Chern classes of quantizable sheaves and characteristic cycles

Abstract: Let A be a formal deformation quantization of the structure sheaf of an algebraic symplectic manifold X. Given a coherent sheaf E on X we define a characteristic class s(E) as a product of the Chern character of E and a certain class associated with the quantization A. We show that if E can be quantized to an A-module then all homogeneous components of s(E) in a certain range of degrees vanish. The proof is based on relating the Chern characters of E and of its quantization. The latter lives in the negative cyclic homology of A and we show that the negative cyclic homology groups of relevant degrees vanish.

In the holonomic case our result says that if the support of the quantizable sheaf E is a (possibly singular) Lagrangian subvariety, then the only nonvanishing Chern class of E is the top degree class which is the Poincare dual of support cycle of E. As an application, let X be a conical symplectic resolution and B the algebra of global sections of a filtered quantization of X. We prove, motivated by a question by Bezrukavnikov and Losev, that the characteristic cycles of finite dimensional simple B-modules are linearly independent.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar