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

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 $2$.

arXiv commentProposition 6.2 contained a mistake and has been replaced. Some computations concerning the action of -1 have been added