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- Title
Smooth embeddings with Stein surface images.
- Authors
Gompf, Robert E.
- Abstract
A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, ℂ2 contains domains of holomorphy (Stein open subsets) that are exotic smoothings of ℝ4, and others homotopy equivalent to S2 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 submanifolds. The main theorem is expressed via Stein handlebodies (possibly infinite), which are defined holomorphically in all dimensions by extending Stein theory to manifolds with non-compact boundary.
- Subjects
SMOOTHNESS of functions; EMBEDDINGS (Mathematics); SET theory; POLYNOMIALS; DOMAINS of holomorphy; HOMOTOPY equivalences; STEIN manifolds; HANDLEBODIES
- Publication
Journal of Topology, 2013, Vol 6, Issue 4, p915
- ISSN
1753-8416
- Publication type
Article
- DOI
10.1112/jtopol/jtt017