Problems in Equivariant Algebraic Geometry

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

The purpose of this project is to use equivariant methods to study problems concerning algebraic varieties with group actions. Part of this project involves moment graphs of varieties with torus actions. For a variety on which a torus acts with finitely many fixed points and curves, one can define a combinatorial object called a moment graph. There has been much recent progress in computing topological invariants, such as cohomology or intersection homology, in terms of the moment graph. The investigator proposes to extend this to K-theory. Schubert varieties are among the most important varieties with this type of torus action; the investigator plans to continue work with Brian Boe on a conjecture that would simplify determining if a point in a Schubert variety is rationally smooth. Moreover, he plans to extend some facts known only for Schubert varieties to the more general setting of varieties with this type of torus action. In addition, the investigator plans to use equivariant methods to study certain interesting varieties: he plans to calculate Chern-Schwartz-MacPherson classes of degeneracy loci, and to calculate interesting invariants (degrees, push-forward measures) of nilpotent adjoint orbits of reductive Lie groups. This project is in the area of mathematics referred to as "algebraic geometry." Algebraic geometry studies geometric objects by describing them as solutions to polynomial equations -- such objects are called "algebraic varieties." Fortunately, although algebraic varieties can be very complicated, many of them have a great deal of symmetry. Mathematicians have been intensely investigating such varieties with extra symmetry for several reasons: The presence of such extra symmetry makes these varieties easier to study, so that these varieties are valuable test cases in developing methods to investigate all varieties. Moreover, varieties with extra symmetry are of great interest in their own right: they play important roles in various areas of mathematics, including number theory, combinatorics, and representation theory. There has been considerable recent progress in developing techniques to understand these kinds of varieties; this project is about extending these techniques, and using them to study particular classes of varieties.