# Contents

## Idea

One may give an abstract definition of an elliptic complex over a complex analytic space: an elliptic pair is the datum of an $\mathbb{R}$-constructible sheaf $F$ and a complex $\mathcal{M}$ of $\mathcal{D}$-modules, such that the intersection

$char(\mathcal{M})\cap SS(F)=T^*_M M\subset T^*M$

of the characteristic variety of $\mathcal{M}$ with the microsupport of $F$ is contained in the zero section of the cotangent bundle.

To such an elliptic pair, one may associate a pair of trace kernels $K_{\mathcal{M}}$ and $K_F$ such that the trace kernel $K_{\mathcal{M}}\otimes K_F$ is well defined (because of the microsupport condition). The index of the elliptic pair is given by the Hochshild class $eu(K_{\mathcal{M}}\otimes K_F)$, and the index theorem essentially says that this class may be computed as the product of the classes of the two kernels in play.

## Generalization

One may try to generalize the microlocal formulation of index theory to global analytic geometry, using derived microlocalization, to get a global analytic index theory that generalizes Kashiwara and Schapira’s approach to the analytic situation over an arbitrary Banach ring.

## Reference

Last revised on January 3, 2015 at 20:40:00. See the history of this page for a list of all contributions to it.