Room P3.10, Mathematics Building

Richard Gonzales, Institut des Hautes Études Scientifiques (IHES)
Equivariant cohomology of spherical varieties

Let $G$ be a connected reductive group. A $G$-variety is called spherical if it is normal and has a dense orbit under some Borel subgroup of $G$. Examples include flag varieties, symmetric varieties, toric varieties and, more generally, group embeddings. The latter are closely related to algebraic monoids. An important topological property of spherical varieties is that they have only a finite number of $G$-orbits and, consequently, also a finite number of $T$-fixed points, for the induced action of a maximal torus $T$ of $G$. Furthermore, under certain assumptions on the singularity type and the number of $T$-invariant curves, the equivariant cohomology of spherical varieties can be understood via GKM-theory, a technique named after Goresky-Kottwitz-MacPherson. In this talk, we give an overview of these notions and, based on previous work by the author, provide a complete description of the equivariant cohomology of rationally smooth group embeddings. Finally, we report on current work extending these results to the study of equivariant Chow rings and K-theory.

joint with String Theory Seminar