Chern classes of quantizable coisotropic bundles

TL;DR

This paper investigates the conditions under which the characteristic classes of coisotropic bundles on algebraic symplectic varieties can be lifted to certain cohomology groups, revealing degree restrictions and special cases for Lagrangian subvarieties.

Contribution

It establishes new criteria for the liftability of characteristic classes of quantizable coisotropic bundles in algebraic symplectic geometry, extending previous work to the holomorphic setting.

Findings

Characteristic class lifts only in degrees 2q to 2(p+q).

For Lagrangian subvarieties, the lift is in a single degree 2q.

Results apply to both algebraic and holomorphic categories.

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 $j_* E$ admits a deformation quantization (as a module) then its characteristic class $\widehat{A}(M) exp(-c(\mathcal{O}_h)) ch(j_* E)$ lifts to a cohomology group associated to the null foliation of $Y$. Moreover, it can only be nonzero in degrees $2q, \ldots, 2(p+q)$. 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.