## Symplectic submanifolds and almost-complex geometry.(English)Zbl 0883.53032

This is a fundamental paper on symplectic manifolds and almost complex geometry. The author develops a general procedure for constructing symplectic submanifolds, extending techniques of line bundles, linear systems, and cohomology in complex geometry to general symplectic manifolds. Recall that if $$(V,\omega)$$ is a symplectic manifold, a submanifold $$W\subset V$$ is called symplectic if the restriction of $$\omega$$ to $$W$$ is nondegenerate.
The main result of the paper is the following existence theorem: Let $$(V,\omega)$$ be a compact symplectic manifold of dimension $$2n$$, and suppose that the de Rham cohomology class $$[\omega/2\pi]\in H^2(V;\mathbb{R})$$ lies in the integral lattice $$H^2(V;\mathbb{Z})$$/Torsion. Let $$h\in H^2(V;\mathbb{Z})$$ be a lift of $$[\omega/2\pi]$$ to an integral class. Then for sufficiently large integers $$k$$ the Poincaré dual of $$kh$$ in $$H_{2n-2}(V;\mathbb{Z})$$ can be realized by a symplectic submanifold $$W\subset V$$.
As a consequence, the author obtains a general existence theorem for pseudo-holomorphic curves: If $$(V,\omega)$$ is any compact symplectic manifold, the following hold: (1) $$V$$ contains symplectic submanifolds of any even codimension; (2) if $$J$$ is a compatible almost complex structure on $$V$$, then there are almost complex structures $$J'$$ arbitrarily close (in $$C^0$$) to $$J$$ such that $$V$$ contains $$J'$$-pseudoholomorphic curves.

### MSC:

 53D35 Global theory of symplectic and contact manifolds 53D05 Symplectic manifolds (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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