Bert van Geemen

Title: Limits of Hodge classes on decomposable abelian fourfolds

Abstract: The Hodge conjecture asserts that the rational vector space of Hodge classes in the cohomology of a smooth 
projective variety is spanned by the classes of algebraic cycles. For abelian fourfolds the conjecture is interesting 
only for those of Weil type, meaning that their endomorphism algebra contains an imaginary quadratic field acting in 
a specific way on the first cohomology group. We determine which algebraic classes on a product JxJ, where J is a 
principally polarized abelian surface, deform to abelian varieties of Weil type and we give some applications. 
Time permitting, we will discuss the relation with recent work of Markman on the Hodge conjecture for some of these 
fourfolds.