We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $g: (\mathbb{ C}^n,0) \rightarrow (\mathbb{ C}^n,0)$ 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: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.
Categories: math.CV math.AG math.DG