Deformation quantization modules on complex symplectic manifolds

We study modules over the algebroid stack WX of deformation quantization on a complex symplectic manifold X and recall some results: construction of an algebra for ⋆-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic WX -modules, finiteness and duality for the composition of “good” kernels. As a corollary, we get that the derived category of good WX -modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this framework. Introduction Let X be a complex manifold, T ∗X its cotangent bundle. The conic sheaf of C-algebras ET∗X of microdifferential operators on T ∗X has been constructed functorially by Sato-Kashiwara-Kawai in [25]. This algebra is associated with the homogeneous symplectic structure and is also naturally defined on the projective cotangent bundle P ∗X . Another (no more conic) algebra on T ∗X , denoted here by WT∗X and defined over a subfield k of C[[τ−1, τ ] has been constructed in [24] (see [5] for related constructions). Its formal version has been considered by many authors after [1] and extended to Poisson manifolds in [22]. In general, neither the algebras EP∗X glue on a complex contact manifold, nor the algebras WT∗X glue on a complex symplectic manifold, although the categories of modules on these non existing algebras make sense. Indeed, one has to replace the notion of a sheaf of algebras by that of an algebroid stack, similarly as one replaces the notion of a sheaf by that of a stack. These constructions are performed in [15], [21], [24] (see also [7] for recent developments and [4, 30, 31] for an algebraic approach). Here, we start by briefly recalling the constructions of the sheaves ET∗X and WT∗X as well as a new sheaf of algebras on T ∗X containing WT∗X , invariant by quantized symplectic transformations, in which the ⋆-exponential is well defined (see 2000 Mathematics Subject Classification. 46L65, 14A20, 32C38, 53D55.