Nonstandard Lagrangian submanifolds in $\CP^n$
River Chiang
We use Hamiltonian actions to construct nonstandard (as opposed to $\RP^n$ and $T^n$) Lagrangian submanifolds of $\CP^n$. First of all, a quotient of $\RP^3$ by the dihedral group $D_3$ is a Lagrangian submanifold of $\CP^3$. Secondly, $\Su(n)/\Z_n$ are Lagrangian submanifolds of $\CP^{n^2-1}$.
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\begin{document}

\title{Nonstandard Lagrangian submanifolds in $\CP^n$}
\author{River Chiang}
\begin{abstract}
We use Hamiltonian actions to construct nonstandard 
(as opposed to $\RP^n$ and $T^n$)
Lagrangian submanifolds of $\CP^n$. First of all, a 
quotient of $\RP^3$ by the dihedral group $D_3$ is a 
Lagrangian submanifold of $\CP^3$. Secondly, $\Su(n)/\Z_n$ 
are Lagrangian submanifolds of $\CP^{n^2-1}$.
\end{abstract}
\maketitle

\comment{only need isotropy orbit with the right dimension}

\comment{$S^3=\So(4)/\So(3)$ in $M^6$?}

\section{Introduction}

Let $(M, \ow)$ be a symplectic manifold. A submanifold $L \subset M$
is \textbf{Lagrangian} if $\dim L = \frac{1}{2}\dim M$ and 
$\ow$ vanishes on $T(L)$.

Using Floer homology, Paul Biran has obtained information on the
topology of Lagrangian submanifolds of some closed manifolds. 
One of Paul Biran's theorems concerns the Lagrangian submanifolds of
$\CP^n$ \cite{B-1, B-2}: 

\begin{theorem}\labell{T:b1}
Let $L$ be a Lagrangian submanifold of $\CP^n$ such that 
$H_1(L;\Z)$ is $2$-torsion (that is, $2H_1(L;\Z)=0$). Then
\begin{enumerate}
\item $H^*(L; \Z_2) \cong H^*(\RP^n; \Z_2)$ as graded vector spaces. 
%\item Let $\gamma$ be a generator of $H^2(\CP^n;\Z_2)$. Then $\beta=\gamma|_L$
%is a generator of $H^2(L;\Z_2)$. Let $\alpha$ be a generator of
%$H^1(L; \Z_2)$. Then $1, \alpha, \beta, \alpha\beta, \beta^2, \dots$
%span the algebra $H^*(L; \Z_2)$.
\item If $n$ is even, the isomorphism in $(1)$ is an isomorphism 
of graded algebras. 
\end{enumerate} 
\end{theorem}

Seidel \cite{Se} also has a theorem with regard to the same subject:

\begin{theorem}\labell{T:se}
Let $L$ be a Lagrangian submanifold of $\CP^n$. Then
\begin{enumerate}
\item $H^1(L;\Z_{2n+2})\neq 0$.
\item $H^1(L;\Z_{2n+2})\ncong (\Z_2)^g$ for any $g \geq 2$.
\item If $H^1(L;\Z_{2n+2}) \cong \Z_2$, then $H^i(L;\Z_2) \cong \Z_2$ 
for all $i=0, \dots, n$.
\end{enumerate}
\end{theorem}


To test the limits of these topological restriction on Lagrangian submanifolds 
of $\CP^n$, we construct new examples using Hamiltonian actions. The 
constructions in this paper relies on the following well-known fact:

\mute{}{By starting with the standard Lagrangian embeddings of $\RP^n$ and
$T^n$ and applying Lagrangian surgery, one can construct many
different Lagrangian submanifolds of $\CP^n$.} 

