Overview of topological properties of real algebraic surfaces
Viatcheslav Kharlamov
These notes reproduce the content of a short, 50-minutes, survey talk given at the Nice University in September, 2004. We added a few topics that have not been touched on in the lecture by lack of time.
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\let\+\sqcup

\title{Overview of topological properties of real algebraic surfaces}
\author{Viatcheslav Kharlamov}
\date{}
\maketitle

\vskip1cm

\begin{abstract}
These notes reproduce the content of a short, 50-minutes,
survey talk given at the Nice University in September, 2004.
We added a few topics that have not been touched on in the lecture
by lack of time.
\end{abstract}

\section{Introduction}\label{intro}




%\address
%Institut de Recherche Math\'ematique Avanc\'ee\newline\indent
%Universit\'e Louis Pasteur et CNRS\newline\indent
%7 rue Ren\'e-Descartes
%\newline\indent 67084, Strasbourg, France
%\endaddress
%
%\email
%kharlam\,\@\,math.u-strasbg.fr
%\endemail
%
%\subjclass % Subject classification
%14J28, 14P25, and 57S25
%\endsubjclass
%
%\keywords % Key words and phrases
%Real algebraic surface, deformation, moduli,
%involution on manifold
%\endkeywords



Topology of real algebraic varieties is a broad subject. Thus,
it is reasonable to specify the "level" of objects and the
goal of study. In what concerns the level, one may distinguish
between affine, projective and abstract varieties, and, from a certain
point of view, it is natural to start with abstract varieties, and
then descend to projective and affine ones. As to the goal, I'll give
preference to those "real results" that require "complex proofs" (even
though in what follows the proofs will almost always stay behind the
scenes) and, moreover, admit "complex statements".

Another major, and traditional, simplification is to consider
{\bf nonsingular} varieties, at least at the first stage. Certainly,
a complete separation from the singular world is never
possible. However, de facto, in all the cases when a complete
understanding was achieved in the nonsingular case, it turned out
that the singular case could be treated, at least in principle, by
similar methods.

In such a setting, it is natural to consider not only algebraic, but
arbitrary K\"ahler compact complex manifolds, and to call a complex
manifold $X$ {\it real}, if it can be equipped with a {\it real
structure}, that is an anti-holomorphic involution $c:X\to X$.
Real points are then, by definition, the fixed points of the real
structure. We denote by $X_\R$ the set of real points, or the {\it real
part}, of $X$. For the sake of symmetry, we often denote $X$ by $X_\C$.


Of course, the principal source of examples is given by nonsingular
varieties defined by systems of real polynomial equations; in these
$c$ is the complex conjugation. Note that by our convention a real
variety is nonsingular if it does not have singular points, be they real
or imaginary.

We consider real algebraic varieties up to the following equivalence
relations: diffeomorphism of real part, diffeomorphism of real
structure, and deformation of variety together with real structure.
As usual, by an {\it elementary real deformation} of a {%\bf nonsingular}
real variety we mean a smooth, differentially locally trivial, family of
{%\bf nonsingular}
real varieties (say, an equivariant deformation in the sense of
Kodaira-Spencer). Two real varieties are called
real deformation equivalent if
there exists a chain of elementary deformations connecting them.
{\it Topology of the real part and that of the real structure are
preserved under deformation.} This phenomenon is one of the main
motivations for the study of topological properties of real varieties.
Note in advance that in many important cases the topology of real structure
(which includes, in fact, the topology of real part) determines the
deformation class.

A fundamental example of deformation is a small variation of a nonsingular
system of polynomial equations (that is, a system whose Jacobian has maximal
rank
at each solution of the system) or, more generally, any variation which is represented in
the total space of systems of a given number of equations (equal to the
codimension of variety) of a given degree in a given
number of variables by a smooth path in the complement of the discriminant
locus, that is a smooth path avoiding singular systems.
However, other deformations may exist as well; thus, surfaces of degree
$\ge 5$ in $\PP^3$ have big deformations failing to be embedded in $\PP^3$
while surfaces of degree $4$ have small K\"ahler deformations failing to be
projective. By contrast, any deformation of a surface of degree $3$ is
realized in $\PP^3$.


Our choice of surfaces as the topic of this talk is motivated by the fact
that this is the first nontrivial case with respect to the above equivalence
relations. In fact, in dimension zero the topology is determined by the natural
number $b_0(X_\R)$ satisfying the following relations
\begin{equation}\label{smip}
b_0(X_\R)\le b_0(X_\C),\quad b_0(X_\R)=b_0(X_\C)\mod
2 %.
\end{equation}
(here and in what follows $b_i(\cdot)$ denotes the rank of $H_i(\cdot\, ;\Z/2\Z)$,
so that $b_0$ is nothing but the number of connected components).
Similarly, in dimension one everything is determined by the number $b_0(X_\R)$
(or, equivalently, by $b_1(X_\R)=b_0(X_\R)$), and the only relations linking
this number with the invariants of $X_\C$ are as follows:
\begin{equation}\label{smic}
b_0(X_\R)\le g(X_\C)+1,\quad \text{and}\quad b_0(X_\R)=g(X_\C)+1\,\, \text{\rm
mod}\,
2 \,\,\text{\it if}\,\, X_\R\,\text{\it divides}\, X_\C.
\end{equation}
Here, $g$ is the usual genus, $g(X_\C)=\frac12 b_1(X_\C)$ if $X$
is irreducible; otherwise, $g(X_\C)+1$ is the sum of $g+1$ over all
irreducible components. The condition that $X_\R$ divides $X_\C$ is
equivalent to the orientability of the quotient $X_\C/c$.
The above relations demonstrate a general phenomenon: $\Top (X_\R)\le
\Top (X_\C)$ (the topology of a real variety is bounded by that of its
complexification).

Moreover, in dimensions less than two topology, and even homology
determines the deformation equivalence classes: $\Top (X_\C)=\Def (X_\C)$
over $\C$ and $\Top (X_\C,c)=\Top (X_\C,X_\R)=\Def (X_\C,c)$
over $\R$. For example, two real irreducible curves are deformation
equivalent, if and only if they have the same number of real components,
$b_0(X_\R)$, and they both either divide $X_\C$ or not (in other words,
either, in both cases, $X_\R$ is homologous to $0$ in $X_\C$, or not).

An advantage of dimension two is that algebraic topology still suffices
to determine $\Top X_\R$, which is no more the case in dimensions $\ge 3$
(that is why the dimensions $\ge 3$ are still very far from being well
understood; cf., however, J.~Koll\'ar's papers \cite{Ko1, Ko2} and
references in \cite{Bour}
for some nontrivial partial results).

Many of the tools used in dimension $\le 2$ can be extended to higher
dimensions, and when consideration of higher dimensions does not lead to
complications we present our results in full generality.

For lack of time, we do not discuss arrangements of curves on surfaces,
construction of surfaces (and curves on surfaces) with prescribed topology,
or enumerative results. We also will not have time to discuss history of
the subject, but it is worth mentioning that many of the results presented
below emerged due to collective efforts of many insiders (in particular,
a group of Russian mathematicians inspired by I.~G.~Petrovsky and
V.~I.~Arnol'd in Moscow, D.~A.~Gudkov in the former Gor'kii, and
V.~A.~Rokhlin in the former Leningrad; the author gratefully dedicates
these notes to the memory of the latter). A reader interested to know
better who, when and how discovered these results is invited to consult
\cite{DK} and the references therein.


\section{Topology of real varieties versus topology of their complexifications}


\subsection{ Smith theory}

Smith theory provides the following relations valid for all
dimensions:
\begin{equation}\label{smith}
\sum b_i(X_\R)\le \sum b_i(X_\C),\quad \sum b_i(X_\R)=\sum b_i(X_\C)\mod
2, %;
\end{equation}
%here and in what follows $b_i(\dots)$ denotes the rank of
where $b_i(\cdot)=\dim H_i(\cdot\, ;\Z/2\Z)$.
Behind these relations there are such
useful tools as Smith's exact sequence and Kalinin's spectral
sequence, see, for example, \cite{DK}. The latter starts with
$$
E^1_*=H_*(X_\C;\Z/2\,\Z), \quad E^2_*=\Ker(1+c_*)/\Im(1+c_*),
$$
and
converges to $H_*(X_\R;\Z/2\,\Z)$ (here and in what follows $c_*$ states for the homomorphism
$H_*(X_\C;\Z/2\,\Z)\to H_*(X_\C;\Z/2\,\Z)$ induced by the real structure
$c:X_\C\to X_\C$). Already the existence of such a
spectral sequence implies (\ref{smith}). According to the above
formula for $E^2_*$, there is a "stronger" inequality
\begin{equation}\label{swan}
\sum b_i(X_\R)\le \dim H_1(\Z/2\,\Z, H_*(X_\C;\Z/2\,\Z)),
\end{equation}
where $H_1(\Z/2\,\Z, H_*(X_\C;\Z/2\,\Z))=\Ker(1+c_*)/\Im(1+c_*)$.

There are two important classes of real varieties enjoying special features,
viz.
\begin{itemize}
\item
$M$-{\it varieties}, i.e. varieties for which $E^1_*=H_*(X_\R;\Z/2\,\Z)$
(which is equivalent to $\sum b_i(X_\R)= \sum b_i(X_\C)$, the extremal case
of (\ref{smith}), or to the existence of $c$-invariant cycles in any homology
class in
$H_*(X_\C;\Z/2\,\Z)$); here $M$ stands for "maximal";
\item
and
$GM$-{\it varieties}, i.e. varieties for which $E^2_*=H_*(X_\R;\Z/2\,\Z)$ (the
extremal case of (\ref{swan}), which is equivalent to
the existence of $c$-invariant cycles in any of the classes in $H_*(X_\C;\Z/2\,\Z)$
fixed by $c_*$; note that all real surfaces with $\pi_1(X_\C)=1$ are $GM$); here
$GM$ stands for "Galois maximal".
\end{itemize}

In dimensions $0$ and $1$ the relations (\ref{smith})\,  yield (\ref{smip})\, and
(\ref{smic}), except for the congruence in (\ref{smic}) which is a {\rm mod} $4$
relation for $\sum b_i$. The latter can be generalized in the
following way: {\it if $X$ is a $GM$-variety of odd dimension, $\dim
X=2k+1$,
and $X_\R$ is homologous to $0$ in $X_\C$, then}
$$
\frac12\sum b_i(X_\R)=\frac12\sum b_i(X_\C)\mod 2.
$$
Note that both parts of the above congruence are integers. Since
$X_\R\sim 0$, for any $x\in H_{2k+1}(X_\C;\Z/2)$ one has
$$(x,c_*x)=(x,[X_\R])=0\mod 2,
$$
which implies that, besides the summands with constant action of
$c$,
any irreducible orthogonal decomposition of
$c_*:H_*(X_\C;\Z/2)\to H_*(X_\C;\Z/2)$ contains
only irreducible components
of rank $4$ (each such component is a permutation of two $\Z/2\,\Z$-planes). Thus,

$$
\sum b_i(X_\R)=
\dim \Ker(1+c_*)/\Im(1+c_*)=
\sum b_i(X_\C) - 4p,
$$
where $p$ is the number of the above irreducible components of
rank $4$.

Smith theory helps to answer the following questions: {\it Is $X_\R$ non empty
(existence of real solutions)? How many connected components
does it have?}  and, more generally, {\it What are the possible values of
$b_i(X_\R)$?}

Already the congruence part of (\ref{smith})\, gives
a simple, often useful, sufficient condition: {\it $X_\R$ is non-empty, if
$\sum b_*(X_\C)$ is odd}. Note that this condition does not depend on the choice
of real structure.

Smith theory is not well adapted to work with individual Betti numbers, and so
in dimensions $>2$ it is hard to get more information and to answer the other
questions using Smith theory alone. Hopefully, in dimension $2$ it is sufficient
to perform homological calculations in Kalinin's spectral sequence
(or in Smith's exact sequence) and to add the information coming from the Lefschetz
trace formula. In particular, in the case of real surfaces with $\pi_1(X_\C)=1$
we get the following formulae:
$$
b_0(X_\R)=\frac12(\sum
b_i(X_\C)
-b_2^{-1}-a),
$$

$$
b_1(X_\R)=
b_2-b_2^{+1}-a,
$$
{\it where $b_2^{\pm 1}$ are the dimensions of the eighenspaces of the involution}
$$
c_*:H_2(X_\C;\R)\to
H_2(X_\C;\R),
$$
and $a$ is the number of nontrivial components
in an irreducible $\Z/2\Z$-vector space decomposition of $c_*:H_*(X_\C;\Z/2)\to H_*(X_\C;\Z/2)$
(each such component is generated by two elements permuted by $c_*$).



