\ $\widehat{L}^{\ast}\rightarrow L$, $\widehat{L}^{\ast
}\rightarrow L^{^{\prime}}$ are bimeromorphic and have connected fibers.
\emph{\newline\newline}The family $L^{^{\prime}}$ has the following
properties: For $t\neq0$ the fiber $L_{t}$ is a minimal rational curve in
$X_{t}\cong(t)\times F\cong F$. $\widehat{L}_{0}^{\ast}$ decomposes into at
least 2 components, one component $\ \widehat{L}_{0}^{\ast(0)}\subset
\widehat{G}_{0}^{\ast}.$lying biholomorphically\ over $l^{^{\prime}}%
=L_{0}^{^{\prime}}$, and at least one component $\widehat{L}_{0}^{\ast(1)}%
$\ through $\widehat{Q}^{\ast}$, lying in the $\mu_{0}$- fiber through
$\widehat{Q}^{\ast}$. \newline\newline\textbf{b.} \emph{Now we are going to
use }$X_{0}$\emph{\ to be k\"{a}hler, i.e. in our case\ projective. }All
$L_{t}^{^{\prime}}\subset\widehat{X}_{t}^{^{\prime}}$ , $t\neq0$, are minimal
rational curves. Consequently for reasons of continuity $L_{0}$ represents the
positive generating element in.$H_{2}(X_{0},\mathbf{Z)}$ $\mathbf{=Z}$ and
cannot decompose as $X_{0}$ is k\"{a}hler, there are no zero homologous
(unions of) curves on $X_{0}$. The induced map $\widehat{L}_{0}^{\ast
(1)}\rightarrow\widehat{\varphi}(\widehat{L}_{0}^{\ast(1)})$ can be supposed
to be biholomorphic, $\widehat{\varphi}(\widehat{L}_{0}^{\ast(1)})$ does not
lie in $S^{\ast}$, but meets $S^{\ast}$ in a finite set of points.. The image
of $\ \widehat{L}_{0}^{\ast(0)}$ in $S^{\ast}$ must be a point. \newline%
\newline\textbf{c. }We have only to discuss the case $\dim S^{\ast}>0$. Let
$fib^{^{\prime}}$ be the fiber of $\widehat{G}_{0}^{\ast}\rightarrow S^{\ast}$
through $\widehat{P}^{^{\prime}}$. Because of the subchapter\ 3.5 on rational
connectedness by chains of lines\ we can choose $l^{^{\prime}}$, such that
$\widehat{\psi}^{-1}[l^{^{\prime}}]$ meets $fib^{^{\prime}}$ in $\widehat
{P}^{^{\prime}}$ pointwise. The image of $\widehat{\psi}^{-1}[l^{^{\prime}}]$
in $S^{\ast}$ is a curve. This curve should be a proper\ component of
$\widehat{\varphi}(\widehat{L}_{0}^{\ast})$, which cannot be, as
$\widehat{\varphi}(L_{0}^{\ast})$ must be irreducible. Therefore $S^{\ast}$
must be a point. This case already has been discussed, as the arguments used
in the $\mathbf{P}^{n}$case work in the case $\dim S^{\ast}=0$ for all flag
manifolds $F$ with $b_{2}=1$ Theorem 1 is proved. \newline\newline This
approach depends heavily on the k\"{a}hlerness of the central fiber $X_{0}$.
We will briefly indicate the approach to theorem 2, specialized to the case of
theorem1. \newline\newline\emph{Remark}. These arguments also work for del
Pezzo manifolds of degree 5 delivering global k\"{a}hler non deformability of
these spaces, a property which is well known. \newline

\subsection{Another proof of $X_{0}=F$ under the assumption that $X_{0}$ is
k\"{a}hler (theorem 1), a sketch.}

As we may replace $\widehat{X}_{0}^{\ast}$ by a canonical resolution
$\widehat{X}_{0}^{\ast\ast}$ of singularities (\cite{bimil}) it is not a
restriction to assume $\widehat{X}_{0}^{\ast}$ to be smooth. Because of
\cite{Hwang1}, p. 331 general fibers of $\widehat{X}_{0}^{\ast}\rightarrow$
$\widehat{X}_{0}^{\prime\prime}$ are biholomorphic to $\mathbf{P}^{m}$. This
can be used to show $\dim S^{\ast}=m-1$, and that $S^{\ast}$ is contained in
all $\mu_{0}$-fibers. In chapter.8 we will discuss this matter. \newline%
\newline There is only to prove something for $F\neq\mathbf{P}^{n}$. We may
suppose that any general fiber $fib=\mathbf{P}^{m}$\ of $\widehat{X}_{0}%
^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ contains a minimal
rational curve. Then $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$ is over general points of $\widehat{X}_{0}%
^{^{\prime\prime}}$ a $\mathbf{P}^{m}$-bundle. Take a general line $l$ in such
a $\mathbf{P}^{m}$, $l$ is in $X_{0}$ a minimal rational curve, i.e. an
irreducible curve of degree $1$ in $X_{0}$, a rational curve, as for general
$s\in\Gamma(X,\mathcal{D})$ the zero set $(s=0)$ meets $l$ analytically just
once (in $l\cap S^{\ast}$); we have $T(\mathbf{P}^{m})|l\backsimeq
\mathcal{O(}2)\oplus\mathcal{O}(1)^{m-1}$, therefore we can assert
$\mathfrak{N}_{l\backslash\mathbf{P}^{m}}\backsimeq$\ $\mathcal{O}(1)^{m-1}%
\,$, there is a vector field in a neighborhood of $l$ which induces on $l$
a\ vector field with a double zero in $l\cap S^{\ast}$.\newline\newline Proof:
Provide $\mathbf{P}^{m}$ with inhomogeneous coordinates $z_{1},...,z_{m}$ ,
consider the vector fields $\varrho_{1}:=$ $z_{1}(z_{1}\cdot\frac{\partial
}{\partial z_{1}}+$ $...$ $+z_{m}\cdot\frac{\partial}{\partial z_{m}})$,
$\varrho_{2}:=z_{1}\cdot\frac{\partial}{\partial z_{2}}$,..., $\varrho
_{m}:=z_{1}\frac{\partial}{\partial z_{m}}$ and their $\Lambda-$product$.$This
gives the assertion. \newline\newline Similarly as in chapter 5, where $\dim
S^{\ast}=0$ was discussed, one shows that the normal bundle of $l$ in $X_{0}$,
is generated by sections having a \ simple\ zero in $l\cap S^{\ast}$ and are
linearly independent in all other points of $l$; invoking the adjunction
formula\ 2.5.$\beta$ and a well known theorem of Grothendieck on splitting of
vector bundles we get $c_{1}(X_{0})=c_{1}(F)=n+1$, a contradiction to
$c_{1}(F)<n+1$; it suffices to show $c_{1}(X_{0})\geq n+1$, suppose that
$(s=0)\cap fib=S^{\ast}$ resp. $\widehat{\varphi}(\widehat{G}_{0}^{\ast}\cap
fib)=S^{\ast}$ and that $(s=0)$ meets $l$ just once and this in $l\cap
\widehat{G}_{0}^{\ast}$:\newline\newline There is a smooth\ $\widehat{X}%
_{0}^{^{\prime\prime}}-$\ neighborhood $U$ of $P"$:=$\widehat{\psi}(fib)$ such
that $\widehat{\psi}^{-1}(U)\cap\widehat{X}_{0}^{\ast}$ can be written as
$fib\times U$ and such that $U$ carries a $(w_{1},...,w_{n-m})-$coordinate
system centered in $P"$. Extend $\frac{\partial}{\partial w_{i}},i=1,...,n-m$,
trivially to vector fields on $fib\times U$, call these extended vector fields
too $\frac{\partial}{\partial w_{i}}$, extend the $\rho_{j}$ too trivially to
$fib\times U$, $j=1,...,m$. The bundle $T(X_{0})|l$ is generated by $\rho
_{1},\varrho_{2},...,\varrho_{m}$, $\widehat{\varphi}_{\ast}(\frac{\partial
}{\partial w_{1}})$,..., $\widehat{\varphi}_{\ast}(\frac{\partial}{\partial
w_{n-m}})$ and equals the bundle$\ \mathcal{O(}2)\oplus\mathcal{O}(1)^{n-1}$
over the rational curve $l$. $T(X_{0})|l$ equals $T(l)\oplus\mathfrak{N}%
_{l/X_{0}}$. The normal bundle $T(X_{0})|l$ /$T(l)=\mathfrak{N}_{l/X_{0}}$ of
$l$ in $X_{0}$ \ is generated over $l$ by (the restrictions\ of) $\varrho
_{2},...,\varrho_{m}$, $\widehat{\varphi}_{\ast}(\frac{\partial}{\partial
w_{1}})$,..., $\widehat{\varphi}_{\ast}(\frac{\partial}{\partial w_{n-m}})$.
This implies the contradictory statement$\ c_{1}(X_{0})=c_{1}(F)=n+1$. - This
finishes this sketch of the proof of theorem 1. q.e.d. \newline

\section{\textbf{Holomorphic liftings of Euler vector fields in the general
case }$m\geq1$\textbf{, the proof of }$m=q$, \textbf{a calculatory approach to
a proof of theorem 2 (and 3). }\newline}

\textbf{a. Some notions; some statements to be proved. }We continue here and
in the following to stick to the notational agreement of the last remark in
1.2.c. - \textbf{\ }Now we will drop the assumption $X_{0}$ being k\"{a}hler
made in subsection 2.of chapter 6. This chapter 7 will be a continuation of
chapter 5. - Let us continue$\ \mu$ to\ be defined by $\tau$ and $s_{0}%
$,...,$s_{n}\in H^{0}(X,\mathcal{D})$ with associated $s_{0}^{^{\prime}}%
$,...,$s_{n}^{^{\prime}}\in H^{0}(X^{^{\prime}},\mathcal{D}^{^{\prime}})$,
$(Y\times F)\cap(s_{0}^{^{\prime}}\neq0)=Y\times\mathbf{C}^{n}$, $(Y\times
F)\cap(s_{0}^{^{\prime}}=0)$ $=Y\times H_{F}$. Suppose that the Euler vector
fields on $Y\times F$ we work with,$\,\ $considered as vector fields on
$Y\times\mathbf{P}^{n}$, have on $(0)\times\mathbf{P}^{n}$ centers $\in
X_{0}^{^{\prime\prime}}$, if nothing else is agreed upon.\textbf{\newline%
}\newline We denoted by $m$ the dimension $m=n-\dim X_{0}^{^{\prime\prime}}%
$\ of a general $\mu_{0}$-fiber. - For our purposes we may suppose (in most
cases) $\widehat{G}$ \ to be normal and that $t$ vanishes along all components
of $\widehat{G}_{0}$ with order one, which can be arranged by a base change.
