Astérisque, n° 345. Deformation quantization modules
On a complex manifold ( X, (...)<sub>X</sub> ), a DQ-algebroid (...)<sub>X</sub> is an algebroid stack locally equivalent to the sheaf (...)<sub>X</sub> [(...)] endowed with a star-product and a DQ-module is an object of the derived category D<sup>b</sup> ((...)).
The main results are :
- the notion of cohomologically complete DQ-modules which allows one to deduce various properties of such a module (...) from the corresponding properties of the (...)-module (...),
- a finiteness theorem, which asserts that the convolution of two coherent DQ-kernels defined on manifolds X<sub>i</sub> x X<sub>j</sub> ( i = 1, 2, j = i + 1), satisfying a suitable properness assumption, is coherent (a non commutative Grauert's theorem),
- the construction of the dualizing complex for coherent DQ-modules and a duality theorem which asserts that duality commutes with convolution (a non commutative Serre's theorem),
- the construction of the Hochschild class of coherent DQ-modules and the theorem which asserts that Hochschild class commutes with convolution,
- in the commutative case, the link between Hochschild classes and Chern and Euler classes,
- in the symplectic case, the constructibility (and perversity) of the complex of solutions of an holonomic DQ-module into another one after localizing with respect to h .
Hence, these Notes could be considered both as an introduction to non commutative complex analytic geometry and to the study of micro-differential systems on complex Poisson manifolds.