We recall that the above definition of $GM$-variety is equivalent to
the existence of an equivariant cycle in each invariant
$\Z/2\,\Z$-homology class, and the definition of $M$-variety
is equivalent to the existence of an equivariant cycle in each
$\Z/2\,\Z$-homology class. The latter happens, for example, if all
$\Z/2\,\Z$-homology classes of $X_\C$ are algebraic and have a real
representative. This is the case for projective spaces and Grassmann
varieties equipped with their tautological real structures. Many
other special varieties also have this maximality property. An
important class of varieties for which the maximality question is
open is provided by the discriminants of polynomials in three or
more (homogeneous) variables. For three variables, this reduces to the
question of maximality of the space of all singular plane curves of
a given degree. In degrees $1$, $2$, and $3$ discriminant is indeed an
$M$-variety, as can be verified by a more or less straightforward
calculation using
the Alexander-Pontryagin duality (one should first compute the Betti
numbers of the space of nonsingular curves). But already in degree $4$,
the question is open.

The relations (\ref{smith})\, apply, in fact, to any
finite-dimensional space with involution, and, in particular,
to singular varieties. In many cases, for example in the case
of projective hypersurfaces, the corresponding upper bounds are
the best known one. These relations can also be applied to
semi-algebraic sets; to do this, it suffices to replace such a set
by its tubular neighborhood and then apply (\ref{smith}) to its boundary,
which is a hypersurface. Y.~Laszlo and C.~Viterbo \cite{LV} recently
addressed the following question: how to bound the total Betti number
of a nonsingular real projective variety $X$ in terms of its degree $d$
and dimension $n$. Combining (\ref{smith})\,
with some inequalities due to Demailly-Peternell-Schneider they proved that
\begin{equation}
\sum b_i(X_\R)\le 2^{n^2+2} d^{n+1},
\end{equation}
while all the previously known estimates (like those of R.~Thom
\cite{Th} and J.~Milnor \cite{Mi}) were of the type
\begin{equation}
\sum b_i(X_\R)\le C d^{2n+1}.
\end{equation}
Using Lefschetz pencil, resp. Morse function type arguments
applied to $X_\C$, resp. $X_\R$, one can improve the leading coefficient
in the Laszlo-Viterbo bound and
get
\begin{equation}\label{coef}
\sum b_i(X_\R)\le d^{n+1} + O(d^n),\quad  \sum b_i(X_\C)\le d^{n+1} +
O(d^n)
\end{equation}
(without appealing to Demailly-Peternell-Schneider
inequalities).
What is an optimal choice of $O(d^n)$? I do not know. At least,
\begin{equation}\label{coeff}
\sum b_i(X_\R)\le \sum b_i(X_\C)\le d^{n+1}
\quad\text{if}\quad d>1,
\end{equation}
and it seems reasonable to expect that $d^{n+1} + O(d^n)$ in
(\ref{coef}) can be replaced by the polynomial
$d^{n+1}+\sum_{k=0}^n a_kd^k$
representing
the total Betti number of a degree $d$ nonsingular hypersurface in
$\C\Bbb P^{n+1}$ (cf. (\ref{total}) for the case $n=2$).
The key point in
the proof of (\ref{coef}) and (\ref{coeff})
is a similar bound,
\begin{equation}\label{dual}
d^*\le d(d-1)^n=d^{n+1}+O(d^n),
\end{equation}
for the degree $d^*$
of the
variety $X^*$ projectively dual to $X$ (I am grateful to F.~Zak who
explained me how such a general bound for $d^*$
is deduced from the, classical, computation of $d^*$ in the special
case of hypersurfaces; he also proposed to replace $O(d^n)$ by $0$
in (\ref{coef})). Indeed, the number of singular fibers of
the pencil of hyperplane sections of $X_\C$ and, respectively, the number of
singular values of the linear Morse function on an affine part of $X_\R$
are bounded by
$d^*$, so that an induction on the dimension $n$ of $X_\R$ gives
a sequence $x_n=\sum b_i(X_\R)$ (resp. $y_n= \sum b_i(X_\C)$)
with the property
$$ x_{n}-2x_{n-1}+x_{n-2}\le  d^*
\quad(\text{respectively}\quad
y_{n}-2y_{n-1}+y_{n-2}\le
d^*)
$$
(here $X=X_n, X_{n-1},\dots, X_0$ is the sequence formed by $X$ and its
consecutive hyperplane sections). Combined with (\ref{dual}), this
yields (\ref{coef}). For $d>2$ and $n>2$ the bound (\ref{coeff}) follows
from the inequalities $y_{n}\le d^*+2y_{n-1}\le d(d-1)^{n}+2d^{n}\le
d^{n+1}$, and if $d=2$ or $n=1,2$, then  it is easy to prove the bound by
an ad hoc argument.

Returning to the Smith bound, let us forewarn that it is impossible
in general
to replace $b_i(\cdot)=\dim H_*(\cdot\,;\Z/2\,\Z)$ in the inequality
(\ref{smith}) by the ordinary Betti numbers $\beta_i(\cdot)=\dim H_*(\cdot\,;\mathbb
Q)$. For example, there exist real Enriques surfaces $X$ with the
real part consisting of two real components, one homeomorphic to a
torus and another to a connected sum of $10$ real projective planes,
while for such real surface $ \dim H_*(X_\C;\mathbb Q)=12<14=
\dim H_*(X_\R;\mathbb Q)$.

\subsection{ Higher order congruences}\label{cong}

Higher order congruences can be found basing on Smith theory and
arithmetic of integral quadratic forms. Here is a typical example:
{\it if $\dim X=2k$ and $X_\R$ is $\Z/2$-homologous to the
middle dimensional Wu class of $X_\C$, then}
$$
\chi (X_\R)=\sigma (X_\C)\mod 8
$$
(here $\chi$ is the Euler characteristic and $\sigma$ is the
signature).
Its one-line proof given below is a model for finding other
higher order congruences. It is based on the Lagrangian property
of the real part and the Lefschetz-Hirzebruch signature formula:
$$
\chi(X_\R)=(-1)^k\sigma(c)=_8 (-1)^k(\Wu_c, c\Wu_c)=
$$
$$
(-1)^k(\Wu_X,
c\Wu_X)=(\Wu_X,\Wu_X)=_8\sigma(X_\C)
$$
(here $\Wu_c$ is the Wu integral characteristic class of the quadratic
form $(x,cy)$ on $H_{2k}(X_\C,\Z)/\text{Tors}$,
$\sigma(c)$ is its signature,
$\Wu_X$ is an integral algebraic representative of the $\Wu$-class of
$X_\C$, and $=_8$ stands for congruence modulo $8$). It should be mentioned that in the case when $X_\C$ is a
complete intersection there are various methods for explicit
computation of $\sigma(X_\C)$ (see, for example,
\cite{DK} and references therein).

There is a series of congruences refining the above one (see, for
example, \cite{DK} and references therein); the two simplest of them
are as follows:

{\it if $\dim X=2k$ and $X_\R$ is a $M$-variety, then}
$$
\chi (X_\R)=\sigma (X_\C)\mod 16;
$$

{\it if $\dim X=2k$ and $X_\R$ is a $(M-1)$-variety, then}
$$
\chi (X_\R)=\sigma (X_\C)\pm 2\mod 16.
$$

Note that $\sigma(X_\C)=(-1)^k\chi (X_\C)\mod4$
(which most easily can be seen from the Hodge decomposition),
which yields an analog of congruence (\ref{smic}) in even dimensions:

$\chi (X_\R)=(-1)^k\chi (X_\C)\mod 4$ {\it if $X_\R$ is
$\Z/2$-homologous to the middle dimensional Wu class of $X_\C$}.

\subsection{An application of Hodge theory and some other inequalities}

As is shown in \cite{Kh}, from the Hodge decomposition and the Lefschetz
formula it follows that
\begin{equation}\label{kh}
\vert\chi(X_\R)-1\vert\le h^{k,k}(X_\C)-1
\quad \text {\it if}\quad \dim X=2k
\end{equation}
(where $h^{k,k}$ is the Hodge number of bidegree $(k,k)$; various
explicit
computations of the Hodge numbers are found in \cite{DK}; an expression
for
$h^{1,1}$ in the case of surfaces in $3$-space is given below in (\ref{h11})).
See \cite{Kh2,Kh3, DK} and references therein for an odd-dimensional version
of this Comessatti-Petrovskii type inequality and
for generalizations to varieties with singularities (naturally,
in the singular case pure Hodge structure is to be replaced by mixed one).
It would certainly be nice to find other applications
of Hodge theory giving more detailed information than (\ref{kh}).
Especially challenging is to somehow relate Hodge theory with Smith theory.

Let me indicate here only a very special amusing
application of (\ref{kh}) to the case of odd dimension.
It concerns plane curves, and, more specifically, line arrangements.
We consider
%To wit, for any general
a generic configuration of $2k$ real lines in the projective
plane. The number of connected components, called cells, of the complement
of the arrangement is equal then to $2k^2-k+1$.
Since the number of lines is real, the cells can be chess-board colored,
and an application of (\ref{kh}) shows that
an upper bound for the number of projective cells of one color is
$\frac32 k(k-1)+1$, so that
%, since the total number of projective cells is
%equal to $2k^2-k+1$
a bound from below for the other color is given by
$\frac12 k(k+1)$.

More special inequalities, not directly related to Hodge theory,
can be obtained using the Lagrangian property of $X_\R$. Thus, in the case of
surfaces one can easily show that

{\it the number $p_-$ of orientable
components of $X_\R$
with $\chi <0$ has an upper bound
$$
p_-\le \frac12 (\sigma^+(X_\C)-1)
$$
{\rm (}where $\sigma^+$ is the positive index of inertia of the
intersection form}{\rm ).}

Note also that in the case of surfaces the inequality (\ref{kh})
can be extended to non-K\"ahler surfaces in the form
$$
2-\dim H^2(X_\C;\R)+2h^{2,0}(X_\C)\le \chi (X_\R)\le 4+\dim
H^2(X_\C;\R)-2h^{2,0}(X_\C),
$$
which is weaker than (\ref{kh}), but differs from it only by $4$ in the
left- and right-hand parts. This difference is due to the absence of
a K\"ahler class in $H^2$ and the asymmetry $H^{1,0}=H^{0,1}-1$ in $H^1$.

It would be interesting to find analogs of (\ref{kh}) for
the signature $\sigma(X_\R)$ instead of $\chi(X_\R)$, of course
under the hypothesis that $X_\R$ is orientable. The best
bound known to me does not involve Hodge theory. It says that
\begin{equation}\label{vi}
\vert\,\sigma(X_\R)\vert \le \frac13 \,c_2^2(X_\C),
\end{equation}
and follows directly from evaluating an algebraic representative of the
second Chern class of $X_\C$ on $X_\R$, viz.
$\inj_*c_2(X_\C)\cap[X_\R]=p_1 (X_\R)\cap[X_\R]=3\sigma(X_\R).$
This bound holds under an additional hypothesis that the tangent,
or cotangent, or some other vector bundle of $X_\C$ with the same $c_2$ is
generated by its sections (this moving condition allows to put an algebraic
representative of the second Chern class in a general position with
respect to $X_\R$ and thus to get (\ref{vi})).
More general and considerably more subtle bounds for arbitrary Pontryagin
numbers can be found in a recent paper by Y.~Laszlo and C.~Viterbo \cite{LV}.

\subsection{Special surfaces}\label{types}
The above tools allow to
understand thoroughly the topology of $X_\R$ for many special
types of surfaces. For example, they lead to a complete
topological classification of $X_\R$, and even of $(X_\C,c)$,
for cubic and quartic surfaces in $\PP^3$. We describe it in terms
of generators: each topological type generates a list of its Morse
simplifications, that is the topological types obtained from the
initial one
by series of Morse surgeries decreasing the total Betti number (removing
a spherical component or contracting a handle).


