Stein structures and holomorphic mappings

Details Stein structures and holomorphic mappings Stein structures and holomorphic mappings

Publishing date

2005-07-11

URL

arxiv.org/abs/math/0507212v4

Journals

Math. Z. 256 (2007), no. 3, 615-646

Science

Mathematics

Field

Complex Variables

Additional info

The original publication is available at http://www.springerlink.com

Type

Scientific paper

Digital Object Identifier

10.1007/s00209-006-0093-0

Abstract

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).

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