Journal of the European Mathematical Society
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Published online: 2016-10-12
Noncritical holomorphic functions on Stein spacesFranc Forstnerič (1) University of Ljubljana, Slovenia
In this paper we prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, every closed discrete subset of a reduced Stein space $X$ is the critical locus of a holomorphic function on $X$. We also show that for every complex analytic stratication with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result 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.
Keywords: Holomorphic functions, critical points, Stein manifolds, Stein spaces, 1-convex manifolds, stratifications
Forstnerič Franc: Noncritical holomorphic functions on Stein spaces. J. Eur. Math. Soc. 18 (2016), 2511-2543. doi: 10.4171/JEMS/647