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Summary: SYMPLECTIC 4-MANIFOLDS AS BRANCHED
COVERINGS OF CP2
DENIS AUROUX
Abstract. We show that every compact symplectic 4-manifold X can
be topologically realized as a covering of CP2
branched along a smooth
symplectic curve in X which projects as an immersed curve with cusps in
CP2
. Furthermore, the covering map can be chosen to be approximately
pseudo-holomorphic with respect to any given almost-complex structure
on X.
1. Introduction
It has recently been shown by Donaldson [3] that the existence of ap-
proximately holomorphic sections of very positive line bundles over com-
pact symplectic manifolds allows the construction not only of symplectic
submanifolds ([2], see also [1],[5]) but also of symplectic Lefschetz pencil
structures. The aim of this paper is to show how similar techniques can be
applied in the case of 4-manifolds to obtain maps to CP2
, thus proving that
every compact symplectic 4-manifold is topologically a (singular) branched
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