Realizing homology classes by symplectic submanifolds
H\^ong-V\^an L\^e
Inselstra\ss e 22-26
D-04103 Leipzig
Germany
2n
math0505562.tex
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\title{Realizing homology classes by symplectic submanifolds}
\author{\bf  H\^ong-V\^an L\^e \\
Max-Planck-Institute for Mathematics in Sciences\\
  Inselstra\ss e 22-26\\
  D-04103 Leipzig, Germany\\
  }

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\maketitle
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\medskip
\begin{flushright}
{\it  \small Dedicated to Anatoly Fomenko on the occasion of his 60th birthday}.
\end{flushright}

\abstract  In this note we prove that a positive multiple of each  even-dimensional integral homology class of a compact symplectic manifold $(M^{2n}, \om)$ can be represented as the difference of the fundamental classes of two  symplectic submanifolds in $(M^{2n}, \om)$.  We also prove  the realizability of some integral homology classes by symplectic submanifolds in $(M^{2n}, \om)$.


\section {Introduction.}

In 1954 Rene Thom prove the following celebrated theorem which relates
the topological structure  with the differentiable structure on compact manifolds.

\medskip

\textbf{1.1. Theorem.} [Thom1954, Theorem II.25]. \textit{For each element $\alpha \in H_k (M^m,\Z)$
of  a compact differentiable manifold $M^m$ there exists a positive number $N(k,m)$
such that the element $N(k,m)\cdot \alpha$ can be realized by a  differentiable submanifold
in $M^m$. }
\medskip

Thom's theorem is optimal in the sense that we cannot replace $N(k,m) = 1$. Namely
Thom showed that for each $k \ge 7$ there is a compact differentiable manifold
$M^m$ and  an element $\alpha \in H_k (M^m, \Z)$ such that $\alpha$ cannot be realized
by the fundamental class of a submanifold in  $M^m$ [Thom1954].

\medskip

As an immediate consequence of Thom's theorem 1.1 we get that  the homology group
$H_*(M^m, \Q)$ is generated by the fundamental classes of differentiable  submanifolds of $M^m$. 
\medskip


In this note we prove the following
weak version of Thom's theorem for compact symplectic manifolds.

 \medskip

{\bf 1.2.  Theorem. }{\it Suppose that $(M^{2n},\om)$ is a compact symplectic manifold.  Then  for each element $\alpha\in H_{2k}(M ^{2n}, \Z)$, $1\le k \le n$, there exists
a positive number $N(\alpha)\in \N^+$ such that $N(\alpha) \cdot \alpha = [S_1 ^{2k}]- [S_2 ^{2k}]$,
where $S_1^{2k}$ and $S_2 ^{2k}$ are symplectic submanifolds in $(M^{2n},\om)$ .}

\medskip

A particular case of Theorem 1.2 is 

\medskip

{\bf 1.3. Theorem.} [Donaldson1996] {\it  Let $(M^{2n}, \om)$ be a compact integral
symplectic manifold, i.e. the cohomology class $[\om]$ belongs to $H^2 (M,\Z)
\subset H^2 (M, \R)$. Then for each $1 \le k \le n$ there exists a number 
$N_1(k)\in \N^+$ such that $N \cdot PD ([\om ^{n-k}])$ can be realized by
a symplectic   submanifold $S_1 ^{2k}$ in $(M^{2n}, \om)$, if $N \ge N_1 (k)$.}

\medskip

Actually in his paper [Donaldson1996] 
Donaldson has stated his result  as in Theorem 1.3 only for $k =n- 1$. Kaoru Ono
has noticed that using a simple  argument we can easily get Theorem 1.3 for $1\le k \le n-2$ as a consequence of [Donaldson1996, Theorems 3, 5]. We shall represent Ono's argument
in section 2. Theorem 1.3 also follows from a result of Auroux [Auroux2000] (see Proposition 2.4 below) which generalizes Donaldson's arguments. 

