We show that the algebraicity of Weil's 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.

Let T : X\ -• X 2 be a strongly equivariant holomorphic embedding of one bounded symmetric domain into another. We show that if σ is an automorphism of C, then τ σ : X° -» X% is also strongly equivariant.

We show that any effective Hodge structure of CMtype occurs (without having to take a Tate twist) in the cohomology of some CM abelian variety over C. As a consequence we get a simple proof of the theorem (due to Hazama) that the usual Hodge conjecture for the class of all CM abelian varieties implies the general Hodge conjecture for the same class. 1[22] S. G. Tankeev, Abelian varieties and the general Hodge conjecture, Izv. Ross.

If the Hodge *-operator on the L2-cohomology of Kuga fiber varieties is algebraic, then the Hodge conjecture is true for all abelian varieties.

No abstract