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

Donaldson, S. K.

J. Differ. Geom. 44, No. 4, 666-705 (1996).

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.

Reviewer: A.Cavicchioli (Modena)

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.)

Keywords:

symplectic manifolds; almost complex geometry; symplectic submanifolds