\medskip

We shall call a homology class $\alpha \in H_{2k} (M,\Z)$ {\bf symplectic},
if $\alpha$ is the  fundamental class of a symplectic  submanifold in $M^{2n}$.
We shall call a homology class $\alpha \in H_{2k} (M,\Z)$ {\bf formal symplectic},
if the pairing $<\alpha,\om^k>$  is positive. 


\medskip 

{\bf 1.4. Conjecture.} {\it For each formal symplectic  class $\alpha \in H_{2k} (M^{2n},\Z)$ there exists a positive number $N_1(\alpha)\in \N^+$ such that
$N_1(\alpha)\cdot \alpha$ is symplectic.} 


\medskip

For $k =1$ and $n \ge 3$ the conjecture 1.4 is true. More precisely we have

\medskip 

{\bf 1.5. Theorem. }{\it Let $(M^{2n}, \om)$ be a compact symplectic manifold with $2n \ge 6$. \\
a) If $\om_{\pi _2 (M^{2n})} \not = 0$, then there exists a symplectic sphere in $(M^{2n},
\om)$.\\
b) The conjecture 1.4 is true for $\alpha \in H_2 (M^{2n},\Z)$.}


\medskip

I am deeply  thankful to  Simon
Donaldson and   J\"urgen Jost for their interests and
supports. I am greatly indebted to Kaoru Ono for his quick
help  and criticism and I thank Dietmar Salamon for pointing out an error in the early version of this note. It is my pleasure to dedicate this note to my teacher Anatoly Fomenko, for he introduced me into  algebraic topology and symplectic geometry. He gave me the book [Rass1958] from his library and urged me to study
the classical works of the French topology school in the last century, and so he influenced  me  over this work directly.
\medskip

\section {Proof of  Theorem 1.2.}

Clearly Theorem 1.2 is a consequence of  the following 

\medskip

{\bf 2.1. Theorem.} {\it Let $(M^{2n}, \om)$ be a compact  symplectic manifold
and $\alpha \in H_{2k} (M^{2n}, \Z)$. Then there exist an integral symplectic
form $\bar\om$ on $M^{2n}$ and  positive integral numbers
$N_2(\alpha) $ and $N_3(\alpha)$ such that $[N_2(\alpha) \cdot \alpha  + N_3(\alpha) PD [\bar\om^{n-k}]]$ and $N_3(\alpha)\cdot PD[\bar \om^{n-k}]$ can be realized by  symplectic submanifolds, respectively $S_i ^{2k}$ of $(M^{2n}, \om)$, $i = 1,2$.}

\medskip

In this section we shall give a proof of Theorem 2.1.
First we prove Theorem 2.1 for  compact integral symplectic manifolds $(M^{2n},\om)$
(and  $\bar \om  = \om$ for this case) with certain estimates for obtained symplectic submanifolds. Next for an arbitrary symplectic  form $\om$ we shall find a rational symplectic  form $\bar\om$ close
enough to $\om$ such that the obtained  $\bar \om$ -symplectic submanifolds
are also $\om$-symplectic submanifolds. 


\medskip

Our needed estimates for symplectic  submanfolds  are expressed in  terms
of $\eta$-transversality and  $C^k$-approximately  (resp. $C^k$-asymptotically) homolomorphic maps. These notions
have been first introduced by Donaldson for mappings from an almost complex
manifold $(M^{2n},J)$ with a Riemannian metric $g$ to $\C$ [Donaldson1996] and later extended by Auroux for
mappings from $(M^{2n}, J)$ with a Riemannian metric $g$ to $\C^k$ (see e.g. [Auroux2000]).

\medskip


{\bf 2.2. Definition.} [Donaldson1996], [Auroux2000].
Given a constant $\eta > 0$, we say that a section $s$ of a vector bundle carrying
a metric and a connection is {\bf $\eta$-transverse to $0$}, if at every point $x$ such that
$|s(x)|\le \eta$ the covariant derivative $\nabla s(x)$ is surjective and admits
 a right inverse of norm less than $\eta^{-1}$.
 
