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.
The main intellectual contribution of the project is a joint theorem with Gregory Pearlstein on the zero locus of an admissible normal function. This result was used by Brosnan, Pearlstein and Schnell to vastly generalize a famous theorem of Cattani, Deligne and Kaplan on the locus of Hodge classes. Cattani, Deligne and Kaplan's theorem can be seen as giving evidence that the Hodge conjecture, one of the most difficult and important unsolved problems in mathematics, might be true. Similarly, the result of Brosnan and Pearlstein can be seen as giving evidence for an important conjecture of Beilinson. The Hodge conjecture asserts a relationship between algebraically defined objects called ``algebraic cycles" and geometrically defined objects called ``cohomology classes." If true, the conjecture would solve several open problems in mathematics. In particular, it would go a very long way toward establishing a conceptual theory of algebraic cycles invented by Grothendieck known as the theory of ``motives." Beilinson's conjecture predicts a way to order algebraic cycles by degree of complexity. This would have implications for several areas in mathematics, including algebraic geometry and number theory. While there are very strong heuristic reasons for believing both the Hodge conjecture and Beilinson's conjecture, it has proved very difficult to come up with evidence. For this reason, several mathematicians have been interested have been interested in the zero locus of normal functions, and, in fact, there are now three separate proofs of the result: the one by Brosnan and Pearlstein, a proof by Schnell and another one by Kato, Nakayama and Usui. Another notable result is Brosnan's joint work with Roy Joshua comparing operations on motivic cohomology. The outcome of this work is a deeper understanding of algebraic cycles. In addition to the above intellectual contributions, the project has contributed to the professional development of five Ph. D. students and one postdoctoral fellow working with the PI.