From 5.2 and from 7.1.b and c. below we get: There is an integer $k>0$, such
that $t^{k}\cdot\widetilde{\chi}_{\widehat{G}}$ (here $\widetilde{\chi
}_{\widehat{G}}=\widehat{\psi}^{\ast}\widehat{\chi}^{^{\prime}}$ )\ is
holomorphic on $\widehat{G}$ and and not zero in a general point of
$\widehat{X}_{0}^{\ast}$. - Let $\widehat{\chi}^{^{\prime}}$ be the
holomorphic vector field on $\widehat{X}^{^{\prime}}=Y\times F$ corresponding
to $\chi^{^{\prime}}$.\newline\newline We call $\widehat{\chi}_{\widehat{G}}%
$\emph{\ :=} $t^{k}\cdot\widetilde{\chi}_{\widehat{G}}$ the \emph{holomorphic
lifting }of $\chi^{^{\prime}}$ resp. $\widehat{\chi}^{^{\prime}}$\ to
$\widehat{G}.$ We have $\widehat{\varphi}_{\ast}\widehat{\chi}_{\widehat{G}%
}=\chi.$ Instead of $\widehat{\chi}_{\widehat{G}}$ we also write
$\widehat{\chi}^{\ast}$ and instead of $\widehat{\chi}_{\widehat{G}}%
|\widehat{X}_{0}^{\ast}$ we use also the notation $\widehat{\chi}_{0}^{\ast}$.
\newline\newline As $k>0$, orbital curves (=closures of orbits) through
general points of $X_{0}$ lie in $\mu_{0}$-fibers.\newline\newline We call
these holomorphic liftings (and their restrictions to $X_{0}$ resp.
$\widehat{X}_{0}^{\ast}$) also Euler vector fields and speak of their centers,
compare 5.1. \newline\newline\textbf{Warning.} We distinguish between liftings
and holomorphic liftings. \newline\newline Let $\sum$ be the collection of all
sections $\widetilde{\sigma}:Y\rightarrow X\setminus H,$ $\sum^{^{\prime}}$ be
the collection of the corresponding sections $\widetilde{\sigma}^{^{\prime}%
}=\mu\circ\widetilde{\sigma}:Y\rightarrow X^{^{\prime}}\setminus H^{^{\prime}%
}.$ Here $H=(s_{0}=0),$ $H^{^{\prime}}=(s_{0}^{^{\prime}}=0)$ are the infinity
hyperplanes. For $\widetilde{\sigma}^{^{\prime}}\in\sum^{^{\prime}}$ let
$\chi^{^{\prime}}(\widetilde{\sigma}^{^{\prime}})$ be the Euler vector field
(resp. vector field of Euler type) on $X^{^{\prime}}=Y\times\mathbf{P}^{n}$
with zeroes along $\widetilde{\sigma}^{^{\prime}}(Y)$ (look at this set as
path of origins $\widetilde{\sigma}^{^{\prime}}(t),$ $t\in Y,$ as the set of
centers of $\chi^{^{\prime}}(\widetilde{\sigma}^{^{\prime}})$)$,$ let
$\chi(\widetilde{\sigma})$ , $\widehat{\chi}_{\widehat{G}}(\widetilde{\sigma
})$ be the corresponding holomorphic liftings on $X,$ resp. $\widehat{G}$, we
are going to construct below, i. e. the meromorphic liftings already
multiplied with an appropriate power of $t,$ such that $\chi(\widetilde
{\sigma}),$ $\widehat{\chi}_{\widehat{G}}(\widetilde{\sigma})$ are holomorphic
and do not vanish identically along $X_{0}$ resp. $\widehat{X}_{0}^{\ast};$ we
will call\ $\chi(\widetilde{\sigma})$ , $\widehat{\chi}_{\widehat{G}%
}(\widetilde{\sigma})$ holomorphic liftings; (for $m=1)$ we introduced the
notion of a holomorphic lifting already in chap. 5. \newline\newline There is
a $q\in\mathbf{N}$ with the following property: There exist $\sigma
_{1},...,\sigma_{q}\in\sum,$ such that $\chi(\sigma_{1}),...,\chi(\sigma_{q})$
are linearly independent in general points of $X_{0}$; for all families of
$q+1$ sections $\sigma_{1},...,\sigma_{q+1}$ in $\sum$ the vector fields
$\chi(\sigma_{1}),...,\chi(\sigma_{q+1})$ are linearly dependent in all points
of $X_{0}.$ The closure $W(P_{0})$\ of an integral manifold (orbit) through a
general $P_{0}\in X_{0}$\ corresponding to $\chi(\sigma_{1}),...,\chi
(\sigma_{q})$ is called an \emph{orbital variety}; these orbital varieties
only depend on the wedge product $\chi(\sigma_{1})\wedge...\wedge\chi
(\sigma_{q}).$\ If $q=1,$ we have an \emph{orbital curve; }this case was
discussed in chap. 5. \newline\newline\emph{We will show that always }%
$m=q$\emph{\ and that the orbital variety through a general.point of }$X_{0}%
$\emph{\ is the (componet of) the }$\mu_{0}$\emph{-fiber through this point.
}\newline\newline\textbf{b. Lifting Euler vector fields. Proof of }$m=q$. \ We
continue to use the notations of 4.2. At the beginning let us assume that $F$
is compact, irreducibel, hermitian, symmetric; this implies that translations
of $F\backslash H_{F}=\mathbf{C}^{n}$ are induced by biholomorphic maps
$F\rightarrow F$. Suppose first $n=4,$ $m=2,$ $\dim X_{0}^{^{\prime\prime}}%
=2$. For reasons of notational simplicity we will assume that $X_{0}%
^{^{\prime\prime}}$ is analytically\ in general points $P^{^{\prime}}$ the
common zero set of $t,$ $f_{1}^{^{\prime}}:=f_{1\delta}^{^{\prime}}$ and
$f_{2}^{^{\prime}}:=f_{2\delta}^{^{\prime}}$, and furthermore we can assume,
that $\psi^{\ast}f_{1}^{^{\prime}},\,\psi^{\ast}f_{2}^{^{\prime}}$ vanish
along $X_{0}^{\ast}$ with the same order $\sigma^{^{\prime\prime\prime}}$;
$\ t,\frac{\ \psi^{\ast}f_{1}^{^{\prime}}}{t^{\sigma^{^{\prime\prime\prime}}}%
},\,\frac{\ \psi^{\ast}f_{2}^{^{\prime}}}{t^{\sigma^{^{\prime\prime\prime}}}}$
and $t,\frac{\ \psi^{\ast}f_{1\widetilde{d}}^{^{\prime}}}{t^{\sigma
^{\prime\prime}}},\,\frac{\ \psi^{\ast}f_{2\widetilde{d}}^{^{\prime}}%
}{t^{\sigma^{\prime\prime}}}$ are local parameters on $G$ with center in a
(chosen) general point $P^{\ast}$ of $X_{0}^{\ast}\subset G$, $P^{^{\prime}}$
its image in $X^{^{\prime}}$\ the map $\psi^{-1}\left[  (f_{1\widetilde{d}%
}^{^{\prime}}=f_{2\widetilde{d}}^{^{\prime}}=0)\right]  \cap U^{\ast
}\rightarrow$ $X^{^{\prime}}$ is biholomorphic over $P^{^{\prime}}$, here
$U^{\ast}$ is a suitable $G$-neighborhood of $P^{^{\ast}}$. $f_{1d}^{^{\prime
}},$ $f_{2d}^{^{\prime}}$ and $f_{1}^{^{\prime}},$ $f_{2}^{^{\prime}}$\ are
analytically independent. In a neighborhood of $P^{\ast}$ we may for our
purposes identify the two pairs$\ \frac{\ \psi^{\ast}f_{1\widetilde{d}%
}^{^{\prime}}}{t^{\sigma^{\prime\prime}}},\,\frac{\ \psi^{\ast}f_{2\widetilde
{d}}^{^{\prime}}}{t^{\sigma^{\prime\prime}}}$ and $\frac{\ \psi^{\ast}%
f_{1}^{^{\prime}}}{t^{\sigma^{^{\prime\prime\prime}}}},\,\frac{\ \psi^{\ast
}f_{2}^{^{\prime}}}{t^{\sigma^{^{\prime\prime\prime}}}}$. The pole order $k$
discussed in 5.1 equals $\sigma^{^{\prime\prime\prime}}$. \newline\newline Now
we make for reasons of simple notations the same simplifying assumptions as in
4.2, 5.2, namely $\sigma^{^{\prime\prime}}=1$, $d=\widetilde{d}$,
$f_{1\widetilde{d}}^{^{\prime}}=f_{1}^{^{^{\prime}}}$, $f_{2\widetilde{d}%
}^{^{\prime}}=f_{2}^{^{\prime}}$. We have to be aware of these simplifications
in interpreting our computations.\newline\newline Denote $z_{1}=\frac
{s_{1}^{^{\prime}}}{s_{0}^{^{\prime}}},$ $z_{2}=\frac{s_{2}^{^{\prime}}}%
{s_{0}^{^{\prime}}}.,....$ Suppose $(z_{1}=0)$ and $(z_{2}=0)$ to be tangent
to $X_{0}^{^{\prime\prime}}$ in the general point $P^{^{\prime}}$ given by
$z_{1}=0,\,$ $...$, $z_{n}=0$ , let the holomorphic vector field $\chi
(\sigma_{1})$ on $X$ correspond to the Euler vector field $\chi^{^{\prime}%
}(\sigma_{1})$ = $z_{1}\frac{\partial}{\partial z_{1}}+...+z_{n}\frac
{\partial}{\partial z_{n}}$ on $X^{^{\prime}}.$ Identify $f_{1}^{^{\prime}},$
$f_{2}^{^{\prime}}$ with their corresponding functions in $z_{1},...,z_{n}.$
In a neighborhood of $P^{^{\prime}}$ we can write $\chi^{^{\prime}}(\sigma
_{1})=g_{1}\frac{\partial}{\partial f_{1}^{^{\prime}}}+\,g_{2}\frac{\partial
}{\partial f_{2}^{^{\prime}}}+z_{3}\frac{\partial}{\partial z_{3}}%
+...+z_{n}\frac{\partial}{\partial z_{n}}.$ Here $g_{j}$, $j=1,2$ can be
described as $z_{1}\frac{\partial f_{j}^{^{\prime}}(z_{1},...z_{n})}{\partial
z_{1}}$ $+...+z_{n}\frac{\partial f_{j}^{^{\prime}}}{z_{n}}$. In a
neighborhood of a general point over $P^{^{\prime}}$ on $X_{0}^{\ast}$ we may
look at $t,\frac{1}{t}\cdot\psi^{\ast}(f_{1}^{^{\prime}})=\widetilde{f_{1}},$
$\frac{1}{t}\cdot\psi^{\ast}(f_{2}^{^{\prime}})=\widetilde{f_{2}}$ as part of
a parametersystem centered in this point. $Mod$ $t$ the vectorfield
$\chi(\sigma_{1})$ can be written as $\psi^{\ast}(g_{1})\cdot\frac{\partial
}{\partial\widetilde{f_{1}}}+\psi^{\ast}(g_{2})\cdot\frac{\partial}%
{\partial\widetilde{f_{2}}}$ in a neighborhood of this point. \newline\newline
Now take a second general holomorphic lifting of an Euler vector field
$\chi(\sigma_{2})$ with the corresponding $\chi^{^{\prime}}(\sigma
_{2})=\overline{h}_{1}\frac{\partial}{\partial f_{1}^{^{\prime}}}+\overline
{h}_{2}\frac{\partial}{\partial f_{2}^{^{\prime}}}+(z_{3}-a_{3})\frac
{\partial}{\partial z_{3}}+...+(z_{n}-a_{n})\frac{\partial}{\partial z_{n}}$
with $\overline{h}_{j}=(z_{1}-a_{1)}\frac{\partial f_{j}^{^{\prime}}%
(z_{1},...,z_{n})}{\partial z_{1}}+...+(z_{n}-a_{n})\frac{\partial
f_{j}^{^{\prime}}}{\partial z_{n}},$ j=1, 2, with $Q^{^{\prime}}:=\mu
(\sigma_{2}(0))$ $=(a_{1},...,a_{n})\in X_{0}^{^{\prime\prime}}$ general.
