# Quantizable coherent sheaves and their Chern classes

## Seminar/Forum

Evan Williams
Peter Hall
Let $$X$$ be a manifold with a symplectic form $$w$$ (in C-infinity, holomorphic or algebraic setting). At least locally the algebra $$O$$ of functions on $$X$$ admits a non-commutative deformation Oh compatible with $$w$$ (as explained in the work of De Wilde-Lecomte, Fedosov, Kontsevich and others). If $$E$$ is a module over $$O$$ (such as sections of a vector bundle on a submanifold $$Y$$ of $$X$$), we study the question whether $$E$$ also admits a deformation to a module $$Eh$$ over $$Oh$$. We prove that in the holomorphic and algebraic cases this imposes rather strong restrictions on the quantum Chern character of $$E$$" built out of the usual Chern character of $$E$$, the A-hat genus of $$X$$ and the Deligne class of $$Oh$$ which encodes information about the choice of $$O_h$$. A version of this result gives a nontrivial condition even in the C-infinity setting, which appears to be new. Joint work with Victor Ginzburg.