Quantizable coherent sheaves and their Chern classes
Seminar/Forum
Let \(X\) be a manifold with a symplectic form \(w\) (in Cinfinity, holomorphic or algebraic setting). At least locally the algebra \(O\) of functions on \(X\) admits a noncommutative deformation Oh compatible with \(w\) (as explained in the work of De WildeLecomte, 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 Ahat 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 Cinfinity setting, which appears to be new. Joint work with Victor Ginzburg.
Presenter

Professor Vladimir Baranovsky, University of California Irvine