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

Abstract

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.