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SYMPLECTIC 4-MANIFOLDS, SINGULAR PLANE CURVES, AND ISOTOPY PROBLEMS
 

Summary: SYMPLECTIC 4-MANIFOLDS, SINGULAR PLANE
CURVES, AND ISOTOPY PROBLEMS
DENIS AUROUX
Abstract. We give an overview of various recent results concerning
the topology of symplectic 4-manifolds and singular plane curves, using
branched covers and isotopy problems as a unifying theme. While this
paper does not contain any new results, we hope that it can serve as an
introduction to the subject, and will stimulate interest in some of the
open questions mentioned in the final section.
1. Introduction
An important problem in 4-manifold topology is to understand which
manifolds carry symplectic structures (i.e., closed non-degenerate 2-forms),
and to develop invariants that can distinguish symplectic manifolds. Ad-
ditionally, one would like to understand to what extent the category of
symplectic manifolds is richer than that of Kšahler (or complex projective)
manifolds. Similar questions may be asked about singular curves inside, e.g.,
the complex projective plane. The two types of questions are related to each
other via symplectic branched covers.
A branched cover of a symplectic 4-manifold with a (possibly singular)
symplectic branch curve carries a natural symplectic structure. Conversely,

  

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

 

Collections: Mathematics