\medskip

{\bf 2.3. Definition.} [Donaldson1996], [Auroux2000]. Let $(M, J)$ be an almost-complex
manifold with a Riemannian metric and let $s$ be a section of an almost complex
vector bundle with metric and connection. Given a sequence $\{ (C_i, c_i), 1\le i \le k\}$ we say that  $s$ is {\bf $C^k$-approximately holomorphic with bounds
$\{ (C_k, c_k)\}$}, if it satisfies the following estimates for all $1\le r \le k$
$$ | \nabla ^r s | \le C_r\cdot c_r ^{ -r /2}, \:  |\nabla^r \bar \p_J|\le  C_r\cdot c_r ^{-{r+1\over 2}},$$
where $\bar \p _J$ is the $(0,1)$-part of the connection.

Moreover, given constants $c_k \to \infty$, we say that a sequence $\{ s_k\}$ of sections is {\bf $C^k$-asymptotically  holomorphic}, if  there exists a sequence
$\{ C_k\}$ such that  each section $c_k$, $k >> 0$, is $C^k$ approximately
holomorphic with bounds $\{ (C_k, c_k)\}$.

If $k =\infty$ we shall only say approximately holomorphic sections, asymptotically
holomorphic sequences.

\medskip

Our proof of  Theorem 2.1 is based on the following result by Auroux.

\medskip

{\bf 2.4. Proposition.} [Corollary 5.1, Auroux2000]. {\it  Let $(M^{2n},\om)$ be a compact symplectic manifold endowed with
an $\om$-tame almost-complex structure $J$, and let $E_k$ be an asymptotically very
ample sequence of locally splittable vector bundles over $(M^{2n}, J)$. Then, for all
large enough values of $k$ there exist asymptotically holomorphic sections $s_k$ of
$E_k$ which are uniformly transverse to 0 and whose zero sets are smooth symplectic submanifolds in $M^{2n}$. }

\medskip

Recall that  a sequence of complex vector bundles $E_k$ with metrics and connections 
over a compact  almost-Hermitian manifold $(M, g, J)$ is
{\bf asymptotically very ample}, if there exist constants $\delta$, $\{ C_r\}, _{r\ge 0},$
and a sequence $c_k \to + \infty$ such that the curvature $F_k$  of $E_k$ satisfies the following
properties\\
1) $<i F_k(v, Jv)\cdot u, u> \ge c_k  g(v,v) |u| ^2 , \forall v \in TX, \forall u \in E_k$,\\
2) $\sup | F^{0,2} _k| _g \le \delta _r c_k ^{1/2}$,\\
3)$\sup | \nabla ^r F_k| _g \le C_r c_k, \:  \forall r \ge 0$.

A sequence of asymptotically very ample complex vector bundles $E_k$ with
metrics  $|.|_k$ and connections $\nabla _k$ is {\bf locally splittable}, if
given any point $x\in X$, there exists over a ball of of fixed $g$-radius around $x$ a decomposition of $E_k$ as a direct sum $L_{k,1}\oplus \cdots \oplus L_{k,m}$ of line
bundles such that the following properties hold:\\
1) the $|.|_k$-determinant of a local frame consisting of unit length local sections
of $L_{k,1}, \cdots L_{k,m}$ is bounded from  below by a fixed  constant independently
of $x$ and $k$,\\
2)denoting by $\nabla_{k,i}$ the connection on $L_{k,i}$ obtained by projecting $\nabla
_{k| L_{k,i}} $ to $L_{k,i}$ and by $\nabla '_k$ the direct sum of the $\nabla_{k,i}$,
the 1-form $\alpha _k = \nabla _k - \nabla _k '$ ( the non-diagonal part of
$\nabla _k$) satisifies the uniform bounds $|\nabla ^r \alpha _k| _g =  O ( c_k ^{r/2})$ for all $r\ge 0$ independently of $x$.

\medskip

{\bf 2.5. Remark-Example.} [Auroux2000]. If $E$ is a fixed complex vector bundle and
$L_k$ are asymptotically very ample line bundles, then the vector bundles $E\otimes L_k$
are locally splittable and asymptotically very ample, so are direct sums of vector
bundles of this type.

\medskip



Proposition 2.4 is a generalization of Theorem 5 in [Donaldson1996] which is
a special case of  Proposition 2.4 for $E^1 = L^{\otimes k}_\om$.  Here $L_\om$ denotes
the complex line bundle over $(M^{2n},\om)$ whose curvature is equal to $\om$.





\medskip

Our proof of  Theorem 2.1 also uses the following Proposition due to Thom [Thom1954].  Let $G$ be
a subgroup of $O(k)$ and $M^n$ be a differentiable manifold. We
say that  a cohomology  class  $\alpha \in H_k (M^n, \Z)$ is realizable w.r.t.  $G$, if
there is a $G$-vector bundle $V^k$ of dimension $k$ over $M^n$ whose Euler class
$e(V^k)$ is equal to $\alpha$ (see [Thom1954]).


\medskip

{\bf 2.6. Proposition.} [Thom1954 Theorem II.25]. {\it For each cohomology class $z \in 
H^k (M^{n}, \Z)$ there exists a number $N(k, n)\in \N^+$ such that the class $N(k,n)\cdot z$
is realizable w.r.t. $SO(k)$. If $k= 2l$, then there exists a number $N_1 (k,n)\ge
N(k,n)$ such that the class $N_1 (k, n)\cdot z$ is realizable w.r.t. $U(l)$. }

\medskip

Actually the proof of Thom in [Thom1954] brings more information on the
vector bundle associated with $N(k,n)\cdot z$.  We have namely the following 

\medskip

{\bf 2.7. Proposition.} {\it  For each cohomology class $\alpha \in 
H^{2k} (M^n, \Z)$ there exists a number  $N_2 (k,n)\in \N^+$  and a complex vector bundle $E^k$ over $M^n$  such that}
$$ c_k (E^k) = N_2 (k, n) \cdot \alpha,$$
$$  c_i (E^k) = 0, \text{ for all } 1 \le i \le k-1.$$

\medskip


{\it Proof.}  
Let $K(\Z, 2k)$ be the Eilenberg-MacLane space and $K(\Z, 2k)^q$  be the $q$-dimensional
skeleton of $K(\Z, 2k)$.
We denote by $f$ a map from $M^n$ to $K(\Z, 2k) ^{n+1}$ such that $f^* (\tau) = \alpha$, where
$\tau \in H^ {2k} (K(\Z, 2k),\Z)$ is the fundamental class of $K(\Z, 2k)$. Let $G_\C (k)$
denote the classifying space of the group $U(k)$ and $c_k \in H^{2k} (G_\C (k), \Z)$
be the top Chern class of the universal complex vector bundle $V^k$ over $G_\C (k)$.
In the proof of [Thom1954 Theorem II.25] Thom has showed that there is  a map 
$$g : K(\Z, 2k)^{n+1} \to  Gr_\C (k)$$
such that  $g ^* (c_k (V^k)) = N_2(k,n)\cdot \tau$ for some positive number $N_2(k,n)$.
Clearly the complex vector bundle 
$E^k = f^* (g^* (V^k))$ satisfies the condition
$$ c_k (E^k) = N_2 (k,n) \alpha.$$
Since the cohomology group $H^i (K(\Z, 2k), \Q)$ is zero for all $i \le 2k-1$, (namely
$H^*(K(\Z, 2k), \Q) = \Q [x],\dim  x = 2k$, see
e.g. [F-F1989 Chapter III, 25.2]),  taking into account the equality
$$ H^i (K(\Z, 2k) ^{n+1},\Q) = H^i (K(\Z,2k), \Q), \: \forall i\le n$$
 it follows that
$$  ( c_i ( f^*(V^k)) = 0 \text { for all }1 \le i \le k-1.$$
Now it is easy to see that $E^k$  satisfies the condition of Proposition
2.7.\QED


\medskip

{\it Proof of Theorem 2.1.} First we shall prove Theorem 2.1 for a compact
integral symplectic  manifold $(M^{2n},\om)$. Given an element
$PD(\alpha )\in H^{2(n-k)} (M^{2n},\Z)$ we choose a complex vector bundle $E^{n-k}$ and a number
$N_2 (n-k, 2n)$  satisfying the condition of Proposition 2.7. The  following Lemma
is an immediate consequence of  Proposition 2.4 and Remark 2.5.

