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 $g:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ 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.

To appear in Proceedings of the AMS. Comments welcome!

Cited by in corpus (1)

• On perturbations of singular complex analytic curves