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Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraicPoisson variety X. We show that any data of first order deformation of thestructure sheaf O_X to a sheaf of noncommutative algebras and of the sheavesO_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 theTor-sheaf Tor^{O_X}_*(O_Y, O_Z). The induced Gerstenhaber bracket on theTor-sheaf turns out to be canonically defined; it is independent of the choicesof deformations involved. There are similar results for Ext-sheaves as well.  Our construction is motivated by, and is closely related to, a result ofBehrend-Fantechi, who considered the case of Lagrangian submanifolds in asymplectic manifold.

Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections