Invisible singularities in complex algebraic geometry

Maurício Corrêa; János Kollár; Stefan Schreieder; Botong Wang

Invisible singularities in complex algebraic geometry

Algebraic Geometry 2026-08-11 v1 Complex Variables

Authors: Maurício Corrêa , János Kollár , Stefan Schreieder , Botong Wang

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.

Keywords

invisible singularities complex algebraic geometry singular fibers topologically smooth projective varieties

Cite

@article{arxiv.2608.10973,
  title  = {Invisible singularities in complex algebraic geometry},
  author = {Maurício Corrêa and János Kollár and Stefan Schreieder and Botong Wang},
  journal= {arXiv preprint arXiv:2608.10973},
  year   = {2026}
}

Comments

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