Modules Over Deformation Quantization Algebroids: An Overview

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This paper is essentially an overview of a forthcoming paper in which we study coherent modules over deformation quantization algebroids on complex Poisson manifolds. First, we construct the convolution of coherent kernels over such algebroids, and prove that this convolution preserves coherency and commutes with duality. Next, we define the Hochschild class of coherent modules and prove that the Hochschild class of the convolution of two coherent kernels is the convolution of their Hochschild classes. Finally, we study with some details the case of symplectic deformations and apply these results to the Euler class of coherent 𝒟-modules.

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