Pulling back singularities of codimension one objects

TL;DR

This paper demonstrates that the preimage of a singular analytic hypersurface under a finite holomorphic map remains singular, extending previous results and also applying to singular holomorphic foliations.

Contribution

It generalizes existing theorems by proving the preservation of singularities under pullback for hypersurfaces and holomorphic foliations.

Findings

Preimage of singular hypersurface remains singular under finite holomorphic maps.

Extension of previous results by Ebenfelt-Rothschild, Lebl, and Denkowski.

Applicable to singular codimension one holomorphic foliations.

Abstract

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