Quatrième de
couverture
On a complex manifold (X, (...)X), a DQ-algebroid (...)X is an algebroid stack locally equivalent to the sheaf (...)X [(...)] endowed with a star-product and a DQ-module is an object of the derived category Db ((...)).
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 Xi x Xj (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.
