The primitive cohomology of the theta divisor of an abelian fivefold

Abstract

The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension g g contains a Hodge structure of level g − 3 g-3 which we call the primal cohomology. The Hodge conjecture predicts that this is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.