Invisible singularities in complex algebraic geometry

Abstract

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fernández~de~Bobadilla and Kollár, a question of Kollár and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes.

Contains the main results from arXiv:2607.05603 (by Corrêa and Kollár) and arXiv:2607.15102 (by Schreieder and Wang)