Suppose that $\ \chi_{1}^{^{\prime\prime}}=g_{1}\frac{\partial}{\partial
f_{1}^{^{\prime}}}+$ $\,g_{2}\frac{\partial}{\partial f_{2}^{^{\prime}}}$ and
$\ \chi_{2}^{^{\prime\prime}}=\overline{h}_{1}\frac{\partial}{\partial
f_{1}^{^{\prime}}}+\overline{\,h}_{2}\frac{\partial}{\partial f_{2}^{^{\prime
}}}$ are linearly dependent in all points of an open nonempty set of
$X_{0}^{^{\prime\prime}}.$ As $\left(  \frac{\partial f_{1}^{^{\prime}}%
}{\partial z_{1}},...,\frac{\partial f_{1}^{^{\prime}}}{\partial z_{n}%
}\right)  $, $\left(  \frac{\partial f_{2}^{^{\prime}}}{\partial z_{1}%
},...,\frac{\partial f_{2}^{^{\prime}}}{\partial z_{n}}\right)  $ are linearly
independent in general points of $X_{0}^{^{\prime\prime}},$ there must be for
\ a general, but fixed $(z_{1},...,z_{n})\in X_{0}^{^{\prime\prime}}$ a linear
relation between the components of arbitrary general $\left(  a_{1}%
,...,a_{n}\right)  \in X_{0}^{^{\prime\prime}},$ implying that $X_{0}%
^{^{\prime\prime}}$ lies in a proper linear subspace of $\left(  0\right)
\times\mathbf{P}^{n}$, compare the reasoning in chapter 5, subsection 5.2: We
are looking for general \ points\ $a=\left(  a_{1},...,a_{n}\right)  \in
X_{0}^{"}$, $z=(z_{1},...,z_{n})\in X_{0}^{^{\prime\prime}}$ such that the
rank of \ $\mathfrak{A:=}(%
\begin{array}
[c]{cc}%
g_{1} & g_{2}\\
\overline{h}_{1} & \overline{h}_{2}%
\end{array}
)$ equals 2. The rank of \ $\mathfrak{A:=}(%
\begin{array}
[c]{cc}%
g_{1} & g_{2}\\
\overline{h}_{1} & \overline{h}_{2}%
\end{array}
)$ equals the rank of the product $%
\begin{array}
[c]{cc}%
(%
\begin{array}
[c]{ccc}%
z_{1} & z_{2} & ...\\
a_{1} & a_{2} & ...
\end{array}
) & (%
\begin{array}
[c]{cc}%
f_{1z_{1}}^{^{\prime}}(z_{1},...) & f_{2z_{1}}^{^{\prime}}\\
f_{1z_{2}}^{^{\prime}} & f_{2z_{2}}^{^{\prime}}\\
... & ...
\end{array}
)
\end{array}
.$We may suppose that for general $z=(z_{1},...,z_{n})\in X_{0}^{^{\prime
\prime}}$ the first two lines in $(%
\begin{array}
[c]{cc}%
f_{1z_{1}}^{^{\prime}}(z_{1},...) & f_{2z_{1}}^{^{\prime}}\\
f_{1z_{2}}^{^{\prime}} & f_{2z_{2}}^{^{\prime}}\\
... & ...
\end{array}
)$ are linearly independent and\ that $(z_{1},...,z_{n})\cdot\left(
\frac{\partial f_{1}^{^{\prime}}}{\partial z_{1}}(z),...,\frac{\partial
f_{1}^{^{\prime}}}{\partial z_{n}}(z)\right)  ^{t}$ is $\neq0$ in $z$, compare
the treatment in subchapter 5.2. We can find $\left(  a_{1},...,a_{n}\right)
\in X_{0}^{^{\prime\prime}}$ such that $(z_{1},z_{2})$, $(a_{1},a_{2})$ are
linearly independent, because otherwise $X_{0}^{^{\prime\prime}}$ would lie in
a hyperplane, which cannot be, we have $\det(\mathfrak{A})\neq0$. With other
words: Suppose that for all choices $\mathbf{O=}(0,...,0),$ $a$ and $z$ of
$X_{0}^{"}$ we have $\det(\mathfrak{A})=0$. $\det(\mathfrak{A})$ is of degree
1 with respect to the $a_{i}$. Fix a general $z\in X_{0}^{^{\prime\prime}}$,
such that $\det(\mathfrak{A})=0$ \ is a non trivial linear equation for the
$a_{i}$, $i=1,...,n$. If $\det(\mathfrak{A})=0$\ for all $a\in X_{0}%
^{^{\prime\prime}}$ , $X_{0}^{^{\prime\prime}}$ has to lie in a proper linear
subspace of $X_{0}^{^{\prime}}$, which cannot be. Thus we have outlined an
inductive proof by describing the beginning and the first inductive step. So
for instance if $m=3$, we consider Euler vector fields with centers in
different general points $\mathbf{O=}(0,...,0),$ $a=(a_{1},...,a_{n})$ ,
$b=(b_{1},...,b_{n})$ of $X_{0}^{^{\prime\prime}},$ where $X_{0}%
^{^{\prime\prime}}$ is defined in general points as $t=f_{1}^{^{\prime}}%
=f_{2}^{^{\prime}}=f_{3}^{^{\prime}}=0,$ the reasoning above gives that the
holomorphic liftings of the first two (properly chosen general points) are
linearly independent in general points of $X_{0},$ here we work as above with
$f_{1}^{^{\prime}},f_{2}^{^{\prime}},$ and we let subsequently $(b_{1}%
,...,b_{n})$ vary with fixed $z$, $a$ similarly as above in order to show
$q=3,$employing that $X_{0}^{^{\prime\prime}}$ cannot lie in a proper linear
subspace of $X_{0}^{^{\prime}},$ by considering $\mathfrak{A=}$ $\left(
\begin{array}
[c]{ccc}%
z_{1} & z_{2} & ...\\
a_{1} & a_{2} & ...\\
b_{1} & b_{2} & ...
\end{array}
\right)  \cdot\left(
\begin{array}
[c]{ccc}%
f_{1z_{1}}^{^{\prime}}(z) & f_{2z_{1}}^{^{\prime}} & f_{3z_{1}}^{^{\prime}}\\
f_{1z_{2}}^{^{\prime}} & f_{2z_{2}}^{^{\prime}} & ...\\
... & ... & ...
\end{array}
\right)  .$\newline\newline We have to look for general\ points $z$, $a,b$ on
$X_{0}^{^{\prime\prime}}$ such that $\det\mathfrak{A\neq}0$. Choose $z$, $a$,
such that $\det\mathfrak{A}=0$ leads to a nontrivial linear\ equation in the
$b_{i}$'s. - Here we discussed $m=q=2$ and $\ $(addressed briefly)$\ m=q=3$%
.The cases $m=3$ or even $m=2$\ can be considered already as the general case,
where we encounter multivector fields $\chi(\sigma_{1})\wedge...\wedge
\chi(\sigma_{m})\mid X_{0}$ and have to show $m=q$. The computations in the
case $m=2$, $m=3$ show how to deal with the general case. - It is clear, that
we can extend these results for special $z,a,b,..$to the case of general
$z,a,b$ because of the identity theorem for holomophic functions.\newline%
\newline In general (if nothing else is agreed upon)\ we will suppose in the
following additionally that the Euler vector\ fields on $X^{^{\prime}}$ resp.
$\widehat{X}^{^{\prime}}$\ are vector fields having their centers on
$X_{0}^{^{\prime}}$ resp. $\widehat{X}_{0}^{^{\prime}}$ in general points
of$\ X_{0}^{^{\prime\prime}}$ resp. $\widehat{X}_{0}^{^{^{\prime\prime}}}$.
