Noncritical holomorphic functions on Stein spaces

Abstract

We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is noncritical outside the exceptional variety.

To appear in J. Eur. Math. Soc. (JEMS)

References in corpus (3)

• Bordered Riemann surfaces in C^2
• Every bordered Riemann surface is a complete proper curve in a ball
• Embeddings of infinitely connected planar domains into C^2

Cited by in corpus (3)

• Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces
• Divisors defined by noncritical functions
• Complete nonsingular holomorphic foliations on Stein manifolds