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The primal cohomology $\mathbb{K}_\mathbb{Q}$ of the theta divisor $\Theta$of a principally polarized abelian fivefold (ppav) is the direct sum of itsinvariant 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 generalppav, $\mathbb{K}_\mathbb{Q}^{-1}$ is a simple Hodge structure of level $2$.

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