\newline\newline Thus it can be claimed that $m=q$ and that (the components
of) general $\mu_{0}$-fibers are orbital varieties for all flag manifolds
$F$\ with $b_{2}=1.$ \newline$\newline$\textbf{c. }Clearly in the general (not
only in the hypersurface) case we have\emph{: All vector fields }$\chi_{0}%
$\emph{\ of this type on }$X_{0}$\emph{ commute with each other (\cite{Siu2},
\S 10). Let }$\gamma_{0}^{\ast(1)}$\emph{ be a general orbital curve of }%
$\chi_{0}$\emph{; then} $\chi_{0}|\gamma_{0}^{\ast(1)}$\emph{ is
additive;}compare (the proof of) \cite{Hwang1}, 2.5, prop. 6.(i).\newline

\section{Theorem 2 (and 3), proof of $X_{0}=F$ without assuming $X_{0}$ to be
k\"{a}hler.\newline}

\subsection{On a result of Siu,the structure of general $\mu_{0}$-fibers.}

\textbf{a. Some notions; some statements to be proved. }We continue to use the
notations intruduced above.\textbf{ }$\mu$ is defined by $\tau$ resp. $t$\ and
$s_{0}$,...,$s_{n}\in H^{0}(X,\mathcal{D})$ with associated $s_{0}^{^{\prime}%
}$,...,$s_{n}^{^{\prime}}\in H^{0}(X^{^{\prime}},\mathcal{D}^{^{\prime}})$,
$(Y\times F)\cap(s_{0}^{^{\prime}}\neq0)=Y\times\mathbf{C}^{n}$, $(Y\times
F)\cap(s_{0}^{^{\prime}}=0)$ $=Y\times H_{F}$. Suppose that the Euler vector
fields on $Y\times F$ we work with,$\,\ $considered as vector fields on
$Y\times\mathbf{P}^{n}$, have on $(0)\times\mathbf{P}^{n}$ centers $\in
X_{0}^{^{\prime\prime}}\cap(s_{0}^{^{\prime}}\neq0)$, $F$ is supposed to be a
flag manifold with Picard number 1, if nothing else is clear from the context.
All essential arguments can easily\ be adapted to del Pezzo manifolds of
dimension $\geq3$ and degree $5$, it will be pointed out, if \ non trivial
different arguments have to be employed. \newline\newline First let us suppose
that $\widehat{X}_{0}^{\ast}$ is a complex manifold. Then any (component of a)
general fiber $fib$\ of $\psi|X_{0}^{\ast}$ resp. $\widehat{\psi}|\widehat
{X}_{0}^{\ast}$ is an orbital variety of the type $W(P_{0})=\mathbf{P}^{m},$
described above in 7.a., which can be identified with a general $\mu_{0}%
$-fiber (component) of $X_{0}\rightarrow X_{0}^{^{\prime\prime}}$. Because a
reformulation of \cite{Siu2}, 10.2 by \cite{Hwang1}, p. 331) says: Let $fib$
be a component of a general $\widehat{\mu}_{0}$-fiber. Given $m$ holomorphic
Euler vector fields on $\widehat{X}_{0}^{\ast}$, i.e. holomorphic liftings,
additive holomorphic vector fields, inducing on the moishezon manifold $fib$
an effective $(\mathbf{C}^{+})^{m}$-action, and if each of these\ vector
fields vanishes on a fixed big Cartier divisor of $fib$, then the general
orbital variety $fib$ is biholomorphic to $\mathbf{P}^{m}$\newline\newline
From 4.1.f one gathers: $G_{0}^{\ast}$ cuts every component of a general fiber
of $X_{0}^{\ast}\rightarrow X_{0}^{^{\prime\prime}}$ in codimension one, i.e.
$\widehat{G}_{0}^{\ast}$\ cuts every component of a general fiber of
$\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ in
codimension one, every component $fib$ of a general $\mu_{0}-$fiber cuts
$S^{\ast}$in codimension $1$; a similar statement holds for the other
components $\widehat{X}_{0}^{\ast\ast\ast}$ $\neq\widehat{X}_{0}^{\ast}$ of
$\widehat{G}_{0}$. The arguments of \cite{Siu2} imply that $fib\cap\widehat
{X}_{0}^{\ast\ast\ast}\cap\widehat{G}_{0}^{\ast}=fib\cap\widehat{G}_{0}^{\ast
}$ analytically.\ Therefore it\ suffices for our purposes to discuss the case
$\widehat{G}_{0}=\widehat{G}_{0}^{\ast}\cup\widehat{X}_{0}^{\ast}$ in order to
describe the procedure (, if we wish,\ otherwise we have to replace
$\widehat{G}_{0}^{\ast}$ by the union of the componenets $\neq\widehat{X}%
_{0}^{\ast}$ of $\widehat{G}_{0}$). $\widehat{G}_{0}^{\ast}\cap\widehat{X}%
_{0}^{\ast}$ is on $\widehat{X}_{0}^{\ast}$ a\ Cartier divisor. All
general\ holomorphic liftings $\widehat{\chi}_{0}^{\ast}$ of Euler vector
fields $\widehat{\chi}^{^{\prime}}$ on $Y\times F$ \ vanish on $\widehat
{G}_{0}^{\ast}\cap\widehat{X}_{0}^{\ast}$. These holomorphic liftings
$\widehat{\chi}_{0}^{\ast}$ are additive (compare \cite{Hwang1}, p. 329). For
any general $s\in H^{0}(X,\mathcal{D})$ and any general fiber $fib$ of
$\widehat{\mu}_{0}$\ the intersection $\widehat{\varphi}^{\ast}((s=0))\cap
fib$ is a big Cartier divisor on $fib$ which equals $\widehat{G}_{0}^{\ast
}\cap fib$. \newline\newline Thus Siu's theorem mentioned above (\cite{Siu2},
10.2 and \cite{Hwang1}, p. 331), says: If each holomorphic lifting $\chi$ of
an Euler vector field $\widehat{\chi}^{^{\prime}}$restricted to such an
$m$-dimensional generic orbital variety (i.e. in our case to a component of a
general $\mu_{0}$-fiber) $fib\subset\widehat{X}_{0}^{\ast}$ vanishes on the
intersection of $fib$ and a fixed Cartier divisor\ (here $\widehat{G}%
_{0}^{\ast}\cap fib\subset\widehat{X}_{0}^{\ast}$, which induces on $fib$ a
big divisor, then the general orbital variety $fib$ is biholomorphic to
$\mathbf{P}^{m}$. \newline\newline Essential is: We can look at $fib$ as a
(component of a) fiber of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$. Let $s\in\Gamma(X,\mathcal{D})$ be general; we may
identify\ set theoretically $fib\cap\widehat{G}_{0}^{\ast}$with $fib\cap
(s=0)$. We need that the reduced analytic set $fib\cap\widehat{G}_{0}^{\ast}$
is a $fib$-Cartier divisor, this is the case if analytically (not only set
theoretically) $\ fib\cap(\varphi^{\ast}s=0)$ $=$ $fib\cap\widehat{G}%
_{0}^{\ast}$; we need that there exist minimal rational curves,i.e. curves\ of
degree 1 in $X_{0}$, i.e. lines in $X_{0}$, which lie in general $\mu_{0}%
$-fibers. In general we do not know whether $\widehat{X}_{0}^{\ast}$ is
smooth.\ Up to now we have not yet\ shown $S^{\ast}\subset fib$. - First let
us desingularize $\widehat{X}_{0}^{\ast}$. \newline\newline\textbf{b}.
\textbf{Canonical resolution of singularities, existence of lines in general
}$\mu_{0}-$\textbf{fibers, related matters}. Suppose that $\widehat{X}%
_{0}^{\ast}$ has singular points. According to \cite{bimil}, theorem 13.4
there exists a canonical resolution $\rho:\widehat{X}_{0}^{\ast\ast
}\rightarrow\widehat{X}_{0}^{\ast}$\ of singularities, even a canonical
resolution $\widehat{X}^{\ast\ast}\rightarrow\widehat{G}$; holomorphic
vectorfields on $\widehat{X}_{0}^{\ast}$ can be lifted to $\widehat{X}%
_{0}^{\ast\ast}$ (see \cite{Siu2}, \S 4, Canonical resolution of singularities
and the liftings of vector fields, p. 262 -263), esp. the holomorphic liftings
$\widehat{\chi}_{0}^{\ast}$ on $\widehat{X}_{0}^{\ast}$\ of Euler vector
fields $\widehat{\chi}^{^{\prime}}$ on $\widehat{X}^{^{\prime}}$\ to
holomorphic vector fields $\widehat{\chi}_{0}^{\ast\ast}:=\rho^{\ast}%
\widehat{\chi}_{0}^{\ast}$ on $\widehat{X}_{0}^{\ast\ast}$, these are
additive, we have $\rho_{\ast}(\widehat{\chi}_{0}^{\ast\ast})=\widehat{\chi
}_{0}^{\ast}$, $\widehat{\varphi}_{\ast}($ $\rho_{\ast}(\widehat{\chi}%
_{0}^{\ast\ast}))=\chi_{0}$,\ certainly $\widehat{\chi}_{0}^{\ast\ast}$ has a
(reduced) zero divisor $(\widehat{s}=0)$\ lying over and\ coming from
$\widehat{G}_{0}^{\ast}\cap\widehat{X}_{0}^{\ast}$; here $\widehat{s}$ is a
section in a suitable line bundle vanishing with order one along the inverse
image of $\widehat{G}_{0}^{\ast}\cap\widehat{X}_{0}^{\ast}$. Let $P^{\ast\ast
}$ be a general point of $(\widehat{s}=0)$, $P=\widehat{\varphi}(\rho
(P^{\ast\ast}))$ its image in $S^{\ast}$. Let $\gamma^{\ast\ast}$ be an
orbital curve of $\widehat{\chi}_{0}^{\ast\ast}$ through\ $P^{\ast\ast}$;
$\widehat{\chi}_{0}^{\ast\ast}|\gamma^{\ast\ast}$ has 2 zeros (we count with
multiplicities), $\frac{1}{\widehat{s}}\widehat{\chi}_{0}^{\ast\ast}$ is a
line bundle valued holomorphic vector field on $\widehat{X}_{0}^{\ast\ast}$ in
an $\widehat{X}_{0}^{\ast\ast}-$neighborhood of $\gamma^{\ast\ast}$ having on
$\gamma^{\ast\ast}$ in $P^{\ast\ast}$ a zero\ with order at most one.\ At
least 2 different orbital curves of $\chi_{0}$ meet in $P$ because of $P\in
S^{\ast}=\widehat{\varphi}(\widehat{G}_{0}^{\ast})$. Therefore the image of
$\frac{1}{\widehat{s}}\widehat{\chi}_{0}^{\ast\ast}$ in $P$ has a zero in $P$,
consequently $(\widehat{s}=0)$ meets $\gamma^{\ast\ast}$ in $P^{\ast\ast}$
with order one. This coincides with the additividy of $\widehat{\chi}%
_{0}^{\ast\ast}|\gamma^{\ast\ast}$. Any general fiber \ $fib$\ of
$\ \widehat{\psi}\circ\rho:\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat
{X}_{0}^{^{\prime\prime}}$ is smooth and decomposes into disjunct components .
