Quantizable coherent sheaves and their Chern classes
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.
Professor Vladimir Baranovsky, University of California Irvine