March 7, 2013

Mathematics Colloquium today

By Reta McDermott

Dima Arinkin, from the University of Wisconsin, Madison, will present "Singular Support of Coherent Sheaves" at 3:30 p.m. Thursday, March 7, in 143 Cardwell Hall.

Abstract: Some geometric objects can be studied microlocally instead of just looking at their support the set of points where the object is nontrivial, one can consider its singular support, which remembers the direction of nontrivial behavior. Examples include the wave front of a distribution, the singular support of a constructible sheaf, and the characteristic variety of a D-module.

The goal of this talk is to sketch such microlocal theory for coherent sheaves. It turns out that the singular support could be non-trivial only if the variety is singular. I will define singular support for coherent sheaves on a singular variety that is locally a complete intersection. The singular support measures the imperfection of a coherent sheaf: it equals 0 if and only if the coherent sheaf has finite Tor dimension, such as the sheaf is perfect. The theory is important for the geometric Langlands program; this relation is going to be discussed in my M-seminar talk.