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We complete Satake's classification of Kuga fiber varieties by showing that if a representation $\rho$ of a hermitian algebraic group satisfies Satake's necessary conditions, then some multiple of $\rho$ defines a Kuga fiber variety.

Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties.

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.

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.

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.

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

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., Comment: 14 pages, published version

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.