The Journal of Geometric Analysis

, Volume 13, Issue 1, pp 77–94 | Cite as

Stein domains in complex surfaces



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.

Math Subject Classifications

32E10 32Q28 32Q55 32V40 

Key Words and Phrases

Stein domains complex surfaces adjunction inequalities symplectic spheres 


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Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  1. 1.Institute of Mathematics, Physics and MechanicsUniversity of LjubljanaLjubljanaSlovenia

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