Joseph Bernstein, Tel Aviv University
Title:Regular Densities and Differential Operators on Singular Varieties.
Abstract:
Let X be an irreducible complex algebraic variety.
I claim that there is a natural (canonical) local construction that associates to X a coherent OX-module ωX that I call the sheaf of Regular Densities on X.
On the open subset Xsm of smooth points of X this sheaf is just the sheaf of densities = volume forms = top degree differential forms. So my claim
is that this familiar sheaf of densities on the subset Xsm has a canonical extension to the whole variety X.
This construction is non-trivial and is based on the theory of D-modules.
In my talk I will describe this construction and discuss some very nice
properties of the sheaf ωX. I will also discuss some generalizations of this construction.
In particular, I will discuss its application to the following general
question: What is a ”correct” notion of a differential operator on a
singular variety X.