\comment{does surgery give this example? does surgery give examples such that 
$H^*(L;\Z_2)$ is not $H^*(\RP^n; \Z_2)$? It seems that $H_1$ is the only 
important thing}

\begin{proposition}\labell{P:iso}
Let a compact Lie group $G$ act on a symplectic manifold $(M, \ow)$ 
in a Hamiltonian fashion with a moment map 
$\Phi\colon M \to \fg^*$. Assume $\alpha \in \fg^*$ is a fixed point of the
coadjoint action. Then for any $m\in \Phi^{-1}(\alpha)$,
the orbit $G\cdot m$ is an isotropic submanifold of $M$. 
\end{proposition}

\begin{proof}
An orbit is smooth. We only need to show that it is isotropic. 
Let $\xi$ and $\eta$ be any vector fields in the Lie algebra $\fg$ of $G$ and
let $\xi_M$ and $\eta_M$ be their induced vector fields on $M$.  
\[ 
\begin{split}
\ow(m)(\xi_M(m),\eta_M(m)) 
&= d\la \Phi(m), \xi \ra (\eta_M(m)) \\
&=\la d\Phi(\eta_M), \xi \ra \\
&=\la \eta_{\fg^*}(\Phi(m)), \xi \ra\\
%&=\la \eta_{\fg^*}(\alpha), \xi \ra \\
&=\la \frac{d}{dt}\bigr|_0\exp t\eta \cdot \alpha, \xi \ra\\
&=\la 0, \xi \ra 
\end{split}
\]
Therefore $\ow(m) = 0$ for all $m\in \Phi^{-1}(\alpha)$. In particular, 
the orbit $G\cdot m$ is a smooth isotropic submanifold of $M$.  
\end{proof}

In particular, any orbit of an abelian action is isotropic as well as
any orbit in the zero level set of a nonabelian action.

\begin{example}
Consider the action of the torus $T^n$ on $\CP^n$ given by
\[
(\lambda_1, \dots, \lambda_n)\cdot[z_0, z_1, \dots, z_n]
= [z_0, \lambda_1^{-1}z_1, \dots,\lambda_n^{-1}z_n].
\] This action is Hamiltonian and its moment map is 
\[
\Phi([z_0, z_1, \dots, z_n])=\frac{1}{2}\left(\frac{|z_1|^2}{\|z\|^2}, \dots, 
\frac{|z_n|^2}{\|z\|^2}\right),
\]where $\|z\|^2 = \sum_{i=0}^n|z_i|^2$. The free orbit $T^n$ is a 
Lagrangian submanifold of $\CP^n$.  
\end{example}

 
Our first example is a Lagrangian submanifold of $\CP^3$. 
Its topology 
has an interesting deviation from Theorems \ref{T:b1}
and \ref{T:se}. 

\begin{theorem}
There exists a Lagrangian submanifold of $\CP^3$, which is a quotient of 
$\RP^3$ by the dihedral group $D_3$.
\end{theorem}

We discuss the topology of this submanifold in 
Section \ref{S:top}.     

We use different actions to construct a second family of examples.
It provides a generalization of the standard 
Lagrangian submanifold $\RP^3$ of $\CP^3$. 

\begin{theorem}
$\Su(n)/\Z_n$ is a Lagrangian submanifold of $\CP^{n^2-1}$.
\end{theorem}

\section{First example}\labell{S:con1}

View $\C^4$ as the space of homogeneous polynomials 
of degree $3$ in two variables. We can define the
irreducible $\Su(2)$ representation $\rho\colon \Su(2) \to \Gl(\C^4)$ by
\[
\rho(A)(p)(z) = p(zA)
\]
where $p \in \C^4$, $A \in \Su(2)$ and $z = (x, y) \in \C^2$.  
The moment map for this representation as a map from $\C^4$ into $\R
\times \C \cong \su(2)^*$ is given by
\[
(u_0, u_1, u_2, u_3) \mapsto
(\frac{3}{2}|u_0|^2+\frac{1}{2}|u_1|^2-\frac{1}{2}|u_2|^2-\frac{3}{2}|u_3|^2,
u_0\overline{u_1}+u_1\overline{u_2}+u_2\overline{u_3}). 
\]
So the zero level set of the moment map must satisfy
\[
\begin{cases}
&3|u_0|^2+|u_1|^2=|u_2|^2+3|u_3|^2,\\
&u_0\overline{u_1}+u_1\overline{u_2}+u_2\overline{u_3}=0.
\end{cases}
\]

This $\Su(2)$ action commutes with the diagonal $S^1$ action on
$\C^4$, and therefore descends to an action on $\CP^3$. 

