Finiteness and index theorems for elliptic pairs and for modules over quantization-deformation rings

Pierre Schapira

University of Paris 6, France

An elliptic pair on a complex manifold X is the data of a coherent module M over the ring D of differential operators and an R-constructible sheaf F on X, the characteristic variety of M not intersecting the microsupport of F outside the zero-section of the cotangent bundle T*X.
If the support of the pair is compact, the complex of solutions of M with values in the sheaf of generalized holomorphic functions associated with F has finite dimensional cohomology over C, and the index is calculated in terms of the Euler class of M and that of F. This result gives a new approach and a generalization of many classical ones,including the Atiyah-Singer theorem.
Next, we consider the case of complex symplectic manifolds. The ring D is replaced with a quantization-deformation ring W (which may not exist globally, but the stack of W-modules exists) and the finiteness and index theorems (partly conjectural) hold when replacing C with a field K containing a parameter h.