lectures

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*.

If the support of the pair is compact, the complex of solutions of

Next, we consider the case of complex symplectic manifolds. The ring