Hodge structures associated to SU(p,1)

TL;DR

This paper investigates Hodge structures linked to abelian varieties with Hodge groups involving SU(p,1), demonstrating that all effective Tate twists of these structures appear in the cohomology of some abelian variety.

Contribution

It establishes a connection between Hodge structures associated with SU(p,1) and their realization in the cohomology of abelian varieties, extending understanding of Hodge structures in this context.

Findings

Effective Tate twists of Hodge structures are realizable in abelian variety cohomology.

Hodge structures associated with SU(p,1) are stable under Tate twists.

The results deepen the link between Hodge theory and the geometry of abelian varieties.

Abstract

Let A be an abelian variety over C such that the semisimple part of the Hodge group of A is a product of copies of SU(p,1) for some p>1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.