\medskip

{\bf 2.8. Lemma.} {\it There  exists a number  $N_3 (E^{n-k})$ such that for
any $N \ge N_3 (E^{n-k})$ there are asymptotically holomorphic  sections of
$E^{n-k} \otimes L^{\otimes N} $ which are uniformly transversal to $0$ and
whose zero set are smooth symplectic submanifolds in $(M^{2n},\om)$.}


\medskip

Now we use the following well-known formula for the Chern classes of tensor
products of complex vector bundles (see e.g. [Hartshorne1977, Appendix A.3]). Denote by
$c_t (E^r)$ the Chern  polynomial of a  complex vector bundle $E^r$ over $X$: $c_t (E^r) =
c_0 (E) + c_1 (E)t + \cdots +c_r (E^r) t^r$. Using the Grothendieck  splitting principle
we write
$$c_t ( E^r) = \Pi _{i =1} ^r (1 + a_i t).$$
Let $F^s$ be another complex vector bundle over $X$ with
$$c_t (E^s) = \Pi _{j =1} ^s (1 + b_j t).$$
Then we have
$$c_t ( E ^r \otimes F^s) = \Pi_{i,j} ( 1 + (a_i + b_j)t).\leqno (2.9)$$
It follows from (2.9) that  for the chosen $E^{n-k}$ satisfying Proposition 2.7 the top Chern class
$$c_{n-k }( E^{n-k}\otimes L^{N_3}) =  N_2(n-k,n)\cdot PD( \alpha) + N_3(\alpha) [\om ^{n-k} ].\leqno (2.10)$$
Combining (2.10) with Lemma 2.8 we get Theorem 2.1 for compact integral symplectic manifold $(M^{2n},\om)$.

\medskip

{\it  Proof of Theorem 2.1 for a general compact symplectic $(M,\om)$.} We need
the following perturbation result.

\medskip 

{\bf 2.9. Lemma.} {\it Suppose that $(M^{2n}, \om, J)$ is a compact symplectic manifold
with a compatible almost complex structure $J$. Then for any given positive numbers $C >0$, $\eta >0$ there exist
a positive number $\delta  > 0$ and an integral symplectic  form $\bar \om$ together with its compatible almost complex structure $\bar J$  over $M^{2n}$ such that the following statement holds.  Let $E^k$ be a complex vector bundle over $(M,\bar \om, \bar J)$ with
metric and connection and $s$ be a section of $(M, g_{\bar J})$ to $E^k$ which
is $\eta$-transversal to 0 and which is $C^2$-approximately holomorphic with
bounds $\{(C,\delta), (C,\delta)\}$.  Then the zero section $s^{-1} (0)$ is a symplectic submanifold w.r.t. both symplectic structures $\om$ and $\bar \om$.}

\medskip

{\it Proof.}  For any symplectic manifold $(M,\om, J)$ with a compatible  almost
complex structure we denote by $G^s_{2k}(M,\om)$ the Grassmanian of $\om$-symplectic
$2k$-planes  in $T_* M$ and by $G^J _{2k}(M,J)$ the Grassmanian of
$J$-invariant $2k$-planes in $T_*M$. Clearly $G^s _{ 2k} (M,\om)$ is an open
neighborhood of $G^J_{2k} (M,J)$. Hence, if $M$ is compact and $\om$ and $J$ are given, there exists a positive number $\eps > 0$
such that if $\rho^v( V^{2k}, G^J_{2k} (M,\om)) < \eps$, then
$V^{2k}$ is a symplectic plane. Here we identify $V^{2k}$ with a point in $G_{2k}(M)$
with $\rho^v$ being the induced metric on  each fiber  $G_{2k} (T_x M), x\in M$. 
Using this open property we can find  a rational symplectic form $ \om_1  $  in a
$C^2$-small neighborhood of  a given symplectic  form $\om$ with  a compatible
almost complex structure $J_1$ which is $C^2$-close to $J$ such that the following property holds. If $V$ is a point in $G_{2k} (T_x M^{2n})$ and $\rho^v_1 (V, G^{J_1}_{2k} (T_x M^{2n})) < (\eps /2)$, then $V$ is symplectic w.r.t. both the symplectic
forms $\om_1$ and $\om$.

 With this at hand it is easy to come to the conclusion of Lemma 2.9.
Namely we take $\bar \om$ to be a multiple of $\om_1$. Then the compatible almost
complex structure $J_1$ is also compatible to $\bar \om$. So we put $\bar J : = J_1$.
Next we note that the metrics $g_1$ and $\bar g$ associated respectively  to $(\om_1, J_1)$ and
$(\bar \om, \bar J)$ induce the same fiber metric $g_1 ^v$ on  each $G_{2k} (T_x M)$,
and any $ \om_1$-symplectic plane (resp. $J^1$-invariant plane)  is also $\bar \om$-symplectic (resp. $\bar J$-invariant plane) and conversely.

Now given positive numbers  $\eta, C$ and $\eps$ we shall choose $\delta >0$ so that from the $\eta$-transversality and $C^2$-approximate holomorphy of a section $s$
from $M^{2n}$ to  a complex vector bundle $E$  with
bounds $\{ (C,\delta), (C, \delta)\}$ we shall get the estimate for tangential space
of $s^{-1} (0)$:
$$\rho_1^v (T_x s^{-1} (0), G^{J_1}_{2n-2k}(T_xM^{2n}))  < {\eps \over 2}.\leqno(2.10)$$
This is possible since the Nijenhuis tensor which measures  non-holomorphy of $s^{-1}(0)$ depends on the ratio of $\bar \p s$ and $\p s$ up to the second order
of $s$ (see
also [Auroux2000]). By our choice of $\bar \om$ and $\bar J$ the equality (2.10) also implies
that $s^{-1} (0)$ is  both $\om$-symplectic and $\bar \om$-symplectic.\QED

  
\medskip

{\it Completion of the proof of Theorem 2.1.} Now given  a compact symplectic
manifold $(M,\om)$ with a compatible almost complex structure $J$  and $\alpha\in
H_2 (M,\Z)$ we shall choose $\bar \om$ and $\bar J$ as in Lemma 9. Let $F^{n-k}$ be a  complex vector bundle  satisfies the condition of Lemma 2.7 and $E_p = F^{n-k}\otimes 
L^{\otimes p}_{\bar \om}$ be the sequence of  asymptotically very ample complex vector bundles
over $(M, \bar \om, \bar J)$. Here $L_{\bar \om}$ is the associated complex line
bundle over $(M,\bar \om, g_{\bar J})$. Let $s_p$ be  sequence of  asymptotically holomorphic sections of $E_p$ from Proposition 2.4. By definition, for any given $\delta > 0$ there exists a number $K >> 0$ such that if $p \ge K$ then $s_p$ is $C^2$-approximately holomorphic with bounds
$\{(C,\delta), (C,\delta)\}$. It follows from
Lemma 2.9 that the zero sections  $s^{-1}(0)$ is also symplectic w.r.t. the symplectic
form $\om$.\QED


\medskip


{\it Another elementary proof of Theorem 1.3 for $1\le k \le n-2$.} [Ono2004]. Clearly Theorem 1.3 is a consequence
of the following statement. There are positive integral numbers $n_1, \cdots, n_k$
and  for each $i = \overline{1,k}$ a section $s_i$ of the line bundle  $L^{n_i}_\om$ such that the section $\hat s: =s_1 \oplus  \cdots \oplus s_k$
intersects to the zero section of $\hat L : =L^{N_1}_\om \oplus \cdots \oplus L^{N_k}_\om$ transversally and $\hat s^{-1} (0)$ is a symplectic submanifold in $(M^{2n}, \om)$,
if $N_i \ge n_i$ for $ 1\le i \le k$.
(Obviously  this statement follows from
 Proposition 2.4 and Remark 2.5.)