{\it There are
$5$ classes of nonsingular cubics generated by $\#_7\R P^2$
and $\,\R P^2\+ S^2$ {\rm (here and in what follows $\# $ stands for a
connected and $\+$ for a disjoint sum)}, and $66$ classes of nonsingular
quartics generated by three {\rm $M$-surfaces}
$\#_{10}(S^1\times S^1)\+ S^2$, $\#_{6}(S^1\times S^1)\+ 5 S^2$,
$\#_{2}(S^1\times S^1)\+ 9 S^2$, two {\rm $(M-2)$-surfaces}
$\#_{7}(S^1\times S^1)\+ 2S^2$, $\#_{3}(S^1\times S^1)\+ 6S^2$,
and a {\rm pair of tori} $2(S^1\times S^1)$.}

Surfaces in $\R P^3$ can also be studied up to different equivalence
relations, such as: {\it ambient isotopy in $\R P^3$}, {\it rigid
isotopy} (i.e., isotopy in the class of nonsingular or, more
generally, equisingular in some appropriate sense surfaces of the same
degree), and {\it rough projective equivalence} (i.e., projective
transformation and rigid isotopy). The difference between the last two
relations is due to the fact that the group $\text{\sl PGL}(4;\R)$ of
projective transformations of $\R P^3$ has two connected components.
Of course, the transformations in the component of unity transform a
surface into a rigidly isotopic one. To what extent the
classifications up to rigid isotopies and up to rough projective
equivalence are topological is an open question for surfaces of
degree $5$ and higher, cf. the discussion in \ref{qs}.

Topologically, the non-spherical component of
the real part of a nonsingular cubic
surface is embedded in $\R P^3$ as the standard $\R P^2$ with unlinked and
unknotted handles attached. Moreover, for cubic surfaces
not only the isotopy equivalence relation, but all
the other relations
mentioned above coincide with the purely topological one.



The embedding of quartic in~$\R P^3$ is also simple: it is isotopic to a
union of ellipsoids and hyperboloids with unknotted and unlinked handles.
With one exception, the components are outside each other; in the exceptional
case the real part consists of two nested spheres. In all other cases the
isotopy type of the real part~$X_\R$ of a real quartic
surface in $\R P^3$ is determined by its topological type and contractibility
or noncontractibility of $X_\R$ in $\R P^3$. It turns out that
in the case of degree~$4$ surfaces all the four classifications
(topological, isotopic, rough projective, and rigid) are different.
Note that the only difference between rough projective
equivalence and rigid isotopy is in chirality which tells
whether or not a surface is rigidly isotopic to its mirror image.
Rough projective equivalence is discussed in \cite{Nik2} and
chirality in \cite{Khchir}.

Any nonsingular degree 4 surface in $P^3$ is a {\it $K3$-surface},
that is a compact complex
surface with $\pi_1=1$ and $c_1=0$. Other examples of $K3$-surfaces
are given by double coverings of a nonsingular quadric in $P^3$ branched
in a transverse section by a quartic, by double coverings of a
nonsingular cubic in $P^3$ branched in a transverse section by a quadric,
by transverse intersections of three quadrics in $P^5$, etc.
Classification of all the real projective $K3$-surfaces up to rough projective
equivalence can also be found in \cite{Nik2}.


The methods used in the study of real $K3$-surfaces are based on the above tools,
including Hodge theory, as well as the Torelli theorem which plays a key role.
In what concerns rigid isotopies and rough projective equivalence,
using the surjectivity of the period map, one can reduce the
study of real structures to a study of arithmetic properties of integral
lattices. Similar methods can be used to study $K3$-surfaces with simple
singularities, but this problem has never been treated systematically.



Starting with degree $5$, our knowledge is much more limited.
It is not even known what are
the extremal values of the Betti numbers of nonsingular quintics.
We only know that the maximal number of connected components is
somewhere in between $23$ and $25$ and that the maximal first
$\Z/2\Z$-Betti number is either $45$ or $47$ (for the surfaces
in the same deformation class one has $\max b_1=47$). The best
known general bounds for the Betti numbers are those given by the
inequalities described in the previous sections. For a
surface in $\PP^3$ and, more generally, for a transversal complete
intersection $X$ in~$\PP^q$, the complex ingredients of these bounds
can easily be found. To wit, if
$X$ is a complete intersection in~$\PP^{q}$ of polydegree
$(m_1,\dots,m_{q-2})$ then
$$
 b_1(X)=0,\qquad b_2(X)=\chi(X)-2,
 $$
 $$
 h^{1,1}(X)=\frac12[b_2(X)-\sigma(X)]+1
 $$
 $$
 \chi(X)=\mu_{q-2}\left(\mu_1^2-\mu_2-(q+1)\mu_1+\tfrac12q(q+1)\right),
 $$
 $$
 \sigma(X)=-\tfrac13\mu_{q-2}(\mu_1^2-2\mu_2-q-1),
$$
where $\mu_i$ is the $i$-th elementary symmetric polynomial in
$(m_1,\dots,m_{q-2})$. In particular, for a surface of degree $m$
in $\PP^3$
\begin{equation}\label{total}
\sum b_i(X_\C)=\chi(X_\C)=m^3-4m^2+6m,
\end{equation}
\begin{equation}\label{h11}
h^{1,1}(X_\C)-1=(m-1)^3-\frac13 m(m-1)(m-2).
\end{equation}


As it was already noticed, the same tools can be applied to singular
objects as well (see, for example, \cite{Kh3} and \cite{Var}). For instance,
one can use them to bound the number of real double points in the following very
simple way. In the
case of surfaces there are two types of such points, viz. solitary
points and nodes (in local coordinates their equations are
$x^2+y^2+z^2=0$ and $x^2+y^2=z^2$ respectively).
One can resolve the nodes (which is differentially equivalent to
replacing a neighborhood of a node by its perturbation $x^2+y^2=z^2+\epsilon^2$)
and replace the solitary points by spheres (which means replacing
a neighborhood of a solitary point in $X_\C$ by its perturbation
$x^2+y^2+z^2=\epsilon^2$). As a result, we obtain a $4$-manifold
diffeomorphic to the minimal desingularization $\tilde X_\C$
of $X_\C$ and an
involution on it such that the fixed point set is diffeomorphic
to a disjoined sum of the minimal desingularization $\tilde X_\R$ of $X_\R$
and $S$  spheres, where $S$ is the number of solitary points
of $X_\R$. Now, applying the Smith inequality, one gets
$$
2S+\sum b_i(\tilde X_\R)\le\sum b_i(\tilde X_\C).
$$
Thus in the case of surfaces of degree $m$ in $\PP^3$

$$
2S\le 2S+\sum b_i(\tilde X_\R)\le m^3-4m^2+6m.
$$
This implies, in particular, that
$S\le \frac12(m^3-4m^2+6m)$. Using the
congruences $\mod 16$ described
in \ref{cong}, this
 can be improved to a sharp bound: {\it the number of solitary
points of a real quartic and, more generally, of any real singular $K3$-surface,
is $\le 10$.} (This may be worth comparing with the upper bound $16$
for the number of complex nodes of a complex quartic. This bound, which
is probably due to R.W.H.T.~Hudson, was extended to any singular $K3$-surface
by V.~Nikulin \cite{Nik} who used arithmetic of integral quadratic forms. As is
well known, probably since Fresnel and Kummer, real quartics with $16$ real
nodes do exist.)



Let me notice that the frontier of our knowledge of surfaces in
$\PP^3$ is similar to the frontier between special surfaces and
surfaces of general type in the Enriques-Kodaira classification of
compact complex surfaces: surfaces of degree $\ge 5$ are of
general type while surfaces of degree $4$ are
%so-called
$K3$-surfaces and surfaces of degree $3$ are rational. This
gives additional motivation to turn to real structures
on complex surfaces in various Enriques-Kodaira classes.




\section{Deformation classes}

Even the above very sketchy discussion shows that
a thorough topological study of surfaces leads unavoidably to
their study up to variation of equations and then to their study
up to deformation (see Introduction for the definition; recall that
we have chosen to work with K\"ahler surfaces).


\subsection{Quasi-simplicity}\label{qs}
As is pointed in \ref{types},
two nonsingular real cubic surfaces are real deformation equivalent
if and only if their real point sets are homeomorphic.
Furthermore, the real structures of two real nonsingular cubic surfaces
are
diffeomorphic if (and only if) the real point sets of the
surfaces are homeomorphic.
This is a manifestation of what we call the quasi-simplicity
property:
a real surface $X$ is called {\it quasi-simple} if it is
real deformation
equivalent to any other real surface $X'$ such that, first, $X'$ is
deformation equivalent to $X$ as a complex surface, and, second, the
real structure of $X'$ is diffeomorphic to the real structure of $X$.


In fact, all rational real surfaces are quasi-simple.
For $\mathbb R$-minimal (i.e., minimal over $\mathbb R$) rational surfaces this result is essentially due
to Comessatti, Manin, and Iskovskikh (see e.g. the survey \cite{Man}).
In full generality this is proved in \cite{DKrat}, where, in addition,
it is shown that the real deformation type of a real rational surface is
determined by certain homological data.

Ruled $\mathbb C$-minimal surfaces of any genus are also quasi-simple, see \cite{W}.
Another class of surfaces whose real deformation theory is well understood
is formed by minimal surfaces of Kodaira dimension $0$. This class consists
of Abelian, hyperelliptic, $K3$-, and Enriques surfaces. They are all
quasi-simple (see \cite{DIK} and \cite{Cata};
recall that, by definition, hyperelliptic and Enriques
surfaces are respectively quotients of Abelian and $K3$-surfaces by
free involutions). Furthermore, quasi-simplicity of hyperelliptic and
Enriques surfaces extends to quasi-simplicity of the quotients of Abelian
and $K3$-surfaces by certain finite group actions, see \cite{Duke}.

Whether elliptic surfaces and irrational ruled non $\mathbb
C$-minimal
surfaces are quasi-simple is, as far as I know, an open question.


It is natural to expect that for surfaces of general type there is no
quasi-simplicity: there should exist examples of real deformation
distinct real surfaces with diffeomorphic real structures.
A challenging problem is to find convenient deformation invariants
which are not covered by the differential topology of $(X_\C, c)$.

Existence of non quasi-simple families of surfaces
of general type does not prevent certain particular classes of surfaces
of general type from being quasi-simple. And examples of quasi-simple
real surfaces of general type do exist. One such example is given by real
Bogomolov-Miyaoka-Yau surfaces, that is, surfaces covered by a ball in
$\C^2$, see \cite{KK}. (Note in passing that in \cite{KK} it is also shown
that there exist diffeomorphic, in fact complex conjugated,
Bogomolov-Miyaoka-Yau surfaces which are not real and thus, being
rigid, are not deformation equivalent as complex surfaces.
These surfaces are counter-examples
to the so called $\rm Diff=Deff$ problem in complex geometry, see
\cite{KK} for precise
definitions and references to counter-examples not related to
the complex conjugation.
This problem is a kind of substitute of
quasi-simplicity for complex varieties. The existence of
$\rm Diff\ne Deff$ examples explains why we need to fix complex deformation
class in the definition of quasi-simplicity of real
varieties.)\footnote{{\it Added in proof.} When this paper had
been
already finished, we with
Vik.~Kulikov have constructed examples of non quasi-simple real surfaces of general type.}





\subsection{ Finiteness}

While the problem of quasi-simplicity is solved for rational
surfaces and is essentially open and very difficult for surfaces of
general type, the situation with finiteness is an opposite one:
finiteness holds both for each complex deformation class of surfaces
of general type (deformation finiteness)
and for any fixed surface of general type (individual finiteness).


To wit, since the composition of two real structures is a biholomorphic
automorphism and since the group of automorphisms of any variety of general
type is finite, there are only finitely many real structures on a
variety of general type (the same argument works for nonsingular hypersurfaces
of degree $\ge 3$ in projective space of dimension $n\ge 3$ with the exception
of $n=3, d=4$).
This is what we call {\it individual finiteness}, which we understand
as finiteness of the number of conjugacy classes of real structures on a given variety
(note that individual finiteness understood in this way extends to hypersurfaces
of degree $4$ in projective spaces of dimension $3$, see \cite{DIK}).


On the other hand, the Hilbert scheme of varieties of general type with given
characteristic numbers is quasi-projective, which implies {\it deformation
finiteness:} real structures on the varieties which, as complex varieties, are
deformation equivalent to a given variety of general type split into a finite number
of real deformation classes (where, according to our definitions, both variety
and real structure are subject to deformation).



Unlike surfaces of general type, a rational surface may have a huge automorphism
group, and, as far as I know, the problem of individual finiteness for rational
surfaces is open. The situation is different with regard to deformation finiteness
of rational surfaces which is an easy consequence of their quasi-simplicity.


In fact, {\it the deformation finiteness holds for any type of
surfaces}. Indeed, the only birational classes of surfaces for which such a
result is not contained in the literature, either explicitly or implicitly,
are elliptic surfaces and irrational ruled surfaces, but for these classes the
proof is more or less straightforward. It would be useful to find a conceptual
proof dealing with all types of surfaces in a unified way.


