Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections

Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf O_X to a sheaf of noncommutative algebras and of the sheaves O_Y and O_Z to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf Tor^{O_X}_*(O_Y, O_Z). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as well. Our construction is motivated by, and is closely related to, a result of Behrend-Fantechi, who considered the case of Lagrangian submanifolds in a symplectic manifold.