Hodge theory, Motives and Vanishing

Purdue University, West Lafayette IN

Investigators

Abstract

The main problems in this proposal concern finding a general framework for Hodge theory, which can be applied to families of algebraic varieties. The proposed framework is a theory of motivic sheaves or motives with parameters. In particular, there would be a theory of motivic local systems. The latter would then lead to a notion of motivic fundamental group, which would provide a link between the topology and arithmetic of varieties. This also connects to Hodge theory in a more traditional sense, as in the study of variations of Hodge structures. A sub-project involves refinements of the Kodaira vanishing theorem using algebraic methods. Hodge theory lies at the core of this proposal. It is the part of algebraic geometry (the study of algebraic varieties or sets of solutions of systems of algebraic equations) that interacts most closely with differential geometry, topology and to a lesser extent, mathematical physics. There are nontrivial and surprising connections with number theory as well; the theory of motives, which is also central to this proposal, arose in part to explain some of these.