Ono has noticed that Theorem 1.3 is also a consequence of Theorem 1 in [Donaldson1996].
Indeed, it suffices to prove the above statement inductively on $k$.  For
$k =1$ the statement  is exactly Theorem 1 in [Donaldson1996].
Assume that the above statement is valid for $k = K$. We shall prove its validity for 
$k = K +1$. We denote by $S_K$ the common zero locus of sections $s_1, \cdots, s_K$
which is by the induction assumption a symplectic submanifold. We note that
the restriction of the line bundle $L_\om$ to $S_K$ has the curvature $\om_{| S_K}$ which
is the symplectic form on $S_K$.
 According to  Donaldson  [Donaldson1996, Theorem 5 and Proposition 3] there exists a number
 $N_0$ such that if  $n_{K+1} > N_0$ there is a section 
 $$ \tilde s_{K+1} : S_K\to L^{n_{K+1}}_\om$$ such that  
$\tilde s_{K+1}$ intersects with the zero section of $L^{n_{K+1}}_\om$ transversally.
Moreover  the zero section  $N:=\tilde s_{K+1}^{-1}(0) $  is a symplectic submanifold
in $S_K$.

Since the fiber $L^{n_{K+1}}_\om$ is contractible, the section $\tilde s_{K +1}$ can be extended to a section $s_{K+1} : M ^{2n} \to L^{n_{K+1}}_\om$. The proof of the above statement is complete, if we can show
that $s_{K+1}$ can be chosen to be  transversal to   the zero section of
a neighborhood $U_\eps(S_K)\subset M^{2n}$.
It is easy to see that the extension of the section from
the submanifold to the ambient space is just done by
pull back the section by the projection of the normal bundle of the symplectic 
submanifold. More explicit  we choose $U_\eps (S_K)$  to be a (geodesic) neighborhood of $S_K$ in $M^{2n}$,
such that there is a diffeomorphism $f$ from $U_\eps (S_K)$ to a neighborhood
$V_\eps (S_K)$ of the zero
section of the  normal bundle $V(S_K)$ of $S_K$ in $M$. By using this diffeomorphism
$f$ we can work now on $V_\eps (S_K)$. It is easy to see (see e.g. [Thom1954]) that 
$$V(S_K) = (L^{n_1}_\om\oplus \cdots \oplus L^{n_K}_\om)_{|S_K}.\leqno (2.11)$$
Then we let the extension of $\tilde s_{K+1} : S_K \to  L^{n_{K+1}}$ to a section
$ \bar s_{K+1} : V_\eps (S_K) \to (f^{-1}) ^* L^{n_{K+1}}$ be defined by
$$ \bar  s_{K+1} (x,l_1, \cdots, l_{K} )  = ( x,l_1, \dots , l_K, \tilde s_{K+1} (x) ) \leqno(2.12).$$
Finally we extend $\bar s_{K+1}$ to a section $s_{K+1}$ over $M^{2n}$. This section
$s_{K+1}$  satisfies the condition of the statement at the beginning of the  (elementary) proof
of Theorem 1.3.\QED



\section {Proof of Theorem 1.5.}
To prove the part (a) of Theorem 1.5 we need the following
\medskip

{\bf 3.1. Lemma.} {\it Suppose that the condition a of Theorem 1.5 is satisfied. Then
there is an embedding $f:S^2\to M$ such that $f^*([\om]) = k[\om_0]$ for some
$k> 0$. Here we denote by $\om_0$ the standard symplectic form on $S^2$.}

\medskip

{\it Proof.} Let $f$ be a smooth map  such that $<[\om], [f(S^2)]> = k > 0$.
Since the space of immersions from $S^2$ to $M^{2n}, n\ge 3,$ is dense in the space of smooth maps [Hirsch1959] we
can assume that $f$ is an immersion.  We can also assume that the intersection point is generic.   Using an  isotopy and the dimension condition we move the  second intersection point
from  the first one.\QED


