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 of the structure sheaf. Consider a smooth coisotropic subvariety $Y$ of codimension $q$ and a vector bundle $E$ on $Y$. We show that if the pushforward of $E$ admits a deformation quantization (as a module), then its “trace density” characteristic class 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.