Some finiteness results are also known
for Klein actions of finite
groups on $K3$- and Abelian surfaces.
In particular,
{\it the number of equivariant deformation classes of $K3$- and Abelian surfaces with
faithful Klein actions of finite groups is finite}, see~\cite{Duke}.

Another, higher-dimensional, generalization of
finiteness of real structures on $K3$-surfaces extends it to
so called holomorphic symplectic (hence hyperk\"ahler) manifolds:
{\it the number of equivariant deformation classes of real structures in a given
deformation class of compact holomorphic symplectic manifolds is finite},
see \cite{inpreparation}.)

The differential topology of $(X_\C, c)$ is preserved under
deformation, and therefore deformation finiteness implies
topological finiteness. Another, more direct, approach to topological
finiteness was  recently developed by Y.~Laszlo and C.~Viterbo.
They proposed to study  finiteness of diffeomorphism types
of  real forms on complex projective varieties of a given degree.
Here one should distinguish between the real and complex degree.
For example, there exists a sequence $X_n$ of complex $K3$-surfaces
of degree $4$ (quartics in $\PP^3$) such that, for appropriate
real structures $c_n$ on $X_n$, their real degrees (the minimal
degree of a real projective embedding $X_n\to\PP^{q_n}$) converge to infinity
(so that these real structures are not induced from $\PP^3$ and, moreover,
can not be induced from $\PP^q$ with bounded $q$).
Of course, varieties of a given real degree split into a finite
number of families. Whether the same is true for real varieties of a given
complex degree is still an open question of utmost importance. But, thanks to
Y.~Laszlo and C.~Viterbo \cite{LV}, we  now have some explicit bounds
for the Pontryagin numbers of varieties of a given real degree and, as a
consequence, some explicit bounds for the number of cobordism classes of such
varieties.




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\end{thebibliography}
\end{document}
...................................

Originally, my interest to the objects
listed in the title was motivated by the first part of
Hilbert's $16^{\text {th}}$
problem, where Hilbert
asks about the number of sheets, their shape, and their position in
$\Rp3$ for surfaces of degree $4$ and higher (due to Schl\"afli and Klein,
the answer was already known for degree~3, see
the precise statement below; for smaller degrees it is elementary).

Hilbert poses his questions for
nonsingular surfaces only. He did not
explain why he
paid special attention to nonsingular objects, their degree and their
topology. One may suspect the following speculation.

Algebraic varieties are naturally organized in families. In modern
terms one
can speak
about Hilbert or Chow schemes, for example. In the case of projective
hypersurfaces, which is
our subject, it is nothing but the space of homogeneous polynomials,
which splits into irreducible components numbered by the degree $d$ of
the polynomials and the number $n+1$ of (homogeneous) variables. Each
component is a projective space of dimension $N_{d,n}=C^d_{n+d}-1$. This
projective space $P^N=\{f(x)\mid\deg f=d\}$, $N=N_{d,n}$, has certain
additional structures that are due to its origin. In particular, it is
equipped with a special
discriminant
hypersurface $D$
that consists of the polynomials $f\in P^N$ such that $\{f=0\}\subset
P^n$ is singular. In fact, it is a part of the stratification
of $P^N$
associated with the universal hypersurface
$$
\Gamma\subset P^N\times P^n, \Gamma=\{(f,x)\vert f(x)=0\}
$$
and its projection $\Gamma\to P^N$. Namely,
$D$ is the critical locus of
the projection, and, as a consequence, over $P^N\setminus D$ we get, in
the modern terminology, a deformation family of nonsingular varieties
$$
\Gamma\supset\Gamma^0\to P^N\setminus D.
$$


Everything said before works equally well over $\C$ and $\R$.
However, there is a principal difference:
over $\C$ the
space $P^N\setminus D$ is connected and all hypersurfaces $\{f=0\}$ with
$f\in P^N\setminus D$ are deformation equivalent, while over $\R$ this is
no
longer the case (except $d=1$) and it is natural to try to understand the
nature of the components of $P^N(\R)\setminus D(\R)$.
First questions are: {\it how many deformation different hypersurfaces of
given degree and dimension
exist}\rom; {\it and how to distinguish deformation different
hypersurfaces}\rom? (Unfortunately,
not very much is known so far in general: most known results concern
curves and surfaces. As to the number of deformations classes, the best,
up to my knowledge, estimate is
$d^{Nn}(n+1)^N(1+o(1))$;
for $n$ fixed and $d\to\infty$ it gives a sequence
growing like $\exp(\gamma_nd^n\log d)$, $\gamma_n\in\R$.
Most likely, this upper bound is not asymptotically sharp and could be
essentially improved as soon as one
understands better the homology of the discriminant hypersurface. For
some related information, see \cite{KO}.)

To
conclude these introductory speculations, let us underline that the
topology of a real hypersurface (which can be understood as the topology
of the real point set,
as in the sixteenth Hilbert problem, or else as that of the complex
conjugation involution) is its main deformation invariant. Thus, the
study of
the topology of a hypersurface is an important ingredient of the
deformation theory.

\head 2.\enspace Typical known results on deformations of
surfaces\footnotemark
\endhead

The\footnotetext{The talks being devoted to surfaces, it is worth noting
however that for curves the deformation problem is solved: real curves
are deformation equivalent, if and only if their real structures are
topologically equivalent, see \cite{Nat}.}
oldest result, which is even older than Hilbert's problem, is due to
Schl\"afli and Klein (1858--1873).


\proclaim{2.1.\enspace Theorem}
Two nonsingular real cubic surfaces are real deformation equivalent
if and only if their real point sets are homeomorphic.
\endproclaim

Since the real part is 2-dimensional, there is no difference between
`homeomorphic' and `diffeomorphic'; the deformation equivalence can be
understood here as a continuous path in $P^N\setminus D$, cf\. above.
(Note also that the analogous result for surfaces of degree $\le 2$ is
trivial; in degree $\le 2$ the topology of the real point set determines
the surface even up to projective transformation, and the transformation
can be chosen preserving the orientation of $P^3$,
i.e., in the component of unity
of the group $PGL(4;\R)$. A simple proof of Theorem 2.1 going back to
Klein is found in \cite{DK2}.)

As a consequence, one
obtains the following.

\proclaim{2.2\enspace Corollary}
The real structures of two real nonsingular cubic surfaces
are
diffeomorphic if \rom(and only if\rom) the real point sets of the
surfaces are homeomorphic.
\endproclaim

In fact, there are $5$ classes of nonsingular cubic surfaces: $\Rp2\+
S^2$, $\Rp2$, $\#_3\Rp2$, $\#_5\Rp2$, and $\#_7\Rp2$. Here, $\#$ stands
for the connected sum and $\+$, for the disjoint sum.

Cubic surfaces are rational. In fact, a similar quasi-simplicity result
holds for all rational surfaces. In what follows we call {\it
quasi-simple} a family of surfaces (or an individual surface) whose real
deformation class\footnote{Here, one can use the general definition of
real deformation equivalence given below.} is determined by the
diffeomorphism type of the real structure.

\proclaim{2.3\enspace Theorem}
Rational surfaces are quasi-simple.
\endproclaim

For minimal surfaces this result is essentially due
to Comessatti,
%\cite{C},
Manin,
% \cite{M}
and Iskovskikh
%\cite{I}
(see,
e.g., the survey \cite{MT}). To prove the theorem for ruled surfaces, as
well as for conic bundles, one can use Kodaira deformations, which
provide the deformation equivalence between ruled surfaces
with rulings of the same parity. For Del Pezzo surfaces the proof is
found in \cite{DIK},
where one can also find a
list of deformation classes of
minimal rational and Del Pezzo surfaces.
A complete proof of theorem 2.3 is given in \cite{DK4}.

In fact, as is proved
in \cite{DK4}, the deformation type of a real rational surface is
determined by certain homological information.

\proclaim{2.4\enspace Proposition}
Two real rational surfaces $X'$, $X''$ are deformation equivalent if and
only if there is a homeomorphism $X'(\R)\to X''(\R)$ preserving the
additional structures described below and one has $E(X')=E(X'')$ where
$E(X)$ is the part of the eighenlattice
$L(X)=\Ker\bigl[(1-\conj_*)\:H_2(X)\to H_2(X)\bigr]$ obtained by removing
from $L(X)$ the summands isomorphic to~$\langle-2\rangle$ \rom(the
condition $E(X')=E(X'')$ holds automatically unless
$b_0(X'_\R)=b_0(X''_\R)=4$ or $5$\rom).

The additional structures to be preserved are\rom:
\roster
\item
a distinguished `essentially nonorientable' component of~$X_\R$, defined
in the case $E(X)=E_8$\rom: this is the only component whose class in
$H_2(X;\Z/2)$ does not belong to the image of $L(X)$\rom;
\item
a cyclic order on the set of the other components, defined whenever
$b_0(X)\ge4$ as follows\rom: two components $F_1$, $F_2$ are neighbors if
and only if there is an element $e\in
H_2(X/\!\conj)$ such that $e^2=-1$ and $e\circ [F_1]= e\circ [F_2]=1\mod 2$.
\endroster
\endproclaim




Next (and practically last for the moment) group of surfaces whose real
deformation theory is well understood is formed by the minimal surfaces
of Kodaira dimension $0$. These are Abelian, hyperelliptic, $K3$-, and
Enriques surfaces. Since they will be our principal
subject, let me recall their definitions and give some examples.

At this point we have to choose the category of surfaces to be
considered, i.e., either algebraic surfaces only or, more generally,
compact complex surfaces. We make the latter choice, as it leads to
simpler and more transparent statements (for rational
surfaces
this makes no difference, since they form a class closed under the
deformation equivalence as it is defined below).

Thus, in what follows we mean by an {\it Abelian surface} the quotient of
$C^2$ by a rank
4 lattice $L\subset C^2$, by a {\it $K3$-surface} a compact complex
surface with $\pi_1=1$ and $c_1=0$, and by an {\it Enriques surface} a
surface with $\pi_1=\Z/2$ whose universal (double) covering is a
$K3$-surface. By hyperelliptic surfaces we mean the (nonsingular)
quotients of Abelian surfaces by free finite group
actions.

Any nonsingular degree 4 surface in $P^3$ and
the double covering of a nonsingular quadric in $P^3$ branched
in a transversal section by a quartic are examples of
$K3$-surfaces.
Consider, for example, the quadric $x^2+y^2=z^2+1$ and pick a section
which is invariant with respect to the involution $x\mapsto x$, $y\mapsto
-y$, $z\mapsto -z$ and which contains none of the (four) fixed points of
the involution; then we can lift the involution to a free involution on
the covering $K3$-surface and thus get an Enriques surface as its orbit
space.


When dealing with a compact complex manifold $X$, we define a {\it real
structure \/} on $X$ as an antiholomorphic involution $c:X\to X$.
Respectively, by a {\it real deformation equivalence \/} we mean the
equivalence generated by local real deformations: given a proper
holomorphic equivariant submersion
$$
f:(M,c)\to (\vert z \vert <1, \conj),
$$
where $c$ is a real structure on $M$ and $\conj$ is the usual complex
conjugation, the real varieties $(X_0,c)$, $(X_t,c)$ with $t\in\R$,
$\vert t\vert<1$, $X_t=f^{-1}(t)$ are said to be {\it real deformation
equivalent}\rom.

