Generic vanishing and minimal cohomology classes on abelian fivefolds

TL;DR

This paper classifies certain special subvarieties of five-dimensional abelian varieties, confirming a conjecture and revealing their existence only in specific geometric contexts like Jacobians and cubic threefolds.

Contribution

It proves that GV-subschemes in five-dimensional ppavs only occur on Jacobians and intermediate Jacobians, extending the understanding of minimal cohomology classes.

Findings

GV-subschemes exist only on Jacobians and intermediate Jacobians

Supports the Pareschi--Popa conjecture in this case

Provides a classification of subvarieties with minimal cohomology class

Abstract

We classify GV-subschemes of five-dimensional ppavs, showing that they exist only on Jacobians of curves and intermediate Jacobians of cubic threefolds, and confirming a conjecture of Pareschi--Popa in this case. The result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor.