The primitive cohomology of theta divisors

TL;DR

This paper surveys the primitive cohomology of theta divisors in principally polarized abelian varieties, discussing known results, general facts, and open problems related to its Hodge structure and the Hodge conjecture.

Contribution

It provides a comprehensive overview of the primitive cohomology of theta divisors, including new general facts and highlighting open problems in the field.

Findings

Primitive cohomology has level g-3 in Hodge structure.

The Hodge conjecture predicts its relation to cohomology of curves.

Several results and open problems are discussed.

Abstract

The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. We survey some of the results known about this primitive cohomology, prove a few general facts and mention some interesting open problems.