Pulling back singularities for analytic complete intersections

Abstract

The following pullback problem will be considered. Given a finite holomorphic map germ $\ddot{I}•:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ and an analytic germ $X$ in the target, if the preimage $Y=\ddot{I}{•}^{-1}(X)$, 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 $\ddot{I}•$, 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).

Some corrections have been made