Holomorphic submersions from Stein manifolds

Abstract

In this paper we prove results on the existence and homotopy classification of holomorphic submersions from Stein manifolds to other complex manifolds. We say that a complex manifold Y satisfies Property S_n for some integer n bigger or equal the dimension of Y if every holomorphic submersion from a compact convex set in C^n of a certain special type to Y can be uniformly approximated by holomorphic submersions from C^n to Y. Assuming this condition we prove the following. A continuous map f from an n-dimensional Stein manifold X to Y is homotopic to a holomorphic submersions of X to Y if and only if there exists a fiberwise surjective complex vector bundle map from TX to TY covering f. We also prove results on the homotopy classification of holomorphic submersions. We show that Property S_n is satisfied when n>dim Y and Y is any of the following manifolds: a complex Euclidean space, a complex projective space or Grassmanian, a Zariski open set in any of the above whose complement does not contain any complex hypersurfaces, a complex torus, a Hopf manifold, a non-hyperbolic Riemann surface, etc. In the case when Y is a complex Euclidean space the main result of this paper was obtained in [arXiv:math.CV/0211112].