Clearly the point $[1, 0, 0, 1]$ is in the zero level set. 
We know that  
\[
(x, y)\begin{pmatrix}
\alpha & \beta\\
-\overline{\beta}&\overline{\alpha}
\end{pmatrix} = (\alpha x- \overline{\beta}y, \beta x + \overline{\alpha}y),
\] where $|\alpha|^2+|\beta|^2=1$. To compute the stabilizer, we set 
$\lambda (x^3+y^3) = (\alpha x- \overline{\beta}y)^3 + (\beta x +
\overline{\alpha}y)^3$ for any possible $\lambda$ in $S^1$. 
Direct computation shows that $\lambda = \pm 1$ when $\beta =0$ and
$\lambda = \pm i$ when $\alpha = 0$. So the stabilizer is 
generated by 
\begin{equation}\label{E:generator}
\begin{pmatrix}
0 & i \\
i & 0
\end{pmatrix} \quad \text{and} \quad
\left\{
\begin{pmatrix}
\alpha & 0 \\
0 &\overline{\alpha}
\end{pmatrix}, \alpha^6=1 \right\}
\end{equation}

Note this $\Su(2)$ action is not effective on $\CP^3$. It induces
an effective $\So(3)$ action and the stabilizer for $[1,0,
0,1]$ is the semidirect product $\Z_2 \ltimes \Z_3$, namely, the 
dihedral group $D_3$. 

Therefore, we have a Lagrangian $L = \So(3)/{D_3} \cong \RP^3/{D_3}$ 
in $\CP^3$. 

In fact, this is the only orbit in the zero level set. 
According to the Guillemin-Marle-Sternberg local 
normal form theorem, a neighborhood of the orbit
of $[1,0,0,1]$ is equivariantly symplectomorphic to a neighborhood of
the zero section in $Y = \So(3)\times_{D_3} \R^3$. It is easy to check that
there exists one single orbit in the zero level set in $Y$. Since the zero 
level set is connected and since the orbit is closed, it follows that the
zero level set in $\CP^3$ consists of one orbit only.

\section{Topology for the first example}\labell{S:top}

Let $\Gamma$ denote the group generated by the elements in 
(\ref{E:generator}). Then the fundamental group of $L$ is
$\pi_1(L) = \Gamma$ since $L= \RP^3/D_3 = S^3/\Gamma$.

Let $b$ denote $\left(\begin{smallmatrix}
0 & i \\
i & 0
\end{smallmatrix}\right)$ and $a$ denote 
$\left(
\begin{smallmatrix}
\alpha & 0 \\
0 &\overline{\alpha}
\end{smallmatrix}\right)$, $\alpha^6=1$. Then $b^2 = -1$ and $ba = a^5b$. 
The commutator subgroup $A = \{\,x^{-1}y^{-1}xy\,|\,x, y \in 
\pi_1(L)\, \}$ is $\{\,1, a^2, a^4\,\}$. By Hurewicz Theorem, 
the first homology group $H_1(L; \Z)$ is $\{\,1, b, b^2, b^3\,\}
\cong \Z_4$.  

Using the Universal Coefficient Theorem and Poincar\'e duality, we
can calculate that
\[
\begin{aligned}
H_0(L;\Z)&=\Z,\qquad &H^0(L;\Z)&=\Z, \qquad &H^0(L;\Z_2)=\Z_2,\\
H_1(L;\Z)&=\Z_4,\qquad &H^1(L;\Z)&=0, \qquad &H^1(L;\Z_2)=\Z_2,\\
H_2(L;\Z)&=0, \qquad &H^2(L;\Z)&=\Z_4, \qquad &H^2(L;\Z_2)=\Z_2,\\
H_3(L;\Z)&=\Z, \qquad &H^3(L;\Z)&=\Z, \qquad &H^3(L;\Z_2)=\Z_2.
\end{aligned}
\]
Also, $H^1(L;\Z_8) = \Z_4$.

The Lagrangian $L$ we found does not 
satisfy the assumption of $2$-torsion as in Theorem \ref{T:b1} and 
Theorem \ref{T:se}(3), 
but $H^*(L; \Z_2)$ is isomorphic to 
$H^*(\RP^3; \Z_2)$ as graded vector spaces. 

