2026/07/16 by Stefan Schreieder, Botong Wang
#math.AG #math.CV
We construct a smooth complex projective variety whose Albanese morphism is a homotopy fiber bundle but not a submersion. The same variety fibers smoothly over the circle, although every holomorphic one-form on it has a zero. A second construction yields smooth complex projective varieties X such that the Aomoto complex of every nonzero holomorphic one-form on every connected finite étale cover of X is exact, while X admits no real closed one-form without zeros. The two constructions build, respectively, on a homology fiber bundle of Corrêa--Kollár that is not a homotopy fiber bundle and on a rational cohomology torus constructed by Debarre--Jiang--Lahoz. Consequently, we disprove Kotschick's conjecture, the remaining implication in the Bobadilla--Kollár conjecture, and a conjecture of the first-named author.