Deformation Quantization Modules Ii. Hochschild Class

This paper is the continuation of [12]. We construct the Hochschild class for coherent modules over a deformation quantization algebroid on a complex Poisson manifold. We also define the convolution of Hochschild homologies, and prove that the Hochschild class of the convolution of two coherent modules is the convolution of their Hochschild classes. We study with some details the case of symplectic deformations. Mathematics Subject Classification: 53D55, 46L65, 32C38 Introduction Set k0 := C[[~]], k := C((~)) = k0[~ ]. In [12], we define a DQ-algebra AX on a complex manifold X as a sheaf of k0-algebras locally isomorphic to (OX [[~]], ⋆), where ⋆ is a star-product, and we define a DQ-algebroid as a k0-algebroid stack locally equivalent to the algebroid associated with a DQ-algebra. (Here, DQ stands for “deformation quantization”.) For a DQ-algebroid AX , we denote by AXa the opposite algebroid A op X and we denote by AX1×X2 the external product of the algebroids AXi (i = 1, 2). An object of D(AX1×Xa 2 ), the bounded derived category of the abelian category of AX1×X 2 -modules, is sometimes called a kernel. ∗The first author is partially supported by Grant-in-Aid for Scientific Research (B) 18340007, Japan Society for the Promotion of Science.