Smooth embeddings with Stein surface images

TL;DR

This paper characterizes open subsets of complex surfaces that can be smoothly perturbed into Stein domains, revealing new exotic structures and embeddings in complex 2-space with applications to Milnor fibers and Akbulut corks.

Contribution

It provides a simple characterization of Stein perturbable open subsets in complex surfaces and constructs various exotic and pseudoconvex embeddings with applications to Milnor fibers.

Findings

Existence of exotic R^4 domains in C^2.

Construction of pseudoconvex embeddings of Brieskorn spheres.

Polynomial formula for Milnor fiber signatures.

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.