Problems in Equivariant Algebraic Geometry

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The purpose of this project is to use equivariant methods to studyproblems concerning algebraic varieties with group actions. Partof this project involves moment graphs of varieties with torusactions. For a variety on which a torus acts with finitely manyfixed points and curves, one can define a combinatorial objectcalled a moment graph. There has been much recent progress incomputing topological invariants, such as cohomology orintersection homology, in terms of the moment graph. Theinvestigator proposes to extend this to K-theory. Schubertvarieties are among the most important varieties with this type oftorus action; the investigator plans to continue work with BrianBoe on a conjecture that would simplify determining if a point ina Schubert variety is rationally smooth. Moreover, he plans toextend some facts known only for Schubert varieties to the moregeneral setting of varieties with this type of torus action. Inaddition, the investigator plans to use equivariant methods tostudy certain interesting varieties: he plans to calculateChern-Schwartz-MacPherson classes of degeneracy loci, and tocalculate 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 geometricobjects by describing them as solutions to polynomial equations-- such objects are called "algebraic varieties." Fortunately,although algebraic varieties can be very complicated, many ofthem have a great deal of symmetry. Mathematicians have beenintensely investigating such varieties with extra symmetry forseveral reasons: The presence of such extra symmetry makesthese varieties easier to study, so that these varieties arevaluable test cases in developing methods to investigate allvarieties. Moreover, varieties with extra symmetry are of greatinterest in their own right: they play important roles invarious areas of mathematics, including number theory,combinatorics, and representation theory. There has beenconsiderable recent progress in developing techniques tounderstand these kinds of varieties; this project is aboutextending these techniques, and using them to study particularclasses of varieties.