Noncritical holomorphic functions on Stein manifolds

By F. Forstnerič. 2002

89 citations recorded by the source corpus.

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/51ba0a90d724ff30fcbdcab8451f40c002531acc

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Exposing Points on the Boundary of a Strictly Pseudoconvex or a Locally Convexifiable Domain of Finite 1-Type (2013, unknown): “1 in [5], for any open set C̃ containing the set C := A ∩ B there exist open sets A′, B′, C ′ with A ⊂ A′, B ⊂ B′ and C ⊂ C ′ ⊂ A′ ∩B′ ⊂ C̃, such that if γ : C̃ → Cn is injective holomorphic, and sufficiently close to the identity, then there exist holomorphic injections α : A′ → Cn, β : B′ → Cn, un”
Holomorphic curves in complex spaces (2006, methodology): “7 in [23] furnishes a new function h̃ (denoted τ in [23]) that is strongly plurisubharmonic on W , while outside of W each level set {̃h = } (for values close to zero) coincides with a certain level set {h = c( )} such that h̃ satisfies the following properties (see Figure 5).”
Bordered Riemann surfaces in C2 (2007, methodology): “Its proof in [19] applies to Cartan pairs in an arbitrary Stein manifold.”
Survey of Oka theory (2010, background): “• There is also a homotopy principle for holomorphic submersions [13].”