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Deformation Quantization Modules

MAA Review

The basic idea of a deformation quantization is to replace the structure sheaf of a complex manifold with a formal deformation of it and then consider modules over this algebra. For technical reasons it is better to replace the structure sheaf with a deformation algebroid stack and work with objects in the corresponding derived category.

The monograph under review contains two previous papers available in preprint form in the ArXiv: "Deformation quantization modules I: Finiteness and duality" and Deformation quantization modules II: Hochschild class", which were further developed in the preprint "Deformation quantization modules".

In the first part, Chapters one to three, the authors introduce a convolution for deformation quantization modules and prove, under the usual properness assumption, that the convolution of two coherent modules is also coherent. This result can be seen as a generalization of a classical theorem of Grauert. The authors also construct a dualizing complex and prove that dualization commutes with convolution, generalizing the classical Cartan-Serre duality theorem.

In the second part, Chapters 4 to 6, the authors construct Hochschild classes of coherent deformation quantization modules and prove that the Hochschild class commutes with convolution. Chapter 7, which is not included in the mentioned ArXiv preprints, is devoted to the study of holonomic modules on complex symplectic manifolds.

Given the clarity of the exposition, the monograph could be used as introduction to the geometry of microdifferential systems in complex analytic manifolds.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.

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