The irreducible components of the primal cohomology of the theta divisor of an abelian fivefold

TL;DR

This paper investigates the structure of the primal cohomology of the theta divisor in a five-dimensional abelian variety, revealing its decomposition into invariant and anti-invariant parts with specific Hodge-theoretic properties.

Contribution

It demonstrates that the invariant part consists of Hodge classes and the anti-invariant part is a simple Hodge structure of level 2 for general cases.

Findings

Invariant part consists of Hodge classes

Anti-invariant part is a simple Hodge structure of level 2

Results apply to very general principally polarized abelian fivefolds

Abstract

The primal cohomology $\mathbb{K}_\mathbb{Q}$ of the theta divisor $\Theta$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\mathbb{K}_\mathbb{Q}^{+1}$, resp. $\mathbb{K}_\mathbb{Q}^{-1}$ under the action of $-1$. For smooth $\Theta$, these have dimension $6$ and $72$ respectively. We show that $\mathbb{K}_\mathbb{Q}^{+1}$ consists of Hodge classes and, for a very general ppav, $\mathbb{K}_\mathbb{Q}^{-1}$ is a simple Hodge structure of level $2$.