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ASYMPTOTICALLY HOLOMORPHIC FAMILIES OF SYMPLECTIC SUBMANIFOLDS
 

Summary: ASYMPTOTICALLY HOLOMORPHIC FAMILIES OF
SYMPLECTIC SUBMANIFOLDS
DENIS AUROUX
Abstract. We construct a wide range of symplectic submanifolds in
a compact symplectic manifold as the zero sets of asymptotically holo-
morphic sections of vector bundles obtained by tensoring an arbitrary
vector bundle by large powers of the complex line bundle whose first
Chern class is the symplectic form. We also show that, asymptotically,
all sequences of submanifolds constructed from a given vector bundle
are isotopic. Furthermore, we prove a result analogous to the Lefschetz
hyperplane theorem for the constructed submanifolds.
1. Introduction
In a recent paper [1], Donaldson has exhibited an elementary construc-
tion of symplectic submanifolds of codimension 2 in a compact symplectic
manifold, where the submanifolds are seen as the zero sets of asymptotically
holomorphic sections of well-chosen line bundles. In this paper, we extend
this construction to higher rank bundles as well as one-parameter families,
and obtain as a consequence an important isotopy result.
In all the following, (X, ) will be a compact symplectic manifold of di-
mension 2n, such that the cohomology class [

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics