 
Summary: SYMPLECTIC 4MANIFOLDS AS BRANCHED
COVERINGS OF CP2
DENIS AUROUX
Abstract. We show that every compact symplectic 4manifold 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
pseudoholomorphic with respect to any given almostcomplex 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 4manifolds to obtain maps to CP2
, thus proving that
every compact symplectic 4manifold is topologically a (singular) branched
