Chern classes of quantizable coisotropic bundles

Abstract

Let $M$ be a smooth algebraic variety of dimension $2(p+q)$ with an algebraic symplectic form and a compatible deformation quantization ${\mathcal{O}}_h$ of the structure sheaf. Consider a smooth coisotropic subvariety $j:Y\to M$ of codimension $q$ and a vector bundle $E$ on $Y$. We show that if $j_{\ast }E$ admits a deformation quantization (as a module) then its characteristic class $\hat{A}(M)exp(-c({\mathcal{O}}_h))ch(j_{\ast }E)$ lifts to a cohomology group associated to the null foliation of $Y$. Moreover, it can only be nonzero in degrees $2q,\dots ,2(p+q)$. For Lagrangian $Y$ this reduces to a single degree $2q$. Similar results hold in the holomorphic category. This is a companion paper of a joint work with Victor Ginzburg on general quantizable sheaves.