Hodge Conjecture For Abelian Varieties Research Articles

Some remarks on the Hodge conjecture for abelian varieties

We show how the classical Hodge conjecture for the middle cohomology of an abelian variety is equivalent to the general Hodge conjecture for the middle cohomology of a smooth ample divisor in the abelian variety. This is best suited to abelian varieties with actions of imaginary quadratic fields.

On the General Hodge Conjecture for Abelian Varieties of CM-type

The General Hodge Conjecture for abelian varieties of CM-type is shown to be implied by the usual Hodge Conjecture for those up to codimension two.

On the standard conjecture for complex Abelian schemes over smooth projective curves

We reduce the Hodge conjecture for Abelian varieties to the question of the existence of an algebraic isomorphism for all and all principally polarized complex Abelian schemes of relative dimension over smooth projective curves. If the canonically defined Hodge cycles are algebraic for all integers , then the Grothendieck standard conjecture on the algebraicity of the operators and holds for . We prove for an Abelian scheme under the assumption that for some geometric fibre of non-exceptional dimension.

General Hodge conjecture for abelian varieties of CM-type

The general Hodge conjecture for abelian varieties of CM-type is shown to be implied by the usual Hodge conjecture for those up to codimension two.