Generic Vanishing, 1-forms, and Topology of Albanese Maps

TL;DR

This paper explores the relationship between cohomology jump loci, singular support, and the topology of Albanese maps on complex varieties, providing new insights into holomorphic 1-forms and fiber bundle structures.

Contribution

It establishes a new equality relating cohomology jump loci and singular support, and applies it to characterize holomorphic 1-forms and Albanese morphisms on complex varieties.

Findings

Identifies a relation between cohomology jump loci and singular support.

Shows the set of 1-forms vanishing somewhere is a finite union of linear subspaces.

Reproves a result linking Albanese morphism behavior to fiber bundle structures.

Abstract

Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety $X$; one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This result is related to Kotschick's conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if $X$ has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients. In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of $H^0(X,\Omega_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1.