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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in deformation quantization, DQ-modules, complex Poisson manifolds, algebroid stacks, convolution of kernels, dualizing complexes, Hochschild homology, Euler classes, and holonomic modules.

Table of Contents

• Modules over formal deformations
• DQ-algebroids
• Kernels
• Hochschild classes
• The commutative case
• Symplectic case and \(\mathcal{D}\)-modules
• Holonomic DQ-modules
• Notation index
• Terminological index
• Bibliography