Symplectic 4-manifolds as branched coverings of ℂℙ 2

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.