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: (\mathbb{C}^n,{\bm 0}) \rightarrow (\mathbb{C}^n,{\bm 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.