\comment{can we say something about generators?}

\comment{add a general statement about extending the theorems?}


\section{Second family of examples$^1$}\labell{S:con2}

\addtocounter{footnote}{+1}
\footnotetext{I would like to thank Mich\`ele Audin for showing me a
beautiful alternative construction of these examples in terms of 
real forms of Lie groups \cite{Au}.}

Consider $\C^{n^2}$ as the set of $n$ by $n$ matrices. $\Su(n)$ acts 
naturally by left multiplication. This action is Hamiltonian and its 
moment map is given by
\[
\Phi(Z) = \frac{i}{2}\,ZZ^* + \frac{1}{2i}\,\text{I}
\] where $I$ is the identity matrix. 

This $\Su(n)$ action 
commutes with the circle action of multiplication by $\lambda\,\text{I}$.
Therefore, it descends to an action on $\CP^{n^2-1}$ where the center $\Z_n$
of $\Su(n)$ acts trivially. Therefore, 
a free orbit in the zero level set of $\C^{n^2}$ descends to an orbit in 
the zero level set of $\CP^{n^2-1}$ with stabilizer $\Z_n$. Since
the dimension of $\Su(n)$ is $n^2-1$, by Proposition
\ref{P:iso}, $\Su(n)/\Z_n$ is a Lagrangian submanifold of $\CP^{n^2-1}$. 

For $n=2$, the above-mentioned orbit $\Su(2)/\Z_2$ is the 
standard Lagrangian $\RP^3$ in $\CP^3$. Indeed it is the projective
space of the real $4$ dimensional space $V$ spanned by the following
matrices: 
\[
\ma{1&0\\0&-1}, \ma{i&0\\0&i}, \ma{0&1\\1&0}, \ma{0&i\\-i&0}.
\]


\comment{topology? the topology of $\Su(n)$ in terms of the topology of 
a product of odd spheres?}


\mute{}{
\section{Open questions}

The Lagrangian in Section \ref{S:con1} comes from an irreducible $\Su(2)$
representation while the Lagrangians in Section \ref{S:con2} come from
reducible $\Su(n)$ actions. We hope to find other nonstandard Lagrangians
by other Hamiltonian $\Su(n)$ actions. 

\comment{need this at all? give out too much?}

\mute{}{
This construction does not readily generalize to similar examples of
Lagrangian submanifolds of $\CP^n$. On the other hand, 
multiplicity free representations should provide other Lagrangians.
\cite{Kn}.} 

We also learned from Paul Biran the following construction: 
%which is due to Audin, Lalonde and Polterovich 
%\cite{A-L-P}. 

\comment{check alp again}

\begin{theorem}\labell{T:alp}
Let $G$ be a compact Lie group and let $(M, \ow)$ be a symplectic manifold. 
Assume $G$ acts on $M$ in a Hamiltonian fashion with moment map $\Phi$.
Denote by $Z$ the preimage of zero of the moment map and by $M_{\text{red}}$  
the reduced space $Z/G$. If $0$ is regular, $Z$ is a Lagrangian submanifold
of $M \times M_{\text{red}}$. 
\end{theorem}

\comment{$G$ acts on $Z$ freely and properly to ensure that
$M_{\text{red}}$ is a manifold?}

\begin{proof}
The dimension of the reduced space is 
$\dim M_{\text{red}}=\dim M - 2\dim G$, so $\dim Z = \dim M - \dim
G = \frac{1}{2}\dim (M \times M_{\text{red}})$.

Let $i\colon Z \to M$ be the inclusion map and $\pi\colon Z \to
M_{\text{red}}\cong Z/G$ be the projection. There exists a unique
symplectic form $\ow_{\text{red}}$ on the reduced space 
$M_{\text{red}}$ such that $\pi^*\ow_{\text{red}}= i^*\ow$.
The two-form $\Tilde{\ow} = \text{pr}_1^*\,\ow - 
\text{pr}_2^*\,\ow_{\text{red}}$ is then a symplectic form on $M\times
M_{\text{red}}$. We have a natural embedding $\varphi\colon Z \to M\times
M_{\text{red}}$ given by 
\[
z \mapsto (i(z), \pi(z)).
\] 
Now 
\[
\begin{split}
\varphi^*\Tilde{\ow} &= \varphi^*\text{pr}_1^*\,\ow -
\varphi^*\text{pr}_2^*\,\ow_{\text{red}} \\
&= (\text{pr}_1 \circ \varphi)^*\ow -
(\text{pr}_2\circ\varphi)^*\ow_{\text{red}} \\
&=i^*\ow - \pi^*\ow_{\text{red}}. 
\end{split}
\]
Therefore $\varphi^* \Tilde{\ow}= 0$ and $\varphi$ is a Lagrangian
embedding. 
\end{proof}



