The main thrust of the project is the construction of a theory of relative motives, or more precisely motivic local systems, over an algebraic variety defined over a field of complex numbers or a subfield. These objects should exist in the abstract, and form a category with good properties (it should be abelian and tannakian). Furthermore they should be realizable by locally constant sheaves on the classical and etale topologies, as well as by variations of mixed Hodge structures. The hope is that this theory should provide a good framework in which to study Hodge theory for families. Such a theory would have a number of ramifications: It would lead to the motivic fundamental group of an algebraic variety which would be related to the usual fundamental group. And it could used in the study of bundles over a curve. There are a couple of subprojects which are separate from the above. These involve the study of vanishing theorems by positive characteristic techniques, and the study of Kaehler-de Rham cohomology.
Algebraic varieties are basic objects in mathematics; they are sets of solutions of systems of algebraic equations. Their study has found interactions with areas as diverse as mathematical physics and cryptography. Indeed Hodge theory, which is the subject of this project, was partly motivated by physical ideas. The goal of this project is to further the understanding of the Hodge theory of varieties by replacing them with simpler objects called motives. The collection of motives would be fine enough to reflect much of the original structure of algebraic varieties, but they would posses a "linear" structure which makes them easier to handle.
