Deformation quantization modules (Q2914245)

On a complex manifold \((X,\mathcal O_X)\), a DQ-algebroid \(\mathcal A_X\) is an algebroid stack locally equivalent to the sheaf \(\mathcal O_X[[\hbar]]\) endowed with a star-product, and a DQ-module is an object in the derived category \(D^b(\mathcal A_X)\). The main results are: NEWLINENEWLINENEWLINE [i.] The notion of cohomologically complete DQ-modules which allows to deduce various properties of such a module \(\mathcal M\) from the corresponding properties of the \(\mathcal O_X\)-module \(\mathbb Z\otimes_{\mathbb Z[\hbar]}^L\mathcal M\),NEWLINENEWLINE[ii.] a fitness theorem, which asserts that the convolution of two coherent DQ-kernels defined on manifolds \(X_i\times X_j\) (\(i=1,2, j=i+1\)), satisfying a suitable properness assumption, is coherent (a non commutative Grauert theorem),NEWLINENEWLINE[iii.] 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 theorem).NEWLINENEWLINE[iv.] the construction of the Hochschild class of coherent DQ-modules and the theorem which asserts that the Hochschild class commutes with convolution,NEWLINENEWLINE[v.] in the commutative case, the link between Hochschild classes and Chern and Euler classes,NEWLINENEWLINE[vi.] 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 \(\hbar\).NEWLINENEWLINEThese notes could be considered both as an introduction to non commutative complex analytic geometry and to the study of microdifferential systems on complex Poisson manifolds.