In the above examples the $K3$ and Enriques surfaces
inherit
a natural real structure from the standard complex conjugation in $P^3$
as soon as the corresponding equations have real coefficients. Small
variations of the coefficients provide local deformations. (Note that
contrary to Enriques surfaces, not all $K3$ surfaces are projective in
general. The reader should be also warned that the above definition of
real deformation equivalence differs slightly from that discussed in
Introduction: every real deformation equivalence in the sense of
Introduction is a real deformation equivalence in the sense of the above
definition, while the
converse is not true; for example, a small deformation of a surface of
degree $4$ in $P^3$ can provide a surface non embedable in $P^3$, and the
same happens for non small deformations of surfaces of degree $>4$. )

\proclaim{2.4\enspace Theorem}
Abelian,
$K3$-, and Enriques surfaces
are quasi-simple.
\endproclaim

For Abelian surfaces this result is essentially due to
Comessatti \cite{C} and (as
long as it concerns
not necessarily algebraic surfaces, that is, without taking into account
a polarization) it follows from fairly elementary analysis of involutions
on lattices. Namely, one considers the universal covering, which is an
affine complex plane, and the covering affine action of the group $L$ of
deck transformations
extended by $\Z/2$ by means of the lifts of the real
structure from the quotient to the covering plane,
$$
0\to L\to G\to \Z/2\to 0.
$$
(Note that the real structure
lifts to a real structure of the covering plane if and only if the
Abelian surface has real points; the existence of such a lift is
equivalent to the existence of a splitting of the above short exact
sequence.)
The deck transformations form a lattice in $\C^2$ and the action of the
extended group of transformations $G$ on $\C^2$ is determined, up to
deformation, by the action of $\Z/2$ on $L$ associated with the extension
and the element $a\in L/(1+c)L$ attached to it. This implies the
quasi-simplicity. (For the similar result in the case of principally
polarized Abelian surfaces, see, ~e.g., \cite {Shi} and
\cite{Si}).\footnote{{\bf Added in proof.} The classification of real
hyperelliptic surfaces up to deformation
has recently been accomplished
by F.~Catanese and P.~Frediani \cite{CF}. As follows from their
classification, the real hyperelliptic surfaces are also quasi-simple.}


For $K3$- and Enriques surfaces the result follows from the global
Torelli theorem, see \cite{N} and \cite{DK}, respectively, or \cite{DIK},
where the proof is given for
both the cases. In fact, in all the three cases (Abelian, $K3$, and
Enriques) the quasi-simplicity can be refined and made more explicit.
Namely, as it follows from the above consideration, the deformation type
of a real Abelian surface is determined by the topology of the real part
with
the exception of one case,
when the real part is empty
and there are $2$ deformation classes
which are distinguished by the action of the real structure in $H^1$ of
the surface. The deformation type of a real $K3$-surface is determined by
the action in $H^2$. For real Enriques surfaces, the real structure lifts
to two real structures on the covering $K3$-surface, and
the deformation type is determined by the induced $\Z/2\times\Z/2$-action
in $H^2$ of the covering $K3$-surface. (Contrary to the case of Abelian
surfaces, a real structure of an Enriques surface lifts always to a real
structure (and, thus, to two real structures) of the universal covering
(which is the covering $K3$-surface), even if the real part of the
surface is empty. Indeed, as is known, the obstruction is the same as for
lifting the real structure into the line bundle associated with the
covering, and in the case of Enriques surfaces this bundle is the
canonical line bundle,
to which the real structure lifts tautologically.)

The above invariants can be simplified further.
%In the case of Abelian surfaces with non empty real part, it is
%the topology of the real part which determines the surface
%up to deformation.
For example, in the case of $K3$-surfaces,
%it is sufficient
%to add
the deformation type is determined by the topology of the real part plus
one more bit of information: the vanishing (or not) of the real part in
the second $\Z/2$-homology group of the surface (note that, in fact,
this is the only possible relation between the classes realized by
separate real components). The additional invariants are more subtle in
the case of Enriques surfaces.
They involve some natural decompositions of the real part in halves and
quarters, quadratic forms on the $\Z/2$-homology of the real part, etc.
The details, the lists of the values of the invariants, and some related
information are found in \cite{DIK}.

As follows from the proofs,
the number of deformation classes is 5 for Abelian surfaces,
75 for $K3$-surfaces, and several hundreds for Enriques surfaces.
(It is worth noting that over $\C$ all
Abelian, as well as $K3$- or Enriques surfaces, are deformation
equivalent.)

As an example of application, let us mention that the following
consequence of the classification of real Enriques surfaces
completes the Bieberbach-Hitchin classification of compact
Ricci flat Einstein 4-manifolds.

\proclaim{2.5\enspace Proposition}
There is only one connected family of compact Ricci flat Einstein
4-manifolds with $\pi_1=\Z/2\times \Z/2$.
\endproclaim

Indeed, as proved by Hitchin \cite{Hi}, any compact Ricci flat Einstein
4-manifold with $\pi_1=\Z/2\times \Z/2$ (or, equivalently,\footnote{As
proved by Hitchin \cite{Hi}, $\vert\sigma\vert \le \frac23 \chi$ for any
compact Einstein 4-manifold, and this bound turns into equality if and
only if the Einstein manifold is Ricci flat.} compact Einstein
$4$-manifolds with $\vert\sigma\vert = \frac23 \chi$ and
$\pi_1=\Z/2\times\Z/2$\rom) is the orbit space of a real structure
without real points on an Enriques surface. According to our
classification, the real Enriques surfaces without real points form a
connected family
(first, direct proof is found in \cite{DK3}).

The deformation classification of real Enriques surfaces happens to
be most advanced
among the other deformation classifications of real objects known for the
moment.
In some sense, it contains a significant number of the others, like that
of cubic surfaces, plane quartics, cubic sections of a quadratic cone,
intersection of two quadrics in $P^4$, generalized (nodal) Del Pezzo
surfaces, curves of bidegree $(2,4)$ on $P^1\times P^1$, etc. Since one
of the methods employed for the study of deformations of Enriques
surfaces reduces the deformation problem to some arithmetic of lattices,
new proofs for the
deformation results listed above can be deduced from the corresponding
lattice calculations.


It is natural to expect that for surfaces of general type there is no
quasi-simplicity: there should exist examples of $\R$-deformation
distinct surfaces with diffirentiably equivalent real structures. The
problem is to find convenient deformation invariants which are not
topological.


This does not exclude the quasi-simplicity for some classes of surfaces
of general type. One such example is given by Bogomolov-Miyaoka-Yau
surfaces, that is,
surfaces covered by a ball in $\C^2$.



\head
3.\enspace Some finiteness results
\endhead


All the deformation classifications mentioned above are finite.
Apparently, this should reflect a general statement: given a differential
type $M$ of a compact 4-manifold, the number of deformation classes of
real surfaces $(X,c)$ with $X$ diffeomorphic to $M$ is finite. It would
be natural to subdivide the proof in two parts and try, first, to prove
that the number of non isomorphic real forms of a fixed complex surface
is finite, and, second, that the map from the moduli space of real
surfaces to the real part of the moduli space of complex surfaces is nice
in some appropriate sense
(e.g., semi-algebraic). Note that the number of complex deformation
classes within $M$ is known to be finite.\footnote{The situation is
changing in higher dimensions. There are known examples of compact smooth
6-manifolds for which the number of deformation classes of complex
structures is infinite. Even in the symplectic category, where it is the
question of deforming the symplectic structures on a given manifold,
there are similar examples.}

Let me mention
a phenomenon
that exposes one of the difficulties of the real deformation theory and
explains the strategy
suggested above. Consider the moduli space $M$ of complex structures on a
given compact manifold $X$ (topological or smooth). When it is well
defined, it is a complex space and it has a natural real structure, which
consists in replacing a complex structure on $X$ by the conjugate one.
Certainly, real varieties $(X,c)$ represent fixed points of this action.
But the converse is wrong, in general. First counter examples appear
already in the case of curves of genus $\ge 3$, see \cite{Shi}. As a
consequence, the moduli space of real objects is extremely rarely the
real part of the moduli space of complex ones.


Contrary to the quasi-simplicity, for the finiteness of the number of
real structures on a given variety the case of general type is an easy
one. Since the composition of two real structures is a biholomorphic
automorphism and since the group of automorphisms of a variety of general
type is finite, there is only finitely many real structures on any
variety of general type. On the other hand, in the case of surfaces the
Hilbert scheme of surfaces of general type with given characteristic
numbers is quasi-projective, which implies directly (without using the
finiteness of the number of real structures on a given surface) the
finiteness of the number of deformation classes of real structures.
Otherwise,
to my knowledge, the problem of finiteness of the number of
conjugacy classes of real structures remains essentially open.

Before discussing the case of surfaces, let me compare the above
finiteness problem with the analogous
problems in other categories.

In the category of compact smooth manifolds, the number of conjugacy
classes of involutions is finite for any manifold of dimension $\le 2$,
while the finiteness fails in higher dimensions. In particular, in
dimension $4$ there are two famous series of examples of infinitely many
pairwise non equivalent smooth involutions on $S^4$: one with various
homology $3$-dimensional spheres as the fixed point set
%diffeomorphic to $S^3$
(see \cite{P}, \cite{DeR}), and the other with the fixed point set
diffeomorphic to
%$S^2$
the standard $2$-sphere (see \cite{Gi}, \cite{Go}, \cite{V}, etc). It is
worth noting that the latter series of examples provide an infinite
number of pairwise non diffeomorphic imbeddings of $R^2$ into $R^4$ with
$R^2$ standard at infinity.

In the category of compact Riemann manifolds, the number of conjugacy
classes of involutions is finite for any manifold of any dimension.

It would be interesting to examine such an intermediate case as that of
compact symplectic manifolds. Here, again it is obvious that in dimension
$\le 2$ the finiteness takes place. Since, as it has already been
mentioned, in the symplectic category in dimension $\ge 6$ there is no
finiteness of deformation classes of symplectic structures, it is
doubtful that some kind of finiteness for the number of anti-involutions
can hold without any boundness hypothesis. It is worth noting that using
Gromov's technique of $J$-holomorphic curves one can prove that any
anti-involution of the standard symplectic $R^4$ is standard as soon as
the involution is standard at infinity. Thus, one may expect, however,
some reasonable finiteness results in $4$-dimensional real symplectic
geometry.

As to algebraic surfaces, in this category we have the following result
proved in \cite{DIK}.

\proclaim{3.1\enspace Theorem}
For any algebraic surface of Kodaira dimension $\ge 1$ and any minimal
algebraic surface, the number of isomorphism classes of real structures
is finite.
\endproclaim

In what concerns this finiteness problem, the case of non minimal rational
surfaces looks to be the most complicated and
enigmatic. However, one may add to the surfaces enumerated in the theorem
the Del Pezzo surfaces and the projective planes blown up at $\le 10$
points. The case of Kodaira dimension $2$ is trivial, because of the
finiteness of the group of automorphisms. The proof in the case of
Kodaira dimension $1$ is based on the Borel-Serre finiteness for real
structures on (connected) algebraic linear groups and the following, well
known, proposition.

\proclaim{3.2\enspace Proposition}
For any algebraic surface $X$, the kernel of the natural representation
of $\Aut'(X)=\Aut/\Aut_0$ in $L=K^\perp_X\subset\Num X$ is finite.
\endproclaim

(Here, $\Num$ stands for the Neron-Severi lattice which is the image of
$\Pic$ in $H^2(X)/\Tors$, $\Aut$ is the group of automorphisms, and
$\Aut_0$ its component of the identity.)

After that, to
finish with the case of Kodaira dimension $1$ it remains to use the
interpretation of the set of conjugacy classes of real structures as the
Galois cohomology set $H^1(\Z/2,\Aut)$, where $\Aut$ is regarded as a
$\Z/2$-space with the action $g\mapsto cgc^{-1}$, $c$ being a fixed real
structure, and to study some standard exact cohomology sequences. (To
apply the Borel-Serre theorem one should use, in addition, the Matsusaka
theorem, which states that $\Aut_0$ is an algebraic linear group.)

Now, the only remaining case is that of minimal surfaces of Kodaira
dimension $0$, i.e., that of hyperelliptic, abelian, $K3$, and Enriques
surfaces. The real structures on abelian surfaces (and abelian varieties
of any dimension)
were treated by A.~Comessatti \cite{C} in
the mid $20^{\text{th}}$. In fact, the finiteness part is not complicated
at all and easily extends to hyperelliptic surfaces.

For $K3$-surfaces (as well as
for Abelian surfaces)
one can prove a stronger statement, which clearly implies the similar
statement for Enriques surfaces (resp., hyperelliptic surfaces).

\proclaim{3.3\enspace Theorem}
Up to isomorphism,  a $K3$-surface admits at most finitely many
finite group Klein actions.
\endproclaim

Here by a group Klein action we mean a group acting by both holomorphic
and antiholomorphic maps. The group in the theorem is not fixed, so that
the theorem states, in particular, that the number of finite groups
acting on a given $K3$-surface is finite. The surface in the theorem is
not
assumed to be algebraic.

The proof looks different in the algebraic and non algebraic cases.
Details are given in \cite{DIK}. Let me underline that in the non
algebraic case the main finiteness ingredient is coming from the
Dirichlet theory of units in algebraic number fields and gives, as a
by-product, the following result (which, in what concerns non algebraic
$K3$-surfaces, we did not find in the literature).

\proclaim{3.4\enspace Proposition}
The group of automorphisms of any $K3$-surface
is finitely generated.
\endproclaim

In the algebraic case the proof of Theorem 3.3 follows more traditional
lines and is based, aside from the global Torelli theorem, on the
finiteness of the kernel of the natural representation of $\Aut$ in
$\Aut\Pic$ (Piatetskii-Shapiro, Shafarevich) and the finiteness of the
Siegel fundamental polyhedron (Siegel, Garland, Raghunathan, Weelenberg).


