Cohomology Theories for Algebraic Varieties

Northeastern University, Boston MA

Investigators

Abstract

The investigator studies the theory of algebraic cobordism, recently constructed by the investigor and F. Morel, as well as generalized cohomology theories for algebraic varieties and schemes, the relation of rational equivalence in the Morel-Voevodsky algebraic-homotopy category, and a refinement of Asakura's arithmetic Hodge cohomology. The investigator attempts to construct a theory of higher algebraic cobordism, along the lines of Bloch's higher Chow groups, examines the category of cobordism motives, and tries to give a theory of algebraic higher elliptic genera. In addition, the investigator examines the generalization of the motivic cohomology to K-theory spectral sequence to other cohomology theories on algebraic varieties, and considers the relationship of this spectral sequence to the slice spectral sequence of Voevodsky, using the notion of rational equivalence in the algebraic homotopy category. Finally the investigator defines a variation of Asakura's arithmetic Hodge cohomology and uses this to construct an infinite sequence of cohomology theories which conjecturely give a better and better approximation to motivic cohomology. The investigator's research involves a mixture of algebraic geometry and algebraic topology. Algebraic geometry is the study of solutions of equations using both algebra and geometry. For example, one can study a circle by examining its equation, or by looking at its geometric properties. Algebraic topology studies spaces by attaching algebraic invariants to them, invariants which can often be computed explicitly. The investigator takes methods and constructions in algebraic topology, and then modifies and refines them so that they can be used to define algebraic versions of the topological invariants. These new algebraic invariants are then applicable for studying subtle properties of solutions of equations.