- In order to have a short notation let us use the same notation for
holomorphic vector fields on $\widehat{X}_{0}^{\ast}$ and the corresponding
vector fields on $\widehat{X}_{0}^{\ast\ast}$. Let us call vector fields on
$\widehat{X}_{0}^{\ast\ast}$ holomorphic liftings if the corrsponding vector
fields on $\widehat{X}_{0}^{\ast}$ are of this type. \newline\newline%
$\widehat{\chi}_{0}^{\ast\ast}|\gamma^{\ast\ast}$ has analytically two zeros,
i.e. a double zero, and this double zero is situated in $\gamma^{\ast\ast}%
\cap\rho^{-1}(\widehat{G}_{0}^{\ast})$. Other orbital curves $\subset fib$ of
$\widehat{\chi}_{0}^{\ast\ast}$ meet $\gamma^{\ast\ast}$ at most in
$\gamma^{\ast\ast}\cap\rho^{-1}(\widehat{G}_{0}^{\ast})$. Have in mind: Let
$l^{^{\prime}}$ be an orbital curve of $\widehat{\chi}_{0}^{^{\prime}}$
through $\psi(\rho(P^{\ast\ast}))$. $(\widehat{\psi}\circ\rho)^{-1}[Y\times
l^{^{\prime}}]\cap fib$ must be connected, any pair of different\ components
of this analytic set meet each other at most in a point of $\rho^{-1}%
(\widehat{G}_{0}^{\ast})$. \newline\newline According to chap. 7 we can find
on $\widehat{X}^{\ast\ast}$ holomorphic liftings $\chi_{1}$,...,$\chi_{m}$ of
Euler vector fields on $X^{^{\prime}}$ with the following properties: There is
a point $P_{0}\in fib$ outside $\rho^{-1}(\widehat{G}_{0}^{\ast})$, such that
$\chi_{1}$,...,$\chi_{m}$ are linearly independent in $P_{0}$. Any set of
$m+1$ holomorphic liftings of this type is linearly dependent in all points of
$fib$. Each $\chi_{i}$ has exactly $\rho^{-1}(\widehat{G}_{0}^{\ast}%
)\cap\widehat{X}_{0}^{\ast\ast}$ as zero divisor. The $\chi_{i}$, $i=1,...,m$,
commute with each other. $fib$ is the topological closure of the corresponding
integral manifold in $fib$, defined by the $\chi_{i}|fib$, $i=1,...,m$.
\newline\newline The arguments of \cite{Siu2}, esp. \S 7-\S 10, imply that
(the components of) general fibers of $\widehat{X}_{0}^{\ast\ast}%
\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are $\mathbf{P}^{m}$'s and that
$fib\cap\rho^{-1}(\widehat{G}_{0}^{\ast})$ can be identified with the infinity
hyperplane of $\mathbf{P}^{m}$\textbf{,} the integral manifold alluded to can
be identified with $fib\backslash$\ $\rho^{-1}(\widehat{G}_{0}^{\ast})$,
compare also \cite{Hwang1}, p. 331.\newline\newline Siu's arguments apply to
$\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}},$
regardless whether the divisor $\widehat{G}_{0}^{\ast}\cap\widehat{X}%
_{0}^{\ast}$ is a Cartier divisor on $\widehat{X}_{0}^{\ast}$ or not, as the
proper inverse image of $\widehat{G}_{0}^{\ast}\cap\widehat{X}_{0}^{\ast}$ on
$\widehat{X}_{0}^{\ast\ast}$ induces a big Cartier\ divisor on (the components
of) any general fiber of $\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$; thus the components of general fibers of $\widehat
{X}_{0}^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are
$\mathbf{P}^{m}$'s and their intersections with $\rho^{-1}(\widehat{G}%
_{0}^{\ast})$ are the infinity hyperplanes. \ Suppose that general\ fibers of
\ $\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are
irreducible; then over general points of $\widehat{X}_{0}^{^{\prime\prime}}$
the space $\widehat{X}_{0}^{\ast\ast}$\ is a $\mathbf{P}^{m}$-bundle.
\newline\newline Now let $fib$ be a general fiber (resp. a component of a
general fiber) of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}%
^{^{\prime\prime}}$; then the induced map $\rho^{-1}(fib)\rightarrow fib$ is
discrete, i.e. $\rho^{-1}(fib)\rightarrow fib$ is the normalization map.
Because suppose that the induced map is not discrete. Then there exists an
irreducible curve in $\rho^{-1}(fib)$ the image of which in $X_{0}$ is zero
homologous. But then the images of all curves $\subset\rho^{-1}%
(fib)=\mathbf{P}^{m}$ in $X_{0}$ are zero homologous, as $H_{2}(\mathbf{P}%
^{m},\mathbf{Z)=Z}$. Consequently the induced map $\rho^{-1}(fib)$
$\rightarrow$ $fib$ is the normalization map. For our purposes we may even
suppose that $\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\ast}}$
is the normalization map; one can assume without doing harm that in a
neighborhood of $fib$ the graph $\widehat{G}^{\ast}$ decomposes only in
$\widehat{G}_{0}^{\ast}$ and $\widehat{X}_{0}^{\ast}$. The validity of these
statements depend decisively on the fact, that holomorphic liftings on
$\widehat{X}_{0}^{\ast\ast}$\ of Euler vector fields are additive on their
orbital curves.\newline\newline As above the general\ additive holomorphic
vector fields $\chi_{0}$\ on these general fibers have only zeros along the
intersection of these fibers with $\widehat{G}_{0}^{\ast}$, $\nu$ is
biholomorphic over $fib\backslash\widehat{G}_{0}^{\ast}$ ($=\mathbf{C}^{m}$).
Different general components of $\mu_{0}$-fibers (considered as a subset of
$\widehat{X}_{0}^{\ast}$)\ meet at most in points of $\widehat{G}_{0}^{\ast}%
$.\newline\newline\textbf{c. On the irreducibility and smoothness of fibers.
}In the above we worked only formally with the assumption that general fibers
of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ should
be irreducible.. - Let us drop the formal assumption that general fibers of
$\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ are
irreducible. We employ Siu's theorem mentioned above. Different irreducible
components $fib1$ of a general fiber $fib$\ of $\widehat{X}_{0}^{\ast\ast
}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ \ meet at most in points over
$\widehat{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$. On $\widehat{X}_{0}^{\ast
}$ there are$\ m$ independent holomorphic liftings $\widehat{\chi}_{0,i}%
^{\ast}$, $i=1$,..., $m$, such that the lifting $\Upsilon$ of$\ \widehat{\chi
}_{0,1}^{\ast}\wedge...\wedge\widehat{\chi}_{0,m}^{\ast}$ to $\widehat{X}%
_{0}^{\ast\ast}$ has a divisorial zero $d^{\ast}$, such that $d^{\ast}$
induces on every component of a general fiber $fib1$ of $\widehat{X}_{0}%
^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$, a $\mathbf{P}^{m}$,
the $(m+1)-$fold \ hyperplane $fib1\cap\rho^{\ast}(\widehat{G}_{0}^{\ast})$.
$\ \frac{1}{d^{\ast}}\Upsilon$ has no zero on $fib1$, $\rho_{\ast}(\frac
{1}{d^{\ast}}\Upsilon)$ has no zero on $\rho(fib1)$, which cannot be if
$\rho(fib1)$meets other\ components. The same reasoning even gives that
general fibers of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}%
^{^{\prime\prime}}$ are smooth, i.e. are $\mathbf{P}^{m}$'s.

\subsection{$S^{\ast}$ is of dimension $m-1$ and is contained in all $\mu_{0}%
$-fibers.}

In chapter 5 we already\ showed $\dim S^{\ast}=0$ for $m=1$. We will have to
make sure $\dim S^{\ast}=m-1$ for all possible $m$. \newline\newline General
$\mu_{0}$-fibers $fib$ are $\mathbf{P}^{m}$'s, according to the above. For our
purposes we may suppose $\widehat{X}_{0}^{\ast}$ to be smooth.\ Because of
4.1.f we already know that $fib\cap\widehat{G}_{0}^{\ast}$ is of dimension
$m-1$. It is enough to show that two different\ general $\mu_{0}%
-$fibers\ which meet in a point of $S^{\ast}$, have in common their
intersections with $S^{\ast}$.- Let $\chi$ be the holomorphic lifting to
$X$\ of a general holomorphic Euler\ vector field on $Y\times F$,
$\widehat{\chi}^{\ast}$ its holomorphic lifting\ to $\widehat{G}$,
$\widehat{\chi}_{0}^{\ast}$ its restriction to $\widehat{X}_{0}^{\ast}$;
$\chi_{0}$ is additive and vanishes along $S^{\ast}$. Let $fib1$, $fib2$
($\mathbf{P}^{m}$'s) be two general fibers\ of $\widehat{X}_{0}^{\ast
}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$, which meet in $P\in S^{\ast}%
$.\newline\newline There exists a bimeromorphic, general fibers preserving map
$\kappa:\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}%
}\times\mathbf{P}^{m}$, we may suppose that the induced restriction\ map
$\kappa|fib1:fib1\rightarrow\kappa\lbrack fib1]$ (concerning the notion of
restrictions of meromorphic maps compare the introduction, for reasons of the
generality of the fibers involved we may suppose that the restriction
$\kappa|fib1$ exists) maps $fib1$\ bimeromorphically onto $\mathbf{P}^{m}$
(i.e.$\kappa\lbrack fib1]=\mathbf{P}^{m}$); we may even\ suppose that $\kappa$
is biholomorphic\ in a neighborhood of the general fiber $fib1$ and that
consequently\ the restriction $\kappa|fib1:fib1\rightarrow\kappa\lbrack
fib1]=\mathbf{P}^{m}$ is biholomorphic, i.e linear: \newline\newline For
general $s\in\Gamma((X,\mathcal{D})$ the zero set$\ \widehat{\varphi}^{\ast
}(s=0)$ has a component $L^{\ast}$ which induces on general fibers $fib$ of
$\widehat{X}_{0}^{\ast}\rightarrow$ $\widehat{X}_{0}^{^{\prime\prime}}$ (these
are $\mathbf{P}^{m}$'s) a hyperplane; here count $L^{\ast}$ with multiplicity
one, although $\widehat{\varphi}^{\ast}s$ may vanish with order $>1$ along
$L^{\ast}$. Let $\mathcal{L}^{\ast}$ be the line bundle on $\widehat{X}%
_{0}^{\ast}$ corresponding to $L^{\ast}$. According to Grauert's image sheaf
theorem ( \cite{Grauert-Remmert}) the image sheaf $\widehat{\psi}%
_{(0)}\mathcal{L}^{\ast}$ is coherent. As $\widehat{X}_{0}^{^{\prime\prime}}$
is projective, the sheaf $\ \widehat{\psi}_{(0)}\mathcal{L}^{\ast}%
\otimes\mathcal{D}^{^{\prime}\widetilde{n}}$ is for large $\widetilde{n}$ over
$\widehat{X}_{0}^{^{\prime\prime}}$ generated by global sections. There exist
elements $\widetilde{s}_{0},...,\widetilde{s}_{m}\in$ $\Gamma(\widehat{X}%
_{0}^{^{\prime\prime}},\widehat{\psi}_{(0)}\mathcal{L}^{\ast}\otimes
\mathcal{D}^{^{\prime}\widetilde{n}})$, which generate (for large
$\widetilde{n}$) $\widehat{\psi}_{(0)}\mathcal{L}^{\ast}\otimes\mathcal{D}%
^{^{\prime}\widetilde{n}}$ in general points $P^{^{\prime}}\in\widehat{X}%
_{0}^{^{\prime\prime}}$. There are $\widetilde{s}_{0}^{\ast},...,\widetilde
{s}_{m}^{\ast}\in\Gamma(\widehat{X}_{0}^{\ast},\mathcal{L}^{\ast}%
\otimes\widehat{\psi}^{\ast}\mathcal{D}^{^{\prime}\widetilde{n}})$ the images
of which are $\widetilde{s}_{0},...,\widetilde{s}_{m}$. The
elements$\ \widetilde{s}_{0}^{\ast},...,\widetilde{s}_{m}^{\ast}$ define a
meromorphic map $\mu^{\ast}:\widehat{X}_{0}^{\ast}\rightarrow\mathbf{P}^{m}$,
with $P^{\ast}\rightarrow(\widetilde{s}_{0}^{\ast}(P^{\ast}),...,\widetilde
{s}_{m}^{\ast}(P^{\ast}))$ for general $P^{\ast}$. The bimeromorphic map
$\widetilde{\psi}\times\mu^{\ast}:\widehat{X}_{0}^{\ast}\rightarrow\widehat
{X}_{0}^{^{\prime\prime}}\times\mathbf{P}^{m}$ has the wanted properties. -
Thus the assertions for $\kappa$ are established and we can continue the proof
of $\dim S^{\ast}=m-1$. \newline\newline In the following we will use: Let
$\mathbf{P}^{m}$, $m\geq2$, be provided with a homogeneous $(p_{0},...,p_{m}%
)$-coordinate system. Let $\eta$ be an additive holomorphic vector field (not
identically zero) on $\mathbf{P}^{m}$, vanishing in all points of $(p_{0}=0)$.
Then $\frac{1}{p_{0}}\cdot\eta$ is $\neq0$ in general points of $(p_{0}=0)$;
all orbital curves of $\frac{1}{p_{0}}\cdot\eta$ through such general points
of $(p_{0}=0)$ lie in $(p_{0}=0)$. \newline\newline Now let $P^{\ast}$ be a
general point of $\widehat{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$,
$P:=\widehat{\varphi}(P^{\ast})\in S^{\ast}$; there are arbitrarily small
$\widehat{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$- neighborhoods $U^{\ast}%
$\ of $P^{\ast},$ such that $U:=\widehat{\varphi}(U^{\ast})$ is an $S^{\ast}%
$-neighborhood of $P$. Let $\widehat{X}_{0}^{\ast}\cap\widehat{G}_{0}^{\ast}$
be given in an $\widehat{X}_{0}^{\ast}$\ neigbourhood be given by $h=0$,
$\frac{1}{h}\cdot\widehat{\chi}_{0}^{\ast}$ is holomorphic in such
neighborhoods $U^{\ast}$ which are small enough. As $\widehat{\chi}_{0}^{\ast
}$ is additive and as general fibers $fib1$ are $\mathbf{P}^{m}$'s ,\ the zero
set of $\widehat{\chi}_{0}^{\ast}|fib1$ is $fib1\cap\widehat{G}_{0}^{\ast}%
$,$\ (\frac{1}{h}\cdot\widehat{\chi}_{0}^{\ast})|U^{\ast}$ exists and does not
vanish in all points of $U^{\ast}$; $\widehat{\varphi}_{\ast}((\frac{1}%
{h}\cdot\widehat{\chi}_{0}^{\ast})|U^{\ast})$ does not vanish in all points of
$U$. We can arrange matters, such that $\widehat{\varphi}_{\ast}((\frac{1}%
{h}\cdot\widehat{\chi}_{0}^{\ast})|U^{\ast})$ has no zero in $U$. But this
implies easily: If such fibers $fib1$, $fib2$ have a point of $U$ in common,
then $fib1\cap S^{\ast}=fib2\cap S^{\ast}$. This gives: All general\ fibers
$fib$\ of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$
which possess a common general\ point of $S^{\ast}$ have a common intersection
with $S^{\ast}$; here have in mind, that we can consider $fib$ as part of
$X_{0}$. Thus we proved $\dim S^{\ast}=m-1$ for $m\geq1$. For $m\geq1$ we
proved $\dim S^{\ast}=m-1$, without using assumptions like$\ X_{0}$ being
projective, or that general fibers of $\widehat{X}_{0}^{\ast}\rightarrow
\widehat{X}_{0}^{^{\prime\prime}}$ contain curves the images of which in
$X_{0}$ are minimal rational curves, i.e. lines. - For notational reasons let
us continue to\ suppose that $\widehat{X}_{0}^{\ast}$ and that general fibers
$fib$ of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}%
$\ are smooth\newline\newline

\subsection{Proof of $X_{0}=F$ without assuming $X_{0}$ to be k\"{a}hler}

\textbf{a. }\emph{The next aim is to show that general }$\mu_{0}$\emph{-fibers
contain minimal rational curves.}\textbf{ }It suffices to discuss the case
$m>1$, i.e. the case $\dim S^{\ast}>0$, as the case $m=1$ is already\ dealt
with. We may suppose that $t$ vanishes along all components of $\widehat
{G}_{0}$ with order $1$, as this can be arranged by a suitable base change.
Next we discuss a situation similar to the one in chap. 6, but the role of
$\widehat{Q}^{\ast}$ differs; we arrange the following situation:\newline%
\newline Choose general\ points $\widehat{P}^{\ast}$, $\widehat{Q}^{\ast}%
\in\widehat{G}_{0}^{\ast},$ such that $\widehat{P}^{`}:=\widehat{\psi}($
$\widehat{P}^{\ast})\neq\widehat{Q}^{^{\prime}}:=\widehat{\psi}^{^{\prime}}($
$\widehat{Q}^{\ast})$ and that $\widehat{P}^{^{\prime}}$, $\widehat
{Q}^{^{\prime}}$ can be joined in $\widehat{X}_{0}^{\prime}$ by a line
$l^{^{\prime}}$ to be specified in the following. $l^{^{\prime}}$ shall
not\ meet $\widehat{X}_{0}^{^{\prime\prime}}$ which can be assumed, see below,
because of $\dim\widehat{X}_{0}^{^{\prime\prime}}\leq n-2$. \newline\newline
Choose a curve $\widehat{C}_{1}^{\ast}$ through $\widehat{P}^{\ast}$, mapped
by $\tau$ biholomorphically onto a neighborhood (which we suppose to be $Y$)
of $0\in Y$; let $\widehat{C}_{1}^{^{\prime}}:=\widehat{\psi}(\widehat{C}%
_{1}^{\ast})$, it will do no harm to suppose $\widehat{C}_{1}^{^{\prime}%
}=Y\times$ $\widehat{P}^{^{\prime}}$. Furthermore let $\widehat{C}^{\ast}$ be
a curve\ through $\widehat{Q}^{\ast}$, mapped by $\tau$ biholomorphically onto
a neighborhood (which we suppose to be $Y$) of $0\in Y$; let $\widehat
{C}^{^{\prime}}:=\widehat{\psi}(\widehat{C}^{\ast})$, it will do no harm to
suppose $\widehat{C}^{^{\prime}}=Y\times$ $\widehat{Q}^{^{\prime}}$%
.\newline\newline Let $L^{^{\prime}}=Y\times l^{^{\prime}}$, $L^{\ast
}=\widehat{\psi}^{-1}[L^{^{\prime}}]$, $L=\widehat{\varphi}(L^{\ast})$.
$L_{0}^{\ast}$ has exactly one component $L_{0}^{\ast(0)}$ with $\widehat
{\psi}(L_{0}^{\ast(0)})=L_{0}^{^{\prime}}$. We can choose $l^{^{\prime}}$ such
that $L_{0}^{\ast(0)}$does not lie in a fiber of $\widehat{G}_{0}^{\ast
}\rightarrow S^{\ast}$. This follows for instance from 3.5; compare for
instance 6.2. If there are no further components of $L_{0}^{\ast}$ (this is
the case if $X_{0}$ is k\"{a}hlerian), the image of $L_{0}^{\ast}=L_{0}%
^{\ast(0)}$ in $X_{0}$\ generates $H_{2}(X_{0},\mathbf{Z)=Z}$. We can identify
$S^{\ast}$ with $fib\cap\widehat{G}_{0}^{\ast}$, where $fib$ is a general
$\mu_{0}$- resp. $\widehat{\mu}_{0}$-fiber; with $fib$ = $\mathbf{P}^{m}$ and
$S^{\ast}$= a hyperplane in $\mathbf{P}^{m}$. \newline\newline The image of
this component $L_{0}^{\ast(0)}$ in $X_{0}$ lies in $S^{\ast}$ and thus can be
moved around in a general $\mu_{0}$-fiber $fib=\mathbf{P}^{m}$, without
changing the homology class in $X_{0}$, and consequently is a positive
multiple of the positive generating element of $H_{2}(X_{0},\mathbf{Z)=Z}$. If
there are no further components of $L_{0}^{\ast}$ (this is the case if $X_{0}$
is k\"{a}hlerian), the image of $L_{0}^{\ast}=L_{0}^{\ast(0)}$ in $X_{0}%
$\ generates $H_{2}(X_{0}$, $\mathbf{Z)=Z}$.. \newline\newline\textbf{b.}
Suppose that there are further components $L_{0}^{\ast(1)}$ of $L_{0}^{\ast}$;
their images in $\widehat{X}_{0}^{^{\prime}}$ are points. We may assume that
$L_{0}^{\ast(0)}$ does not meet $\widehat{X}_{0}^{\ast}$. This can be
arranged, as $F$ is a flag manifold with $b_{2}=1$, esp. homogeneous, and as
$\dim\widehat{X}_{0}^{^{\prime\prime}}\leq n-2$. Therefore the line
$l^{^{\prime}}$ can be chosen, such that this line does not meet $\widehat
{X}_{0}^{^{\prime\prime}}$:\newline\newline Suppose $l^{^{\prime}}%
\nsubseteq\widehat{X}_{0}^{^{\prime\prime}}$ and that there are intersection
points of $l^{^{\prime}}$and $\widehat{X}_{0}^{^{\prime\prime}}$; it is no
restriction to suppose these to be smooth points of $\widehat{X}_{0}%
^{^{\prime\prime}}$. Let $(\widetilde{f}^{^{\prime}}=0)$ be a hypersurface
section of\ $\widehat{X}_{0}^{^{\prime}}$ of degree $\delta$, which meets
$l^{^{\prime}}$ pointwise and contains $\widehat{X}_{0}^{^{\prime\prime}}$.