\begin{example}
Let $S^1 = \{\lambda \in \C : |\lambda|=1\}$ act on $(\C^{n+1},
\ow_{\text{std}})$ by multiplication.
%\[
%\lambda \cdot (z_1, z_2, \dots, z_n) = (\lambda z_1, \lambda z_2,
%\dots, \lambda z_n). 
%\] 
This action is Hamiltonian with a moment map 
$\Phi(z) = -\frac{1}{2}|z|^2+\frac{1}{2}$.
The zero level set $\Phi^{-1}(0)$ is $S^{2n+1}$ and the 
reduced space $\Phi^{-1}(0)/S^1$ is $\CP^{n}$. By Theorem
\ref{T:alp}, there exists a Lagrangian sphere $S^{2n+1} \subset \CP^n
\times \C^{n+1}$. 
\end{example}




\comment{We do not know if there are similar examples of Lagrangian submanifolds in
$\CP^n$. 
This construction can not readily generalize to $\Su(n)$ actions on 
$\CP^m$ for all $m$ since the dimensions don't match. Even when the
dimensions do match, we don't have $\RP^n \subset \CP^n$}

\comment{but don't know yet how to compute other actions and whether 
there will be something recognizable}

\begin{remark}
We can extend Theorem \ref{T:alp} with the shifting trick. 
Let a compact Lie group $G$ act on a symplectic manifold $(M, \ow)$
with a moment map $\Phi$ and let $\orbit \neq \{0\}$ be a
coadjoint orbit in $\Phi(M)$. Consider
the action of $G$ on the manifold $M' = M \times \orbit$ with
symplectic structure $\ow' = \ow \times (-\ow_\orbit)$
where $\ow_\orbit$ is the standard symplectic form on the coadjoint
orbit $\orbit$. This action is Hamiltonian and the
corresponding moment map is given by
\[
\Phi'(m, \mu) = \Phi(m) - \mu.
\] 
It is easy to see that ${\Phi'}^{-1}(0) \cong \Phi^{-1}(\orbit)$. 
By Theorem \ref{T:alp}, we have a Lagrangian embedding 
$\Phi^{-1}(\mathcal{O}) \subset M \times \mathcal{O} \times
M_{\text{red}}$. Here $M_{\text{red}}$ denotes $\Phi^{-1}(\orbit)/G$.   
If $G=\Su(2)$ or $\So(3)$, this implies that 
$\Phi^{-1}(\mathcal{O}) \cong G \times_{S^1} W$ is a 
Lagrangian submanifold of $M \times M_{\text{red}} \times \CP^1$ where
$W$ is a level set of a Hamiltonian circle action. The cohomology of
$\Phi^{-1}(\mathcal{O})$ is computable by means of
$S^1$-equivariant cohomology. 
\end{remark}

\comment{notation for $\ow \times (-\ow_\orbit)$?}

Along these lines of thoughts, we
hope to provide other examples to support Theorems \ref{T:b1} and \ref{T:se}
as well as the following two Theorems of Paul Biran \cite{B-1, B-2}:

\comment{for example, know the reduced space of complexity two space 
from the results of Hui, need to use equivariant cohomology, especially 
need to know the Kunneth formula, but should be able to work out at 
least the trivial example when the level set is $\CP^2 \times \R$?}

\comment{but it is hard to say if there is anything interesting}

\begin{theorem}\labell{T:b2}
Let $L \subset \CP^n \times \CP^n$ be a Lagrangian submanifold with 
$H_1(L; \Z) = 0$. Then $H^*(L; \Z_2) \cong H^*(\CP^n; \Z_2)$ as 
graded algebras. 
\end{theorem}

\begin{theorem}\labell{T:b3}
Let $X^{2m}$ be a $2m$ dimensional closed symplectic manifold with 
$\pi_2(X)=0$. Suppose that $\CP^n \times X^{2m}$ has a Lagrangian sphere,
where $n\geq 1$ and $n+m \geq 2$. Then $m \equiv n+1 \pmod{2n+2}.$
\end{theorem}
}



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  \bibitem[B-2]{B-2} P. Biran, {\em Geometry of symplectic intersections.} 
	 Proceedings of the International Congress of Mathematicians 
	(Beijing 2002), Vol. II, 241--255.

