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

Yajnaseni Dutta, Feng Hao, Yongqiang Liu

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 $H^0(X,Ω_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1.

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

Abstract

Given a bounded constructible complex of sheaves on a complex Abelian variety, we prove an equality relating the cohomology jump loci of 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 ; 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 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 . We show that this is indeed the case for forms admitting zero locus of codimension 1.

Paper Structure

This paper contains 13 sections, 22 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. linearity of 1-forms admitting zeros
  3. Preliminaries
  4. 1-forms and associated local systems
  5. Constructible Complexes of Sheaves and Perverse Sheaves
  6. Constructible complex of sheaves on Abelian varieties
  7. Linearity and comparison
  8. Proof of \ref{['cor general type']}
  9. (Logarithmic) 1-forms with codimension one zeros
  10. Arapura's result about cohomology jump loci
  11. Projective case
  12. Quasi-projective case
  13. Logarithmic Generalization of A Theorem of Spurr

Key Result

Theorem 1.2

Let $A$ be a complex Abelian variety. For any $\mathcal{F} \in D^b_c(A)$, we have the equality where $\operatorname{SS}(\cdot)$ denotes the singular support (see Definition singular support for complex) of constructible complexes in the cotangent space $T^*A\simeq A\times H^0(A, \Omega_A^1)$ and $\pi\colon T^*A\to H^0(A, \Omega_A^1)$ is the natural projection. In particular, $\pi(\operatornam

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Remark 1.9
  • Theorem 1.11: (see \ref{['Thm:Proj-codim1']})
  • ...and 42 more