\head 4.\enspace``To have and have not'' (added in proof).
\endhead

Additional
natural question is: {\it Can any complex nonsingular surface
be deformed to a real surface}\rom? As is shown in \cite{KK}, the
answer is in the negative.

\proclaim{4.1\enspace Theorem}
There exist rigid surfaces without any real structure.
\endproclaim

Here by rigid surface we mean a surface having no nontrivial local
deformations. In fact, the surfaces in our examples are strongly rigid
(in the sense of Mostow): $X$ is strongly rigid if any
homotopy equivalent surface $Y$ is isomorphic to either $X$ or $\bar X$.
In our principal example the surface is a $\Z/5\times\Z/5$ Galois
covering of $P^2$ ramified in a configuration of 9 lines dual to the
inflection points of a cubic. In this, slight, modification of Hirzebruch
constructions (we select additional ramification weights on the lines) we
control the automorphism group of the covering surface. We prove that the
group is exhausted by the deck transformations and, moreover, the surface
has no any anti-holomorphic automorphism. The strong rigidity follows, as
usually, from the relation $c_1^2=3c_2$ (which implies that the universal
covering is the complex ball; certainly, the surfaces in question
are different from $\C P^2$).

As a consequence, we get new counter examples to the (complex) \, Dif=Def
\, conjecture. More precisely, we get the following result.

\proclaim{4.2\enspace Proposition}
There exist smooth oriented 4-manifolds for which the moduli space
$M_{\text{dif}}$ of complex structures with given underlying smooth
structure (and given orientation)
consists of 2 points interchanged by complex conjugation.
\endproclaim

After \cite{KK}, F.~Catanese \cite{Ca} worked out different,
non rigid, examplpes where also the complex conjugation is
interchanging the components of the moduli space. His surfaces
are covered by the bi-disc $D\times D\subset \C^2$.

Other counter examples given in \cite{KK}
are the fake projective planes
(that is the surfaces of general type with $q=p_g=0$ and $K^2=9$).
It is easy to build from the surfaces considered some examples
of varieties $X$ of any dimension $\ge 3$ having the similar
property, i.e., the varieties such that
 $X$ and $\bar X$ belong to different components
of the moduli space. For example, it is sufficient to consider
products of any of our surfaces (rigid examples) or products of
fake projective planes with a torus (non rigid examples).
It follows from the well known properties of the Albanese
map and Siu's rigidity theorem.




{\bf Acknowledgements.} I am grateful to quite a number of people with
whom I had useful discussions concerning the subject, and to my various
coauthors with whom the above results were obtained. My additional
gratitude is to A.~Degtyarev, who helped me to improve this short survey
text.

\widestnumber\key{DIK5}
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\end





In this paper we consider nonsingular real algebraic surfaces.  From
the na\"\i{}ve point of view, these are just surfaces given in a real
projective space by a nonsingular system
of polynomial equations with real coefficients. However, at certain
stage it becomes natural, and even necessary, to enhance and extend
this notion.  First, as polynomial equations also make sense
over~$\C$, one can consider complexification. The resulting complex
surface, given by the same equations in the corresponding complex
projective space, is invariant under the complex conjugation
involution, and the original real surface is its fixed point set.
Then one can take the surface out of the ambient space, considering
it as an abstract analytic manifold, and thus arrive to the notion of
complex analytic manifold equipped with a {\it real structure\/}; the
latter, by definition, is just an antiholomorphic involution on the
manifold, and it is this involution (and, in particular, its fixed
point set) that is subject of the study.  (Note that one could as
well consider algebraic surfaces over~$\C$ endowed with a Galois
involution. In both the cases a real structure on a surface~$X$ is an
involutive isomorphism $X\to\,\wb{\!X@!@!@!}@,@,@,$. However, we
prefer to deal with complex analytic manifolds as, on one hand, the
analytic category is wider, and on the other hand, the problems we
are interested in and the tools we are using are topological. Above
all, in all the cases considered below the topology of real
structures happens to be the same in both the categories.)

Apart of the main question, to study the real structure (involution)
up to homeomorphism or diffeomorphism, there are several other,
more visual, levels of investigation. In particular, the so-called
purely real approach concerns only the topology of the real point set
of the variety (i.e., the fixed point set of the real structure). The
level of study being fixed, the question still can be posed in
various ways. The first desire is to classify the real structures (or
fixed point sets, or whatever is chosen for the subject). Then one
has to confine oneself to a certain class of complex surfaces, say,
one or several related deformation families.

Chronologically, the first family considered from this point of view
were cubic surfaces in~$\Rp3$, which were subjected to
different classifications.  Probably, the first classification taking
into account the real structure was given by Schl\"afli~\cite{S1},
who in~1858 introduced his famous 5~kinds of generic (i.e.,
nonsingular) cubic surfaces. It is rather difficult to believe
that he had no idea about the shape of the real part of these
surfaces; however, it was not until~1872 (see~\cite{S2},~\cite{S3})
that we could find in his papers any related remarks. Probably, in
spite of Riemann's input, the topological setting in that time was
still neither current nor respected.

Apparently, it was Klein who first explicitly posed and solved all
the basic questions concerning topology of real cubic surfaces.
In~1873 (see~\cite{Kl}) he showed that Schl\"afli's classification
coincides with the topological classification of the real parts of
cubic surfaces.  Furthermore, he showed that the moduli space of
cubic surfaces with a given topological type of the real part is
connected, which in fact gives the complete topological
classification of Galois involutions on cubic surfaces: two such
involutions are equivalent if and only if their fixed point sets
(i.e., the real parts of the surfaces) are homeomorphic.

Cubic surfaces occupy a special position among other surfaces: from
the complex point of view they form one of the infinitely many
components of the moduli space (or, in other words, belong to a
particular deformation type) of rational surfaces.  First results on
the classification of general real rational surfaces were obtained by
Enriques~\cite{Enr} in~1897. The classification was completed
in~1912 by Comessatti (see~\cite{Co1}, \cite{Co2}), who extended
Klein's results to arbitrary real rational surfaces and described the
topology of the real parts for each (complex) deformation type (see
Theorem~2.1 in Section~2).\footnote{The description of the
connected components of the modili spaces is also contained, but, as
far as we know, not explicitly stated in Comessatti's works
\cite{Co1}--\cite{Co3}:  with one exception, within one complex
deformation type the moduli space of minimal real rational surfaces
with a given topological type of the real part  is connected.} In the
late sixties Manin~\cite{M1},~\cite{M2} and
Iskowskikh~\cite{Isk1}--\cite{Isk3} put these results into the modern
framework, completed some statements, gave new proofs, and
generalized the results to 2-extensions of fields other than~$\R$.

It is worth mentioning that it is due to his solution
of this classification problem that Comessatti found a nontrivial
bound for the number of components of a real rational surface, which
he later generalized to his famous estimate on the Euler
characteristic: in the modern terminology this result states that the
Euler characteristic of the real part of a real algebraic surface is
bounded by the Hodge number $h^{1, 1}$ of its complex part.

Another natural direction of developing the subject is the study of
real quartics in~$\Rp3$, which was started by Rohn and Hilbert.
(Hilbert even included the corresponding questions in his famous list
of problems.) After a period of relative oblivion, in the late 1960's
they were made a subject of study by Utkin, who followed the approach
of Rohn and Hilbert (which relates quartic surfaces to plane sextics)
and used the classification of sextics just obtained by Gudkov. The
topological classification of the real parts of real quartics was
completed in~1976 by Kharlamov~\cite{Kh}. Once again the solution of
a classification problem stipulated the discovery of new general
phenomena:  a series of congruences on the Euler characteristic of
the real part of a real surface (Gudkov, Arnold, and Rokhlin
congruences and their generalizations).

Quartic surfaces also belong to a special class: they are all
so-called $K3$-surfaces. Complex analytic $K3$-surfaces form a
connected moduli space, where quartics constitute a connected
subspace.  From the differential point of view all the complex
$K3$-surfaces are diffeomorphic to one another.  In fact, the
topological classification of the real parts of general $K3$-surfaces
coincides with and follows from the topological classification of the
real parts of real quartics:  all the prohibitions on the real part
are of purely topological nature, and all the nonprohibited types are
realized by quartics.

More advanced classification of real $K3$-surfaces was done in~1979
by Niku\-lin~\cite{N1}, who found and rather explicitly described the
connected components of the moduli space. According to Nikulin, two
real $K3$-surfaces belong to one component if and only if their
Galois involutions are topologically equivalent, and the action of
the Galois involution is determined up to diffeomorphism by some
simple numerical topological invariants.

Following the Enriques classification of complex algebraic surfaces,
there remains only five special birational\mnote{just `birational'?}
classes of surfaces: abelian surfaces, surfaces with a pencil of
rational curves, hyperelliptic surfaces, surfaces with a pencil of
elliptic curves of canonical (Kodaira) dimension~1, and Enriques
surfaces.

Abelian surfaces were classified by Comessatti (see~\cite{Co3} for
the precise statements and further references.\mnote{like this?} Some
results on the topology of hyperelliptic surfaces and real surfaces
with a real pencil of rational curves and the classification of
singular fibres of real pencils of elliptic curves were obtained by
Silhol~\cite{Si}.

The topological classification of the real parts of real Enriques
surfaces, as well as of some canonical structures that they inherit
from the complexification, was started by V.~Nikulin~\cite{N2} and
recently completed by the authors~\cite{DK1},~\cite{DK2}. Similar to
what happened during the investigation of other special classes of
surfaces, as a by-product of this study we discovered some new
topological properties of the Galois involution. The purpose of this
paper is to present these results, with an emphasis on the relatively
new tools applied and the veritable information which they give about
surfaces more general than the Enriques surfaces.

The paper is organized as follows: In~\S2 we cite some results which
answer some of the questions posed above. In~\S3 we present a
specific tool which we used to classify real Enriques surfaces and
state some of its properties (see~\cite{DK2}). This tool, so called
{\it Kalinin's spectral sequence}, which we know mainly due to
O.~Viro and I.~Kalinin, unfortunately is not widely known to the
specialists in real algebraic geometry and, in view of its general
nature, rather belongs to topology of periodic transformation groups.
In~\S4 we prove two new results on topology of real algebraic
surfaces, which, on one hand, were originated by the classification
of real Enriques surfaces and, on the other hand, illustrate
applications of Kalinin's spectral sequence.

\subhead
Acknowledgements
\endsubhead
We would like to thank {\it Mathematisches Forschungs\-insitut
Oberwolfach\/}:  an essential part of this project was elaborated
during our RiP stay in this institute.

\head
2.\enspace Some answers
\endhead

Below, when describing the topological type of the real part of a
surface, we denote by~$S_g$ the orientable surface of genus~$g$ and
by~$V_q=\#_{q}\Rp2$, the nonorientable surface of genus~$q$. We use
any of $S=S_0=V_0$ for the 2-sphere.

We start with reproducing Comessatti's result on the classification
of minimal real rational surfaces.

\proclaim{2.1.\enspace Theorem \rm(Comessatti~\cite{Co1--Co3})}
Each minimal real rational surface is one of the following\rom:
\roster
\item
real projective plane $P^2$\rom:\quad$\RX=V_1$\rom;
\item
real quadric $P^1\times P^1$\rom: there are four types\rom: $S$,
$S_1$, and two nonequivalent empty surfaces\rom;
\item
ruled rational surfaces $\Sigma_m$, $m\ge2$\rom:
\smallskip
\halign{\hbox to.5\hsize{\indent\indent\indent#\hfil}&#\hfil\cr
$m$ even\rom:\quad$\RX=\varnothing$ or $S_1$,&
$m$ odd\rom:\quad$\RX=V_2$\rom;\cr}
\smallskip
\item
real conic bundles over $P^1$ whose reducible fibers are all real and
consist of pairs of complex conjugated exceptional curves\rom:\quad
$\RX=mS$, where $2m\ge4$ is the number of reducible fibers\rom;
\item
Del Pezzo surfaces of degree $d=K^2=1$ or~$2$\rom:
\smallskip
\halign{\hbox to.5\hsize{\indent\indent\indent#\hfil}&#\hfil\cr
$d=1$\rom:\quad$\RX=V_1\+4S$,&
$d=2$\rom:\quad$\RX=3S$ or $4S$.\cr}
\endroster
\endproclaim

\remark{Remark}
The two nonisotopic real structures on $X=P^1\times P^1$ with
$\RX=\varnothing$ is the exception mentioned in the introduction.
\endremark

\remark{Remark}
The Del Pezzo surface of degree~2 with $\RX=3S$ can also be
represented as a conic bundle over~$P^1$ with six reducible fibers.
\endremark

In order to state other results, we need the following notion:

\definition{2.2.\enspace Definition}
A {\it Morse simplification\/} is a Morse transformation which
decreases the total Betti number, i.e., either removing a spherical
component ($S\to\emptyset$) or contracting a handle
($S_{g+1}\to S_g$ or $V_{p+2}\to V_p$). A particular complex
deformation family being fixed, a {\it topological type\/} (i.e., a
class of surfaces with homeomorphic real parts) is called {\it
extremal\/} if it cannot be obtained from another topological type by
a Morse simplification.
\enddefinition

\remark{Remark}
Note that a Morse simplification may not correspond to a Morse
simplification in a continuous family of complex surfaces. As a
result, the notions of extremal topological type and extremal (in the
obvious sense) surface may be different. E.g., according to Viro and
Kharlamov~\cite{Vi}, any surface whose real part is
$\operatorname{mod}2$ homologous to zero in the complexification is
extremal, though it may have nonextremal topological type.
\endremark

In order to illustrate this notion we list all (not only minimal)
topological types of Del Pezzo surfaces of degree~1 and~2.
(Certainly, this result follows immediately from Comessatti's
classification).

