Smooth embeddings with Stein surface images

Abstract

A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, complex 2-space C^2 contains domains of holomorphy (Stein open subsets) that are exotic R^4's, and others homotopy equivalent to the 2-sphere but cut out by smooth, compact 3-manifolds. Pseudoconvex embeddings of Brieskorn spheres and other 3-manifolds into complex surfaces are constructed, as are pseudoconcave holomorphic fillings (with disagreeing contact and boundary orientations). Pseudoconcave complex structures on Milnor fibers are found. A byproduct of this construction is a simple polynomial expression for the signature of the (p,q,npq-1) Milnor fiber. Akbulut corks in complex surfaces can always be chosen to be pseudoconvex or pseudoconcave submanifods. The main theorem is expressed via Stein handlebodies (possibly infinite), which are defined holomorphically in all dimensions by extending Stein theory to manifolds with noncompact boundary.

26 pages, 1 figure. Version 2 has minor stylistic changes for clarity, remark expanded at end of Section 4; accepted for publication by the Journal of Topology

References in corpus (3)

• Stein surfaces as open subsets of C^2
• Nuclei and exotic 4-manifolds
• Creating Stein surfaces by topological isotopy

Cited by in corpus (8)

• Minimal genera of open 4-manifolds
• Exotic smoothings via large R^4's in Stein surfaces
• The trace embedding lemma and spinelessness
• Group actions, corks and exotic smoothings of R^4
• On contact type hypersurfaces in 4-space
• Creating Stein surfaces by topological isotopy
• H-principle for complex contact structures on Stein manifolds
• Some 3-dimensional transverse C-links (Constructions of higher-dimensional C-links, I)