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