Superconnections and special cycles on Shimura varieties and generalizations

Mercredi 15 novembre 2017 14:00-17:00 - Luis Garcia - IHES

Résumé : Shimura varieties are of central importance to number theory and their geometry and arithmetic is a topic of intensive research. They come with a very rich collection of subvarieties known as special cycles. In the 1980’s Kudla and Millson gave very general theorems describing the span of these special cycles in cohomology in terms of automorphic forms. Their proofs involve producing explicit differential forms that are Poincare dual to the special cycles. I will describe a different approach to defining these differential forms that uses the formalism of superconnections and takes advantage of theorems proved by Bismut, Gillet and Soule. This allows for simplified proofs of several of the main theorems of Kudla-Millson and also to extend this construction to arithmetic quotients of period domains. I will also discuss work in progress on other applications of this construction, such as constructing Green forms for these special subvarieties (joint work with S. Sankaran) or extending Kudla-Millson modularity theorems to certain variations of Hodge structure not parametrized by Shimura varieties.
The talk will start with a general introduction to special cycles and no number theory background will be assumed.

Lieu : Salle 113-115

Superconnections and special cycles on Shimura varieties and generalizations  Version PDF