# Algebraic Geometry Seminar

#### Equivariant motives and Sheaves on moment graphs.

**Speaker:**
Kirill Zaynullin, University of Ottawa

**Location:**
Warren Weaver Hall 317

**Date:**
Tuesday, November 14, 2017, 3:30 p.m.

**Synopsis:**

Goresky, Kottwitz and MacPherson showed that the equivariant cohomology of varieties equipped with an action of a torus $T$ can be described using the so called moment graph, hence, translating computations in equivariant cohomology into a combinatorial problem. Braden and MacPherson proved that the information contained in this moment graph is sufficient to compute the equivariant intersection cohomology of the variety. In order to do this, they introduced the notion of a sheaf on moment graph whose space of sections (stalks) describes the (local) intersection cohomology. These results motivated a series of paper by Fiebig, where he developed and axiomatized sheaves of moment graphs theory and exploited Braden-MacPherson’s construction to attack representation theoretical problems.

In the talk we explain how to extend this theory of sheaves on moment graphs to an arbitrary algebraic oriented equivariant cohomology $h$ in the sense of Levine-Morel

(e.g. to K-theory or algebraic cobordism). Moreover, we show that in the case of a total flag variety $X$ the space of global sections of the respective $h$-sheaf also describes

an endomorphism ring of the equivariant $h$-motive of $X$.

This is a joint work in progress with Martina Lanini.