Tate twists of Hodge structures arising from abelian varieties of type IV

TL;DR

This paper demonstrates that for specific abelian varieties of type IV, all effective Tate twists of Hodge structures in their cohomology appear in the cohomology of some abelian variety, supporting the general Hodge conjecture.

Contribution

It establishes the occurrence of all effective Tate twists of Hodge structures in cohomology for certain abelian varieties of type IV, advancing understanding of the Hodge conjecture.

Findings

All effective Tate twists of Hodge structures occur in cohomology.

The general Hodge conjecture is verified for certain non-simple abelian varieties of type IV.

Provides new insights into the structure of Hodge structures in abelian varieties.

Abstract

We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.