Hodge theoretic and algebraic approaches to the theory of motives

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The proposed research concerns two topics within the theory of motives and algebraic cycles. The first is Hodge theory. Motivated by work of Mark Green and Phillip Griffiths on the Hodge conjecture and by work of Richard Hain and David Reed on algebraic cycles, the PI and Gregory Pearlstein have defined a sequence of metrized line bundles called biextension line bundles associated to Hodge classes in smooth,projective complex varieties. The main goal of the proposed research is to understand the metrics and the asymptotics of the metric at infinity in the hope of gaining insight into the geometry of moduli spaces and into the Hodge conjecture.The second part of the proposed research concerns cohomological invariants associated to algebraic groups. These are invariants associating to any torsor for an algebra group G over a field F a class in the Galois cohomology of F. Although they seem difficult to compute explicitly, cohomological invariants are very natural objects, and one would hope that they give full information about the torsors for an algebraic group. By an observation of Burt Totaro, the cohomological invariants of a group G are computatable in terms of the motivic cohomology of the classifying space of G. The PI intends to use Totaro's observation to compute cohomological invariants of the spinor group and related groups. The unifying theme in both proposed topics is to understand to what extent problems in algebraic geometry can be linearized and studied using cohomology. The Hodge conjecture, which motivates the first proposed topic, asks if cohomology determines algebraic cycles. Similarly, the second proposed topic asks to what extent cohomological invariants determine torsors. Since linear invariants are usually more tractable than non-linear ones, both topics are of fundamental importance in algebraic geometry and related subjects.