Salman Abdulali

Abelian varieties and the general Hodge conjecture

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.

ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE

We show that the algebraicity of Weils Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

Hodge structures on abelian varieties of type III

We show that the usual Hodge conjecture implies the general Hodge conjecture for certain abelian varieties of type III, and use this to deduce the general Hodge conjecture for all powers of certain 4-dimensional abelian varieties of type III. We also show the existence of a Hodge structure M such that M occurs in the cohomology of an abelian variety, but the Tate twist M(1) does not occur in the cohomology of any abelian variety, even though it is effective.

Tate twists of Hodge structures arising from abelian varieties

We consider the category of Hodge substructures of the cohomology of abelian varieties, and ask when a Tate twist of such a Hodge structure belongs to the same category.

Hodge Structures Associated to SU(p, 1)

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.