\medskip

{\it Proof of Thereom 1.5.a.} First we  find  a map covering  $F : T_* S^2 \to  T_* M^{2m}$ which is fiber-wise
symplectic. The existence of such a map is equivalent to the existence of a section
of the bundle $Iso_{sym} (T_*S^2, T_*M^{2m})$ over $S^2$. The fiber of this bundle
is $Symp(2m) / Symp (2m-2)$  which is $2m-1$ connected  [Gromov1986]. So there is no obstruction for  a section.
Now Theorem 1.5.a follows immediately from the  h-principle of Gromov and  the observation, that we can make a $C^1$-perturbation of a symplectic immersion  of
$S^2$ to get a symplectic embedding, since the dimension of $M^{2n}$ is at least $6$.

\medskip


{\bf 3.2. H-principle for symplectic immersions.} [Gromov1986, 3.4.2.A] {\it Let $g$ be 
an arbitrary  (possibly singular) closed smooth 2-form on a smooth manifold $V$
and let $F_0 : (T_*V, g) \to (T_*M, \om)$ be a fiberwise injective isometric homomorphism
for which $(\pi F) ^* [\om] = [g]$.  Let us denote by $i$ the embedding $V\to T^*V$ as the zero section, and by $\pi$ the projection  $T^* M \to M$.
If $\dim V  < \dim M$, then the map
$\pi \circ F \circ i: V \to M ^{2n}$ admits a fine $C^0$- approximation by isometric  smooth immersions
$f: (V,g) \to (M^{2n}, \om)$ whose differentials $Df: T_*V \to T_*M$ are homotopic to $F_0$ in the space of fiberwise injective
isometric homomorphisms.}

\medskip

 To prove (b) we replace Lemma 3.1 by the Thom existence theorem [Thom1954, Theorem II.25].
 The rest of our argument repeats the proof of (a).\QED

\medskip


We close this note with the following questions. It is easy to see that any symplectic curve $\Sigma$ in a symplectic manifold
$(M^{2n}, \om)$ is a $J$-holomorphic curve for some compatible almost complex
structure $J$ on $(M^{2n},\om)$. When can we  ensure that  the holomorphic  curves in Theorem 3 are generic in Gromov's
sense? If it is the case, we can obtain the existence of true holomorphic curves representing different homology classes in certain algebraic projective manifolds.


\medskip

 
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\end{document}

math0505562.extracted_bib
\begin{thebibliography}{99999}
\bibitem[Auroux2002]{Auroux2000}{\sc D. Auroux},
\textit{Estimated transversality in symplectic geometry and projective maps}, in
: Symplectic Geometry and Mirror Symmetry, Seoul 2000, 1-30.
\bibitem[Donaldson1996]{Donaldson1996}{\sc S. K. Donaldson},
\textit{Symplectic submanifolds and almost-complex geometry}, J.D.G. 44 (1996), 666-705.
\bibitem[F-F1989]{F-F1989}{\sc A. T. Fomenko, D.B. Fuks}, {\it Course on homotopic topology,} Nauka, Moscow 1989 (in Russian).
\bibitem[Gromov1986]{Gromov1986}{\sc M. Gromov}, \emph{Partial Differential relations,} Springer 1986.
\bibitem[Hartshorne 1977]{Hartshorne1977}{\sc R. Hartshorne}, {\it Algebraic
geometry}, Springer 1977.
\bibitem[Hirsch1959]{Hirsch1959}{\sc M. W. Hirsch}, {\it Immersions of manifolds}, T.A.M.S,
v.93,(1959), 242-276. 
\bibitem[Ono2004]{Ono2004}{\sc K. Ono}, {\it Private communication}, 2004.
\bibitem[Rass1958]{Rass1958} {\it  Fiber spaces and their applications}, (Selected
papers), Moscow 1958 (in Russian).
\bibitem[Thom1954]{Thom1954} {\sc R. Thom}, {\it  Quelques  properties globales des variates
differentiables}, Comm. Math. Helv. vol 28, (1954), 17-86.
\end{thebibliography}