Stein Domains in Complex Surfaces

2002/01/31 by Franc Forstneric
Mathematics · #math.CV #msc:32E10 #msc:32Q28 #msc:32Q55 #msc:32V40

paper · pdf

published as J. Geom. Anal., 13 (2003), 77-94 · Journal of Geometric Analysis, to appear

arxiv created 2003/03/03 · arxiv updated 2009/11/30

Abstract

Let S be a closed connected real surface and f a smooth embedding or immersion of S into a complex surface X. Assuming that the number of complex points of the immersion (counted with algebraic multiplicities) is non-positive we prove that f can be uniformly approximated by an isotopic immersion g whose image g(S) in X has a basis of open Stein neighborhoods which are homotopy equivalent to g(S). We obtain precise results for surfaces in the complex projective plane CP² and find an immersed symplectic sphere in CP² with a Stein neighborhood. Conversely, the generalized adjunction inequality for embedded oriented real surfaces in complex surfaces shows that the existence of a Stein neighborhood implies non-positivity of the number of complex points.

Related

• Riemann surfaces in Stein manifolds with density property
• Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces
• On nonimbeddability of topologically trivial domains and Thin Hartogs figures of P₂(ℂ) into Stein spaces
• A classification of taut, Stein surfaces with a proper \R-action
• Immersions of surfaces in almost complex 4-manifolds