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 Fern\'andez~de~Bobadilla and Koll\'ar, a question of Koll\'ar and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes.