Stein domains in complex surfaces

Abstract

Let S be a closed connected real surface and π: S→X a smooth embedding or immersion of S into a complex surface X. We denote by I(π) (resp. by I±(π) if S is oriented) the number of complex points of π (S)∪X counted with algebraic multiplicities. Assuming that I(π)≤0 (resp. I±(π)≤0 if S is oriented) we prove that π can be C0 approximated by an isotopic immersion π1: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π1 (S). We obtain precise results for surfaces in\(\mathbb{C}\mathbb{P}^2 \) and find an immersed symplectic sphere in\(\mathbb{C}\mathbb{P}^2 \) with a Stein neighborhood.