Pulling back singularities for analytic complete intersections

TL;DR

This paper proves that for finite holomorphic maps, if the preimage of a geometric complete intersection is smooth, then the original is also a complete intersection, extending previous results to this broader class.

Contribution

It provides an affirmative solution to the pullback problem for geometric complete intersections under finite holomorphic maps.

Findings

Preimage smoothness implies original is a complete intersection.

Extends previous results from hypersurfaces to complete intersections.

Builds on and generalizes earlier work by Ebenfelt-Rothschild, Lebl, Denkowski, Giraldo-Roeder, and Jelonek.

Abstract

The following pullback problem will be considered. Given a finite holomorphic map germ $\phi : (\mathbb{C}^{n}, 0) \to (\mathbb{C}^{n}, 0)$ and an analytic germ $X$ in the target, if the preimage $Y = \phi^{-1}(X)$, taken with the reduced structure, is smooth, so is $X$. The main aim of this paper is to give an affirmative solution for $X$ being a geometric complete intersection. The case, where $Y$ is not contained in the ramification divisor $Z$ of $\phi$, was established by Ebenfelt-Rothschild (2007) and afterwards by Lebl (2008) and Denkowski (2016). The hypersurface case was achieved by Giraldo-Roeder (2020) and recently by Jelonek (2023).