$l^{^{\prime}}$ meets $(\widetilde{f}^{^{\prime}}=0)$ in $\delta$ points,
counted with multiplicities; suppose there is a common intersection point
$O^{^{\prime}}$\ of $l^{^{\prime}}$ and $\widehat{X}_{0}^{^{\prime\prime}}%
$.\ Loosely speaking, one can move by biholomorphic maps $\widehat{X}%
_{0}^{^{\prime}}\rightarrow$ $\widehat{X}_{0}^{^{\prime}}$, arbitrarily close
to the identity map, the line $l^{^{\prime}}$\ away from $\widehat{X}%
_{0}^{^{\prime\prime}}$ in a neighborhood of $O^{^{\prime}}$, because of
$\dim\widehat{X}_{0}^{^{\prime\prime}}\leq n-2$, without increasing the number
of the remaining common intersection points of $l^{^{\prime}}$, $\widehat
{X}_{0}^{^{\prime\prime}}$. The same idea in a slightly different phrasing: In
order to have a concrete general example, suppose $\dim\widehat{X}%
_{0}^{^{\prime\prime}}=n-2$. We can arrange the situation, where there is a
$2$-dimensional irreducible\ variety $V\subset F$ containing $l^{\prime}$ and
meeting $\widehat{X}_{0}^{^{\prime\prime}}$ at most pointwise\ in a finite
number of points, say $\rho$, counted with multiplicity. If we move $V$ a
little bit, this number $\rho$ does not change. Via moving $V$ by suitable
biholomorphic maps $F\rightarrow F$ arbitrarily close to the identity can get
in the situation, where $l^{^{\prime}}$ and $\widehat{X}_{0}^{\prime\prime}$
are disjunct. \newline\newline Consequently because of 4.1.e. the
"intersection number" $\frac{1}{t^{\sigma}}\widehat{\varphi}^{\ast}%
\mathcal{D}_{0}|L_{0}^{\ast(1)}$ is zero; here the vanishing order$\ \sigma$
has the meaning of 4.1.e; for general $s\in H^{0}(X,\mathcal{D})$ the
holomorphic section $\frac{1}{t^{\sigma}}\widehat{\varphi}^{\ast}s|L_{0}%
^{\ast(1)}$ has no zeroes and is holomorphic, because we may assume
$\widehat{\psi}^{-1}[(s^{^{\prime}}=0)]\cap L_{0}^{\ast(1)}=\varnothing$, here
$s^{^{\prime}}\in\Gamma(\widehat{X}^{^{\prime}},\widehat{\mathcal{D}%
}^{^{\prime}})$ is associated to $s$. Therefore the image $\widehat{\varphi
}(L_{0}^{\ast(1)})$ is zero homologous in $X_{0}$. As for reasons of
continuity the image of $L_{0}^{\ast}$ in $X_{0}$ represents the generating
element of $H_{2}(X_{0},\mathbf{Z)}$, the only possibility is $\sigma=1$;
$L_{0}^{\ast(0)}$ is of degree one in $X_{0}$, lies in all $\mu_{0}-$fibers,
let $fib=\mathbf{P}^{m}$ be a general one, all $\mathbf{P}^{m}$ lines in $fib$
are lines in $X_{0}$. $.$\newline\newline\emph{Remark}. In the above we
supposed $F$ to be a flag manifold with Picard number 1. The homogenity of $F$
allowed to choose $l^{^{\prime}}=L_{0}^{^{\prime}}$, such that $L_{0}%
^{^{\prime}}$ does not meet $\widehat{X}_{0}^{^{\prime\prime}}$. If $F$ is a
del Pezzo manifold $F_{n}$ of dimension $n\geq3$ and degree 5, it seems that
we have to look for another argument. We have only to discuss the spaces
$F_{4}$ and $F_{5}$. For $F_{4}$ one has to discuss the case $\dim\widehat
{X}_{0}^{^{\prime\prime}}=2$ and for $F_{5}$ the cases $\dim$ $\widehat{X}%
_{0}^{^{\prime\prime}}=2$ and $3$. As any general hyperplane section of
$F_{n}$ delivers $F_{n-1}$ (if $4\leq n\leq6$), we can cut down $F_{n}$ to
$F_{3}=V_{5}$ and $\widehat{X}_{0}^{^{\prime\prime}}$ to a curve or a finite
set of points. Therefore it suffices to solve the following problem: Given a
curve $\overline{\gamma}\subset V_{5}$, find a line $l^{^{\prime}}$ in $V_{5}%
$, which does not meet $\overline{\gamma}$. - Any point of $V_{5}$ lies on a
line ; each point lies on only finitely many lines (naturally one knows more,
this number of lines is $\leq3$, see \cite{Furushnaka}, p. 118), the family of
lines in $V_{5}$ is a $\mathbf{P}^{2}$, esp. of dimension $2$; therefore we
have plenty of choices to choose a line $l^{^{\prime}}$ with the wanted
property, as the family of lines meeting $\overline{\gamma}$ is at most of
dimension 1, a general line $l^{^{\prime}}$ will not meet $\overline{\gamma}%
$.- Now let us go back to $F=$ a flag manifold with $b_{2}=1$. \newline%
\newline c.\emph{ Therefore we know that general }$\mu_{0}$\emph{-fibers
contain minimal rational curves. }- Some of the following arguments can be
found in chap. 6.3, it seems to be more convenient not just to refer to chap 6
but to reproduce the relevant reasoning a second time.-\ There is only to
prove something for $F\neq\mathbf{P}^{n}$. Suppose that any general fiber
$fib=\mathbf{P}^{m}$\ of $\widehat{X}_{0}^{\ast}\rightarrow\widehat{X}%
_{0}^{^{\prime\prime}}$ contains a minimal rational curve. Then $\widehat
{X}_{0}^{\ast}\rightarrow\widehat{X}_{0}^{^{\prime\prime}}$ is over general
points of $\widehat{X}_{0}^{^{\prime\prime}}$ a $\mathbf{P}^{m}$-bundle. Take
a general line $l$ in such a $\mathbf{P}^{m}$, $l$ is in $X_{0}$ a minimal
rational curve, i.e. an irreducible curve of degree $1$ in $X_{0}$, a rational
curve, as for general $s\in\Gamma(X,\mathcal{D})$ the zero set $(s=0)$ meets
$l$ analytically just once (in $l\cap S^{\ast}$); we have $T(\mathbf{P}%
^{m})|l\backsimeq\mathcal{O(}2)\oplus\mathcal{O}(1)^{m-1}$, therefore we can
assert $\mathfrak{N}_{l\backslash\mathbf{P}^{m}}\backsimeq$\ $\mathcal{O}%
(1)^{m-1}\,$.\newline\newline Proof: Provide $\mathbf{P}^{m}$ with
inhomogeneous coordinates $z_{1},...,z_{m}$ , consider the vector fields
$\varrho_{1}:=$ $z_{1}(z_{1}\cdot\frac{\partial}{\partial z_{1}}+$ $...$
$+z_{m}\cdot\frac{\partial}{\partial z_{m}})$, $\varrho_{2}:=z_{1}\cdot
\frac{\partial}{\partial z_{2}}$,..., $\varrho_{m}:=z_{1}\frac{\partial
}{\partial z_{m}}$ and their $\Lambda-$product$.$This gives the assertion.
\newline\newline Similarly as in chapter 5, where $\dim S^{\ast}=0$ was
discussed, one shows that the normal bundle of $l$ in $X_{0}$, is generated by
sections having a zero in $l\cap S^{\ast}$ and are linearly independent in all
other points of $l$; invoking the adjunction formula\ 2.5.$\beta$ and a
theorem of Grothendieck on splitting of vector bundles we get $c_{1}%
(X_{0})=c_{1}(F)=n+1$, a contradiction to $c_{1}(F)<n+1$; it suffices to show
$c_{1}(X_{0})\geq n+1$, suppose that $(s=0)\cap fib=S^{\ast}$ resp.
$\widehat{\varphi}(\widehat{G}_{0}^{\ast}\cap fib)=S^{\ast}$ and that $(s=0)$
meets $l$ just once and this in $l\cap\widehat{G}_{0}^{\ast}$:\newline\newline
There is a smooth\ $\widehat{X}_{0}^{^{\prime\prime}}-$\ neighborhood $U$ of
$P":=\widehat{\psi}(fib)$ such that $\widehat{\psi}^{-1}(U)\cap\widehat{X}%
_{0}^{\ast}$ can be written as $fib\times U$ and such that $U$ carries a
$(w_{1},...,w_{n-m})-$coordinate system centered in $P"$. Extend
$\frac{\partial}{\partial w_{i}},i=1,...,n-m$, trivially to vector fields on
$fib\times U$, call these extended vector fields too $\frac{\partial}{\partial
w_{i}}$, extend the $\rho_{j}$ too trivially to $fib\times U$, $j=1,...,m$.
