Pink's Theory of Hodge Structures and the Hodge Conjecture over Function Fields

Hartl Urs, Juschka Ann-Kristin

Abstract

In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian t-modules and t-motives of Greg Anderson to Hodge-Pink structures. This allows to associate with each uniformizable t-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analog of the Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished work of Pink and the first author, the second author proved in her Diploma thesis that the Hodge-Pink group equals the motivic Galois group of the t-motive as defined by Papanikolas and Taelman. This yields a precise analog of the famous Hodge Conjecture, which is an outstanding open problem for varieties over the complex numbers. In this report we explain Pink'€™s results on Hodge structures and the proof of the function field analog of the Hodge conjecture. The theory of t-motives has a variant in the theory of dual t-motives. We clarify the relation between t-motives, dual t-motives and t-modules. We also construct cohomology realizations of abelian t-modules and (dual) t-motives and comparison isomorphisms between them generalizing Gekeler'€™s de Rham isomorphism for Drinfeld modules.