Pulling back singularities of codimension one objects

Abstract

We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map g : ( C n , 0 ) → ( C n , 0 ) g: (\mathbb {C}^n,\mathbf {0}) \rightarrow (\mathbb {C}^n, \mathbf {0}) is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal. Geom. 15 (2007), no. 2, 491-507], Lebl [ArXiv preprint https://arxiv.orglabs/0812-2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1–2, 83–91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.