Urs Hartl: The Hodge conjecture over function fields

Oberseminar 14.11.2013

Over a global function field one has a neutral Tannakian category of mixed motives, namely the uniformizable mixed $t$-motives defined by G. Anderson. R. Pink clarified the concept of Hodge structures in equal characteristic and defined Hodge realizations of $t$-motives, by using the theory of $\sigma$-bundles on the rigid analytic punctured open unit disc. We prove the analog of the famous Hodge conjecture in this situation, namely that the Hodge realization functor induces an isomorphism of the motivic Galois group of a $t$-motive onto its Hodge group.