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 is the critical locus of a holomorphic function on . 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.
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Franc Forstnerič, Noncritical holomorphic functions on Stein spaces. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2511–2543DOI 10.4171/JEMS/647