Mathematische Zeitschrift

, Volume 256, Issue 3, pp 615-646

First online:

Stein structures and holomorphic mappings

  • Franc ForstneričAffiliated withInstitute of Mathematics, Physics and Mechanics, University of Ljubljana Email author 
  • , Marko SlaparAffiliated withInstitute of Mathematics, Physics and Mechanics, University of Ljubljana

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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).


Stein manifolds Complex structures Holomorphic mappings

Mathematics Subject Classification (2000)

32H02 32Q30 32Q55 32Q60 32T15 57R17