Algebra and Geometry Seminar
Some geometric objects can be studied `microlocally': instead of just looking at their support (the set of points where an object is non-trivial), one can consider their `singular support', which remembers the `direction' of non-trivial behavior'. Examples include the wave front of a distribution, the singular support of a constructible sheaf, and the characteristic variety of a D-module.
In this talk, I will sketch two such `microlocal' theories.
First, I will discuss the singular support for coherent sheaves. It turns out that this theory is non-trivial only for coherent sheaves on a singular variety. The singular support measures the `imperfection' of a coherent sheaf: it equals zero if and only if the coherent sheaf has finite Tor dimension (i.e., the sheaf is perfect). Going one step up in the categorification hierarchy, I will then consider the singular support for categories over a variety.
The project is motivated by the geometric Langlands correspondence; I hope to sketch the relation with the Langlands correspondence in the talk.