The bundle $T(X_{0})|l$ is generated by $\widehat{\varphi}_{\ast}(\rho
_{1}),\widehat{\varphi}_{\ast}(\varrho_{2}),...,\widehat{\varphi}_{\ast
}(\varrho_{m})$, $\widehat{\varphi}_{\ast}(\frac{\partial}{\partial w_{1}}%
)$,..., $\widehat{\varphi}_{\ast}(\frac{\partial}{\partial w_{n-m}})$ and
equals the bundle$\ \mathcal{O(}2)\oplus\mathcal{O}(1)^{n-1}$ over the
rational curve $l$. $T(X_{0})|l$ equals $T(l)\oplus\mathfrak{N}_{l/X_{0}}$.
The normal bundle $T(X_{0})|l\backslash T(l)$ $=\mathfrak{N}_{l/X_{0}}$ of $l$
in $X_{0}$ \ is generated over $l$ by (the restrictions\ of) $\widehat
{\varphi}_{\ast}(\varrho_{2}),...,\widehat{\varphi}_{\ast}(\varrho_{m})$,
$\widehat{\varphi}_{\ast}(\frac{\partial}{\partial w_{1}})$,...,
$\widehat{\varphi}_{\ast}(\frac{\partial}{\partial w_{n-m}})$, these vector
fields have in common just one point $\in S^{\ast}$. This implies the
contradictionary statement$\ c_{1}(X_{0})=c_{1}(F)=n+1$, because $l$ generates
$H_{2}(X_{0},\mathbf{Z)=Z}$ and\ the degree of $l$ with respect to anti
canonical.line bundle of $X_{0}$ equals $n+1$ On the other hand we suppose
here $c_{1}(X_{0})=c_{1}(F)<n+1$\ Thus theorem 2 is proved.\newline%
\newline\textbf{d. }It is clear, that the above can be adapted to a proof of
theorem 3. Compare in this respect the remark in subsection b above,\ 3.5 and
the properties of del Pezzo manifolds of degree 5 listed in 3.6. Thus we get a
reduction to $\dim S^{\ast}=0$, i.e. to the hypersurface case, where general
holomorphic liftings of Euler vector fields have divisorial liftings. But
because of \cite{Hwang1}, \cite{Hwang2} and/or chapter 5 above this can only
happen for $F=\mathbf{P}^{n}$.

\section{\textbf{Appendix. Some notations and symbols which occur often in the
same or similar sense. }}

\textbf{Some notations in connection with the bimeromorphic map }$\mu$. See
(for instance) 1.1, 1.2.a and 4.1. $Y=\{|t|<1\},$ $\tau:X\rightarrow Y$ is a
deformation map, $\dim X_{0}=n,$ $X^{^{\prime}}=Y\times\mathbf{P}^{n}$,.
$\widehat{X}^{^{\prime}}=Y\times F$, $F=$ flag manifold with $b_{2}=1$ or a
del Pezzo manifold of degree 5 and dimension $n\geq3$, $\mu:X\rightarrow
X^{^{\prime}}$ and $\widehat{\mu}:X\rightarrow Y\times F$ are certain
bimeromorphic maps with graphs $G$ and $\widehat{G}$, $X_{0}^{\ast}$= the
component of $G_{0}$ which is a proper modification of $X_{0}$ i.e. is the
graph of the induced meromorphic map $\mu_{0}:X_{0}\rightarrow\mathbf{P}^{n}$,
$G_{0}^{\ast}$= the component of $G_{0}$ which is a modification of
\ $X_{0}^{^{\prime}}=\mathbf{P}^{n}$, $\widehat{X}_{0}^{\ast}$= the component
of $\widehat{G}_{0}$ which is a modification of $\widehat{X}_{0}$ i.e. is the
graph of the induced meromorphic map$\ \mu_{0}:X_{0}\rightarrow F$,
$\widehat{G}_{0}^{\ast}$= the component of $\widehat{G}_{0}$ which is a
modification of \ $\widehat{X}_{0}^{^{\prime}}=F$, $\varphi:G\rightarrow X$
and $\psi:G\rightarrow X^{^{\prime}}$and $\widehat{\varphi}:\widehat
{G}\rightarrow X$ and $\widehat{\psi}:\widehat{G}\rightarrow\widehat
{X}^{^{\prime}}$ are the projection maps, $X_{0}^{^{\prime\prime}}=\psi
(X_{0}^{\ast})$, $\widehat{X}_{0}^{^{\prime\prime}}=$ $\widehat{\psi}($
$\widehat{X}_{0}^{\ast}),\ d=\deg X_{0}^{^{\prime\prime}}$ in $X_{0}%
^{^{\prime}}=\mathbf{P}^{n}$, $\ m$ = dimension of a general fiber $fib$ of
$\widehat{\psi}|$ $\widehat{X}_{0}^{\ast}:$ $\widehat{X}_{0}^{\ast}%
\rightarrow$ $\widehat{X}_{0}^{^{\prime\prime}}$, $S^{\ast}=\varphi
(G_{0}^{\ast})$ $=$ $\widehat{\varphi}(\widehat{G}_{0}^{\ast})$, $\widehat{S}$
= indeterminancy set of $\widehat{\mu}$ (which lies in $X_{0}$), $S$ =
indeterminancy set of $\mu$, $\mathcal{D}$ is the positive generating line
bundle on $X$ which induces for $t\neq0$\ on $X_{t}$ $=F$ \ the hyperplane
bundle, $\mathcal{D}^{^{\prime}}$ and $\widehat{\mathcal{D}}^{^{\prime}}$are
the corresponding line bundles on $X^{^{\prime}}$ resp. $\widehat{X}%
^{^{\prime}}$. $s_{0}$, ..., $s_{n}$, .., $s_{N}^{^{\prime}}$ are sections in
$\mathcal{D}$ such that $\widehat{\mu}:X\rightarrow\widehat{X}^{^{\prime}}$ is
defined by $P\rightarrow(\tau(P);s_{0}(P):...:s_{n}(P)$:...: $s_{N}(P))$ for
general $P\in X;$ similarly the bimeromorphic map $\mu:X\rightarrow
X^{^{\prime}}=Y\times\mathbf{P}^{n}$is defined by $\tau$ and $s_{0}$, ...,
$s_{n}$. $H_{F}$ is a hyperplane (i.e. a hypersurface generating
$H_{2n-2}(F,\mathbf{Z)=Z}$) which among other properties fulfills $F\backslash
H_{F}=\mathbf{C}^{n}$ and delivers $\mu_{F}:F\rightarrow\mathbf{P}^{n}$,
compare for instance 3.2. Furthermore the del Pezzo manifolds $F_{3}=V_{5}$,
$F_{4}$, $F_{5}$, $F_{6}$ are dealt with in 3.6. $\sigma$, $\sigma^{^{\prime}%
}$, $\sigma^{^{\prime\prime}}$, $\sigma^{^{\prime\prime\prime}}$ can represent
vanishing orders as in 4.1 and later subsections, $\sigma$, $\sigma^{^{\prime
}}$, $\widehat{\sigma}^{^{\prime}}$, $\sigma^{\ast}$, $\sigma_{1}$,
$\sigma_{1}^{^{\prime}}$, $\widehat{\sigma}_{1}^{^{\prime}}$\ represent
sections over $Y$ as in 5.1, there will be no confusion. $f_{\delta}%
^{^{\prime}}$, $f_{1\delta}^{^{\prime}}$, $f_{2\delta}^{^{\prime}}$ are local
functions, see 4.2. - "Hypersurface case" means that $X_{0}^{^{\prime\prime}}$
is a hypersurface in in $(0)\times\mathbf{P}^{n}$, $i.e.m=1$. Concerning
$\varkappa:\widehat{X}_{0}^{\ast\ast}\rightarrow\widehat{X}_{0}^{^{\prime
\prime}}$ see 8.2. \newline\newline\textbf{Some notations in connection with
Euler vector fields, }see for instance 1.2, 3.1.a, 3.1.c, 3.2, 3.6, 5.1.a,
5.1.c. $\widehat{\chi}^{^{\prime}}=\widehat{\chi}^{^{\prime}}(\sigma)$ is an
Euler vector field on $Y\times$ $F$ with center $\sigma(Y)\subset$
$Y\times(F\backslash H_{F})=Y\times\mathbf{C}^{n}$, here $\sigma=\sigma(t)$ is
a section over $Y$, $\widehat{\chi}^{^{\prime}}$ induces an Euler vector field
$\chi^{\prime}$ on $Y\times\mathbf{P}^{n}$, $\chi^{\prime}$ resp.
$\widehat{\chi}^{^{\prime}}$ can be lifted to a meromorphic vector field
$\widetilde{\chi}$ on $X$ with a pole set of order $k\geq1$ along $X_{0}$, the
holomorphic vector field (called too an Euler vector field)$\ \chi=t^{k}%
\cdot\widetilde{\chi}$ is called the holomorphic lifting of $\chi^{^{\prime}}$
resp. $\widehat{\chi}^{^{\prime}}$to $X$, $\widetilde{\chi}_{\widetilde{G}}$
and $\widehat{\chi}_{\widehat{G}}$,$=\widehat{\chi}^{\ast}=t^{k}\cdot$
$\widetilde{\chi}_{\widetilde{G}}$ are corresponding vector fields on
$\widehat{G}$. "$\chi$" can also represent a character as in 3.2.
$\widehat{\gamma}^{^{\prime}}$, $\widehat{\gamma}_{0}^{^{\prime}}$,
$\widehat{\gamma}^{^{\ast}}$, $\widehat{\gamma}_{0}^{^{\ast}}$, $\gamma$,
$\widehat{\gamma}_{0}^{^{\ast}(0)}$and $\widehat{\gamma}_{0}^{^{\ast}(1)}%
$represent orbital curves resp. families of in general)\ orbital curves.
$\widehat{\gamma}_{0}^{^{\ast}(1)}$ can in general be identified with
$\gamma_{0}^{^{\ast}(1)}$and lies in general in a general $\mu_{0}$-fiber
$fib$; but in different situations one should check the definitions of
$\widehat{\gamma}_{0}^{^{\ast}(0)}$ and $\widehat{\gamma}_{0}^{^{\ast}(1)}$.
The meaning of $N_{1}$, $\alpha(i),$ $e_{\alpha(i)}$, $e_{\alpha_{k}}%
$,$\ \alpha_{1}$, ..., $\alpha_{l}$, $h_{\alpha(i)}$, $\alpha_{k}=\alpha_{1}$,
$\omega_{k}$, $\widetilde{h}_{1}$, $h_{1},$\ reflections $\sigma_{\alpha_{k}}$
in root spaces can be found in chapter 3, esp. in 3.1, 3.2. in later chapters
$h_{1}$ (in chap. 7 for instance) \ also is used as the symbol of a
function.\textbf{.}

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