\proclaim{2.3.\enspace Theorem}
The topological types of the real parts of Del Pezzo surfaces of
degree $d=1$ and~$2$ are those \rom(and only those\rom) which may be
obtained by a series of Morse simplifications from the following
extremal types\rom:
\smallskip
\indent\indent
$d=1$\rom:\quad$V_9$, $V_3\+S$, $V_2\+V_1$, and $V_1\+4S$\rom;

\indent\indent
$d=2$\rom:\quad$V_8$, $V_2\+S$, $V_1\+V_1$, $4S$, and $S_1$.
\endproclaim

Finally, we list the topological types of the real parts of real
$K3$- and Enriques surfaces.

\proclaim{2.4.\enspace Theorem \rm(Kharlamov~\cite{Kh})}
The topological types of the real parts of $K3$-surfaces are those
and only those which may be obtained by a series of Morse
simplifications from the following extremal types\rom:
\roster
\item
$M$-surfaces\rom:\quad$S_{10}\+S$, $S_6\+5S$, $S_2\+9S$\rom;
\item
$(M-2)$-surfaces\rom:\quad$S_7\+2S$, $S_3\+6S$\rom;
\item
Pair of tori\/\rom:\quad $2S_1$.
\endroster
\endproclaim

\proclaim{2.5.\enspace Theorem \rm(see~\cite{DK1})}
There are\/ \rom{87} topological types of real Enriques surfaces.
Each of them can be obtained by a sequence of Morse simplifications
from one of the\/ \rom{22} extremal types listed below. Conversely,
with the exception of the two types $6S$ and $S_1\+5S$, any
topological type obtained in this way is realized by a real Enriques
surface.

The\/ \rom{22} extremal types are\/\rom:
\bgroup
\twocoljot1pt\firstindent4\parindent
\roster
\item $M$-surfaces\/\rom:
\twocol
\label(a)\chi(\er)=8\:   &\label(b)\chi(\er)=-8\:\cr
        4\V(1)\+2S,     &\V(11)\+\V(1),\cr
        \V(2)\+2\V(1)\+3S,      &\V(10)\+\V(2),\cr
        \V(3)\+\V(1)\+4S,       &\V(9)\+\V(3),\cr
        2\V(2)\+4S,             &\V(8)\+\V(4),\cr
        \V(4)\+5S,              &\V(7)\+\V(5),\cr
        \V(2)\+S_1\+4S,         &2\V(6),\cr
                                &\V(10)\+S_1;\cr
\endtwocol
\item[2] $(M-2)$-surfaces with $\chi(\er)=0$\rom: \twocol
        \V(4)\+2\V(1),          &\V(5)\+\V(1)\+S,\cr
        \V(3)\+\V(2)\+\V(1),    &\V(4)\+\V(2)\+S,\cr
        \V(6)\+2S,              &2\V(3)\+S,\cr
        \V(4)\+S_1\+S,          &2\V(2)\+S_1;\cr
\endtwocol
\item[3] Pair of tori\/\rom:\quad $2S_1$.
\endroster
\egroup
\endproclaim

\remark{Remark}
Nikulin's classification of real $K3$-surfaces contains the following
result: a real $K3$-surface~$X$ is determined up to equivariant
diffeomorphism by the topological type of its real part~$\RX$ and its
{\it type\/} (i.e., whether the fundamental class~$[\RX]$ is or is
not homologous to zero in $H_2(\C X;\Z/2)$).
\endremark

Since the fundamental group of a complex Enriques
surface~$\C E$ is~$\Z/2$, its real part inherits an interesting
additional structure: the set of its connected components naturally
splits into two halves, $\R E=\R E\index1\cup\R E\index2$. Each half
is covered by the real part of one of the two real structures on the
covering $K3$-surface. The study of this
decomposition was started by
V.~Nikulin~\cite{N2} as part of his attempt to classify real Enriques
surfaces. The complete classification of triads
$(\R E;\R E\index1,\R E\index2)$ up to homeomorphism is given
in~\cite{DK2}:

\proclaim{2.6\. Theorem}
Each half of a real Enriques surface may be either~$S_1$, or~$2V_2$,
or $\ga V_g\+a V_1\+b S$, $g>1$, $a\ge0$, $b\ge0$, $\ga=0,1$. With
the exception of the types $kS$ and $V_{2r}\+kS$ any decomposition
into halves satisfying the above condition is realizable. The
exceptional topological types admit only the distributions listed in
Figure~\rom1.
\endproclaim

\midinsert
\line{\hfil\openup0pt\vbox{\halign
{#\qquad&\quad#\quad&\quad#\quad&\quad#\cr
\graph5{\(0,0)\(0,1)\(0,2)\(0,3)\(0,4)\(1,0)\(1,1)\(1,2)\(1,3)\(1,4)
 \(2,0)\(2,1)\(3,0)\(3,1)\(4,0)\(4,1)\(2,2)}&
\graph5{\(0,0)\(0,1)\(0,2)\(0,3)\(0,4)\(1,0)\(1,1)\(1,2)\(1,3)\(1,4)}&
\hbox to5\step{\hfil\graph2{\(0,0)\(0,1)\(0,2)}\hfil}&
\hbox to5\step{\hfil\graph1{\(0,0)\(0,1)}\hfil}\cr
\noalign{\vglue-2pt}\cr
\capt{\{aS\}\+\{bS\},}&\capt{\{V_4\+aS\}\+\{bS\}}
&\capt{\{V_6\+aS\}\+\{bS\}}&\capt{\{V_8\+aS\}\+\{bS\},}\cr
\capt{\{V_2\+aS\}\+\{bS\}}&&&\capt{\{V_{10}\+aS\}\+\{bS\}}\cr}}\hfil}
\botcaption{Figure 1}
Exceptional topological types
\endcaption
\endinsert

\remark{Remark}
It seems reasonable to conjecture that a real Enriques surface~$E$ is
determined up to equivariant diffeomorphism by the topological type
of its real part~$\R E$, the above decomposition into halves, and its
{\it type\/} (i.e., whether the fundamental classes~$[\R E]$,
$[\R E\index1]$, and $[\R E\index2]$ are
homologous to zero, $w_2(\C E)$, or none of these in
$H_2(\C E;\Z/2)$), cf. the remark on $K3$-surfaces.
\endremark

\head
3.\enspace Tools
\endhead

In this section we introduce our primary tool---so called {\it
Kalinin's spectral sequence\/}, which was originally constructed by
I.~Kalinin~\cite{Ka} as a stabilized version of the Borel-Serre
spectral sequence for equivariant cohomology. This sequence starts at
the homology $H_*(Y)$ of a topological space~$Y$ with involution~$c$
and converges to the total homology $H_*(\Fix c)$ of the fixed
point set of~$c$. (Unless stated explicitly, {\it all the
homology and cohomology groups have coefficients\/}~$\Z/2$.)
The resulting filtration $\CF^*$ on $H_*(\Fix c)$ and the isomorphisms
$\vr_p$ between the limit term of the spectral sequence and
$\Gr_{\CF}H_*(\Fix c)$ were discovered by O.~Viro geometrically before
Kalinin's work and were primarily related to the Smith exact sequence. As
it is shown in~\cite{DK2}, Kalinin's spectral sequence can be derived
from the Smith exact sequence as well.

Below we give a geometrical description of Kalinin's spectral
sequence and Viro homomorphisms~$\vr_*$ and state their main
properties. Proofs of these results and/or further references can be
found in~\cite{DK2}. Since homology groups are more transparent and
easier to deal with, we decided to use the homology language (though
we cannot certainly help mentioning cohomology when speaking
about multiplications and Poincar\'e duality). As our approach is
geometrical, we have to appeal to the notion of chain.  Depending on
the nature of~$Y$, one may work with singular, simplicial, smooth, or
any other kind of chains considered in algebraic topology. To assure
convergence of the spectral sequence, $Y$ must satisfy certain
conditions, which, strictly speaking, depend on the homology theory
chosen (e.g., sheaf theories and locally compact finite dimensional
spaces).  However, in this paper all the results are applied to the
best possible topological spaces---smooth manifolds, so they do not
depend on this choice.

Thus, let us fix a good (see above) topological space~$Y$ with
involution~$c$ and denote by~$\Fix c$ the fixed point set
of~$c$.  Consider the partial homomorphisms
$\bv_r\:H_*(\Fix c)\dasharrow H_r(Y)$ and the $\Z$-graded spectral
sequence $(\rH_*,\rd_*)$ defined as follows:
\roster
\item
$\bv_0$ is zero on $H_{\ge1}(\Fix c)$ and its restriction to
$H_0(\Fix c)$ coincides with the inclusion homomorphism\rom;
\item
$\vr_p$ is defined on a\/ \rom(nonhomogeneous\/\rom)
element~$x\in H_*(\Fix c)$ represented by a cycle $\sum x_i$
(where $x_i$ is the $i$-dimensional component of~$x$)
if and only if there exist some chains $y_i$ in~$Y$, $1\le i\le p$, so
that $\partial y_1=x_0$ and $\partial y_{i+1}=x_i+(1+c_*)y_i$ for
$i\ge1$.  In this case $\vr_p x$ is represented by the class of
$x_p+(1+c_*)y_p$ in $H_p(Y)$\rom;
\item
$H^1_*=H_*(Y)$ and $\Rd1_*=1+c_*$\rom;
\item
$\rd_p$, considered as a partial homomorphism
$H_p(Y)\dasharrow H_{p+r-1}(Y)$, is defined on a cycle $x_p$ in~$Y$ if
and only if there are some chains
$y_p=x_p,y_{p+1},\dots,\allowmathbreak y_{p+r-1}$ so that
$\partial y_{i+1}=(1+c_*)y_i$.
In this case $\rd_p x_p=(1+c_*)y_{p+r-1}$.
\endroster

\proclaim{3.1.\enspace Theorem}
The  homomorphisms~$\bv_*$ and spectral sequence $(\rH_*,\rd_*)$
are natural with respect to equivariant maps. Furthermore,
$\rH_*$ and $\rd_*$ do form a spectral sequence \rom(i.e.,
$\rd_p$ are well defined homomorphisms $\rH_p\to\rH_{p+r-1}$
and\/ $\RH{r+1}_p=\Ker\rd_p/\Im\rd_{p-r+1}$\rom), and this
sequence converges to $H_*(\Fix c)$ via~$\bv_*$, i.e., $\bv_p$
induces an \rom(honest\rom) isomorphism
$\CF^p/\CF^{p+1}\to\iH_p$, where
$\CF^p=\operatorname{Domain}\vr_p=\Ker\vr_{p-1}$.
\endproclaim

There is an obvious cohomology version $\rH^*\Rightarrow H^*(\Fix c)$
of the spectral sequence, which is dual to the homology one. The cup-
and cap-products in~$Y$ naturally extend to, respectively, a
$\Z/2$-algebra structure in~$\rH^*$ and $\rH^*$-module structure
in~$\rH_*$. If $Y$ is a connected $N$-manifold and $\Fix c\ne\emptyset$,
then the fundamental class $[Y]$ survives to~$\iH_N$ and the
multiplication $\cap[Y]\:\rH^p\to\rH_{N-p}$ is an isomorphism
(Poincar\'e duality), which, in the usual way, defines homology
intersection pairing $\circ\:\rH_p\otimes\rH_q\to\rH_{p+q-N}$. The
relation between this pairing and the ordinary intersection pairing
in~$\Fix c$ is given by the following statement:

\proclaim{3.2.\enspace Theorem}
Let $Y$ be a smooth closed $N$-dimensional manifold with a smooth
involution $c\:Y\to Y$, and let $F=\Fix c$ be the fixed point set
of~$c$. Denote by~$w(\nu)$ the total Stiefel-Whitney class of the
normal bundle~$\nu$ of~$F$ in~$Y$.  Then for
$a\in\CF^p$ and $b\in\CF^q$ one has
$w(\nu)\cap(a\circ b)\in \CF^{p+q-N}$, and
$$
\bv_p a\circ\bv_q b=\bv_{p+q-N}[w(\nu)\cap(a\circ b)].
$$
\endproclaim

The (homology) Steenrod operations $\Sq_t\:H_*(Y)\to H_{*-t}(Y)$ also
extend to $\rH_*$. In order to describe their relation to the
ordinary Steenrod operations in~$H_*(\Fix c)$, let us introduce the
{\it weighed Steenrod operations\/}
$\hSq_tx=\sum_{0\le j\le t}\binom{P-p}{t-j}\Sq_jx$, where
$x\in H_p(Y)$ and $P>p+t$ is a power of~2. (The binomial coefficients
do not depend on~$P$, see, e.g., Lemma~I.2.6 in~\cite{SE}.) Then one
has:

\proclaim{3.3.\enspace Theorem}
If $x\in\CF^p$ and $t\ge0$, then \smash{$\hSq_tx\in\CF^{p-t}$} and
$$
\Sq_t\bv_px=\bv_{p-t}\hSq_tx.
$$
\endproclaim

We conclude this section with the description of Viro homomorphisms
(in dimensions up to~2) in the case when $Y$ is a real algebraic
surface and $c=\conj$ is the real structure. Let
$C_1, C_2,\dots, C_k$ be the components of $\RY$. Denote
by~$\<C_i>$ and~$[C_i]$ their classes in $H_0(\RY)$ and fundamental
classes in $H_2(\RY)$ respectively. Then the values $\bv_0\<C_i>$,
$\bv_1\ga$ (with $\ga\in H_1(\RY)$), and $\bv_2[C_i]$ are always well
defined and coincide, essentially, with the inclusion homomorphisms.
The value $\bv_1\<C_i-C_j>$ is also well defined and is represented
by the equivariant circle $(1+\conj_*)y_1$, where $y_1$ is an arc
in~$\CY$ connecting a point in~$C_i$ with a point in~$C_j$. If, under
some appropriate choice of~$y_1$, this circle is homologous to zero,
$(1+\conj_*)y_1=\partial y_2$, then $\bv_1\<C_i-C_j>=0$ and
$(1+\conj_*)y_2$ represents~$\bv_2\<C_i-C_j>$. Similarly, if
$\bv_1\ga=0$, i.e., $\ga=(1+\conj_*)y_1$ for some cycle~$y_1$
in~$\CY$, then there exists a chain~$y_2$ in~$\CY$ such that
$\partial y_2=\ga+(1+\conj_*)y_1$, and $(1+\conj_*)y_2$
represents~$\bv_2\ga$. Finally, if $\bv_1\ga=\bv_1\<C_i-C_j>$, then
$\bv_2(\ga+\<C_i-C_j>$ is defined and is represented by
$(1+\conj_*)y_2$, where $\partial y_2=\ga+(1+\conj_*)y_1$ and $y_1$
is an appropriately chosen arc connecting two points in~$C_i$
and~$C_j$.

Elements of the form $\bv_2[C_i]$, $\bv_2\<C_i-C_j>$,
$\bv_2\ga$, and $\bv_2(\ga+\<C_i-C_j>)$ span~$\iH_2(\CY)$ (i.e.,
$\iH_2(\CY)$ consists of their linear combinations
$\sum\bv_2x_i$; with an abuse of
language we let $\sum\bv_2x_j=\bv_2\sum x_j$ provided that the latter
is well defined, even if the summonds are not well defined).
According to Theorem~3.2, Kalinin's intersection form on~$\iH_2(\CY)$
is the one given by Table~1. In the table, the intersection
$\ga\circ\gb$ is regarded as an element of~$H_0(Y_\R)$, and
$(\ga\circ\gb)[\RY]$ and $(\ga\circ\gb)[C_i]$ are, respectively, the
total intersection number and its portion falling into~$C_i$. \
$\delta_{ij}$~stands for the Kronecker symbol:  $\delta_{ii}=1$ and
$\delta_{ij}=0$ if $i\ne j$.  The intersection form extends linearly
to the classes $\bv_2(\ga+\bv_2\<C_i-C_j>)$, as if $\bv_2\ga$ and
$\bv_2\<C_i-C_j>$ were well defined.

\midinsert
\offinterlineskip
\def\sp{\quad}
\let\a\ga
\let\b\gb
\def\d[#1]{\delta_{#1}}
\def\\{\cr\noalign{\hrule}\omit\cr}
\let\c\circ
\topcaption{Table 1}\endcaption
\line{\hfill\vbox{
\halign{\vrule height11pt depth5pt\hfil\sp$#$\sp\hfil\vrule&&
 \hfil\sp$#$\sp\hfil\vrule\cr\noalign{\hrule}\omit\cr
                &\bv_2\<C_i-C_j>&\bv_2\a        &\bv_2[C_i]\\
\bv_2\<C_k-C_l> &0              &0              &\d[ik]+\d[il]\\
\bv_2\b         &0              &(\a\c\b)[\RY]  &(\b\c\b)[C_i]\\
\bv_2[C_k]      &\d[ik]+\d[jk]  &(\a\c\a)[C_k]  &\d[ik]\chi(C_i)\\
}}\hfill}
\endinsert

The Bockstein homomorphism $\Sq_1\:\iH_2(\CY)\to\iH_1(\CY)$
is given by Theorem~3.3: $\Sq_1\bv_2[C_i]=\bv_1w_1(C_i)$ and
$\Sq_1\bv_2(\ga+\<C_i-C_j>)=\bv_1\ga$.

\head
4.\enspace Other results
\endhead

The classification results cited in Section~2 can be considered as an
advanced experimental part of the study of the topology of real
algebraic surfaces. As any experiment, it serves not only to confirm
the applicability of a certain general theory but also to help to
discover new phenomena. Several examples have already been mentioned
in Section~1: the classification of real rational surfaces
(Theorem~2.1) led Comessatti to his inequality on the Euler
characteristic, and the Arnold-Gudkov-Rokhlin congruence~\cite{Ro}
was first observed for real $K3$-surfaces
(Statement~(1) of Theorem~2.4).\footnote{More precisely, the
congruence was conjectured by D.~Gudkov based on his classification
of plane sextics, which are closely related to $K3$-surfaces. Note
that the experimental material known to Ragsdale and Comessatti could
already reveal some congruences, but they both did not notice them
and put attention only to the inequalities.} The classification of
real Enriques surfaces also gives considerable material for
observations. Even a glance at the list of extremal types given by
Theorem~2.5 and at the complete list generated from it reveals
several regularities. One can notice, for example, that all the
$M$-surfaces are nonorientable and that the orientable
$(M-2)$-surfaces (appearing in the derived
list) satisfy the same congruences as $M$-surfaces.  In~\cite{DK2} we
established (and made use of) several general results of this kind.
(References to other related results known in the literature are also
found in~\cite{DK2}.) Below we suggest a slightly different
interpretation and generalization of this phenomenon.  We hope that
for a good observer the classification of real Enriques surfaces may
provide materiel for other discoveries.

\proclaim{4.1.\enspace Theorem}
Let $X$ be a complex closed surface with antiholomorphic involution,
and let $w_2(\CX)=0$. Then\rom:
\roster
\item
if $H_1(\CX)=0$, then $\RX$ is orientable\rom;
\item
if $H_1(\CX;\Q)=0$ and $\RX$ is nonorientable, then $X$ is an
$(M-d)$-surface, $d\ge2$, and
{\advance\normalparindent\rosteritemwd\roster
\letitem a
if $d=2$, then $\chi(\RX)\equiv\s(\CX)\pmod{16}$\rom;
\letitem b
if $d=3$, then $\chi(\RX)\equiv\s(\CX)\pm2\pmod{16}$\rom;
\letitem c
if\hbox{ $d=4$ }and $\chi(\RX)\equiv\s(\CX)+8\pmod{16}$, then
the image of $[\RX]$ belongs to
$\Tors H_2(\CX;\Z)\otimes\Z/2\subset H_2(\CX)$.
\endroster}\endroster
\endproclaim

\proclaim{4.2.\enspace Theorem}
Let $X$ be a complex closed surface with antiholomorphic involution,
and let $w_2(\CX)\ne0$. Then\rom:
\roster
\item
if $X$ is an $M$-surface, then $\RX$ is nonorientable\rom;
\item
if, besides, $w_2(\CX)\in\Tors H_2(\CX;\Z)\otimes\Z/2$ and $\RX$ is
orientable, then $X$ is an $(M-d)$-surface, $d\ge2$, and
{\advance\normalparindent\rosteritemwd\roster
\letitem a
if $d=2$, then $\chi(\RX)\equiv\s(\CX)\pmod{16}$\rom;
\letitem b
if $d=3$, then $\chi(\RX)\equiv\s(\CX)\pm2\pmod{16}$\rom;
\letitem c
if\hbox{ $d=4$ }and $\chi(\RX)\equiv\s(\CX)+8\pmod{16}$, then
the image of $[\RX]$ belongs to
$\Tors H_2(\CX;\Z)\otimes\Z/2\subset H_2(\CX)$.
\endroster}\endroster
\endproclaim

The congruence part (Statement~\therosteritem2) of Theorems~4.1
and~4.2 is proved similar to the well known Arnold-Gudkov-Rokhlin
type congruences. Let $\Discr H^+$ be the discriminant form of the
lattice $H^+$ of the $\conj_*$-invariant elements of
$H_2(\CX;\Z)/\Tors$. Then in both the theorems it suffices to prove
that, under the hypotheses, in~(a), (b), and~(c) one has
$\dim\Discr H^+=0$, $1$, and~$\le2$ respectively. This, in turn,
would follow from the fact that either $\dim\iH_1(\CX)<\dim H_1(\CX)$
or portion of $\Tors H_2(\CX;\Z)\otimes\Z/2\subset H_2(\CX)$ dies
in~$\iH_2(\CX)$. (We address an interested reader to~\cite{DK2,
\S6}.) It is this assertion that is actually proved below.

\demo{Proof of Theorem~4.1}
\therosteritem1 If there is an element $\ga\in H_1(\RX)$ with
$\ga^2=1$, then $(\bv_2\ga)^2=1$. ($\bv_2\ga$ is well defined since
$H_1(\CX)=0$.) This contradicts to the assumption that the
intersection form in~$H_2(\CX)$ and, hence, in~$\iH_2(\CX)$ is
even.

\therosteritem2 By assumption, there is an element $\ga\in H_1(\RX)$
with $\ga^2=1$. Similar to~\therosteritem1 one concludes that for any
such element  $\bv_1\ga\ne\bv_1C$ for any $C\in H_0(\RX)$ with
$\bv_0C=0$. (In particular, $\bv_1\ga\ne0$.) Furthermore, any
nonorientable component~$C_i$ of~$\RX$ is of even genus, i.e.,
$w_1^2(C_i)=0$. Now it is easy to see that $\bv_1\ga$ does not belong
to the image of $\Sq_1\circ\bv_2$. On the other hand,
$\Sq_1\:H_2(CX)/H_2(\CX;\Z)\otimes\Z/2\to\Tors_2H_1(\CY;\Z)=H_1(\CY)$
is an isomorphism; hence, $\Im\bv_2$ does not cover
$H_2(\CX)/H_2(\CX;\Z)\otimes\Z/2$.
\qed
\enddemo

\demo{Proof of Theorem~4.2}
\therosteritem1 If $\RX$ is orientable, there is no
element~$\ga\in H_*(\RX)$ with $(\bv_*\ga)^2=1$; hence, $w_2(\CX)$
dies in $\iH_2(\CX)$ and the surface is not maximal.

\therosteritem2 According to~\therosteritem1, portion of
$\Tors H_2(\CX;\Z)\otimes\Z/2$ (at least~$w_2(\CX)$) dies in
$\iH_2(\CX)$.
\qed
\enddemo

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\end{thebibliography}