Abstract:

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

中文翻译：

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阿贝尔五倍体θ除数的原始同调的不可约成分

摘要：

主极化阿贝尔五倍数（ppav）的theta除数$ \ Theta $的原始同调$ \ Bbb {K} _ {\ Bbb {Q}} $是其不变和反不变部分$ \ Bbb的直接和{K} _ {\ Bbb {Q}} ^ {+ 1} $，分别为。$ \ Bbb {K} _ {\ Bbb {Q}} ^ {-1} $在$ -1 $的作用下。对于平滑的$ \ Theta $，它们的维度分别为$ 6 $和$ 72 $。我们显示$ \ Bbb {K} _ {\ Bbb {Q}} ^ {+ 1} $由Hodge类组成，对于非常普通的ppav，$ \ Bbb {K} _ {\ Bbb {Q}} ^组成{-1} $是$ 2 $级的简单Hodge结构。