%  \bibitem[B-3]{B-3} P. Biran, {\em Lagrangian barriers and symplectic
%     embeddings.} Geom. Funct. Anal. \textbf{11} (2001), no. 3,
%   407--464.
   
%  \bibitem[B-4]{B-4} P. Biran, {\em Lagrangian non-intersections.} In
%   preparation.
   
%  \bibitem[B-5]{B-5} P. Biran, {\em Algebraic degenerations and Floer
%     homology.} In preparation.
   
%  \bibitem[B-C-1]{B-C-1} P. Biran and K. Cieliebak, {\em Symplectic
%     topology on subcritical manifolds.} To appear in Comment. Math.
%   Helv.
   
%  \bibitem[B-C-2]{B-C-2} P. Biran and K. Cieliebak, {\em Lagrangian
%     embeddings into subcritical Stein manifolds.} To appear in Israel
%   J. Math.
   
%  \bibitem[E-G]{E-G} Y. Eliashberg and M. Gromov, {\em Convex
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%   Providence, RI, 1991.
   
%  \bibitem[F-1]{F-1} A. Floer, {\em Witten's complex and
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%   \textbf{30} (1989), no. 1, 207--221.
     
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\end{thebibliography}



\end{document}

math0303262.extracted_bib
\begin{thebibliography}{9999999}
   
  \bibitem[Au]{Au} M. Audin, Unpublished notes. 

  \bibitem[A-L-P]{A-L-P} M. Audin, F. Lalonde and L. Polterovich, {\em
     Symplectic rigidity: Lagrangian submanifolds.} In Holomorphic
   curves in symplectic geometry.  Edited by M. Audin and J.
   Lafontaine. Progress in Mathematics, \textbf{117}. Birkh\"{a}user
   Verlag, Basel, 1994.

  \bibitem[B-1]{B-1} P. Biran, {\em Homological uniqueness of 
     Lagrangian submanifolds.} In preparation.

  \bibitem[B-2]{B-2} P. Biran, {\em Geometry of symplectic intersections.} 
	 Proceedings of the International Congress of Mathematicians 
	(Beijing 2002), Vol. II, 241--255.

%  \bibitem[B-3]{B-3} P. Biran, {\em Lagrangian barriers and symplectic
%     embeddings.} Geom. Funct. Anal. \textbf{11} (2001), no. 3,
%   407--464.
   
%  \bibitem[B-4]{B-4} P. Biran, {\em Lagrangian non-intersections.} In
%   preparation.
   
%  \bibitem[B-5]{B-5} P. Biran, {\em Algebraic degenerations and Floer
%     homology.} In preparation.
   
%  \bibitem[B-C-1]{B-C-1} P. Biran and K. Cieliebak, {\em Symplectic
%     topology on subcritical manifolds.} To appear in Comment. Math.
%   Helv.
   
%  \bibitem[B-C-2]{B-C-2} P. Biran and K. Cieliebak, {\em Lagrangian
%     embeddings into subcritical Stein manifolds.} To appear in Israel
%   J. Math.
   
%  \bibitem[E-G]{E-G} Y. Eliashberg and M. Gromov, {\em Convex
%     symplectic manifolds.} In {\em Several complex variables and
%     complex geometry}, Part 2 (Santa Cruz, CA, 1989), 135--162, Proc.
%   Sympos. Pure Math., \textbf{52}, Part 2, Amer. Math. Soc.,
%   Providence, RI, 1991.
   
%  \bibitem[F-1]{F-1} A. Floer, {\em Witten's complex and
%     infinite-dimensional Morse theory.} J. Differential Geom.
%   \textbf{30} (1989), no. 1, 207--221.
     
%  \bibitem[F-2]{F-2} A. Floer, {\em Morse theory for Lagrangian
%     intersections.} J. Differential Geom. \textbf{28} (1988), no. 3,
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\end{thebibliography}