Deformation Quantization Modules

On a complex manifold $(X,\mathcal{{O}}_X)$, a $\mathrm{{DQ}}$-module is a module (in the derived sense) over an algebroid stack locally equivalent to the sheaf $\mathcal{{O}}_X[[\hbar]]$ endowed with a star-product. The book treats relative finiteness, duality and index theorems for $\mathrm{{DQ}}$-modules, showing in particular the functoriality of the Hochschild class in this framework and studying in detail holonomic modules in the symplectic case. Hence, these notes could be...
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On a complex manifold $(X,\mathcal{{O}}_X)$, a $\mathrm{{DQ}}$-module is a module (in the derived sense) over an algebroid stack locally equivalent to the sheaf $\mathcal{{O}}_X[[\hbar]]$ endowed with a star-product. The book treats relative finiteness, duality and index theorems for $\mathrm{{DQ}}$-modules, showing in particular the functoriality of the Hochschild class in this framework and studying in detail holonomic modules in the symplectic case. Hence, these notes could be considered both as an introduction to noncommutative complex analytic geometry and to the study of microdifferential systems on complex Poisson manifolds.