Pierre Schapira "Microlocal Euler classes and index theorem"

Pierre Schapira (IMJ)
Microlocal Euler classes and index theorem

I will show how to adapt the formalism of Hochschild homology for coherent sheaves on a complex manifold to a wide class of sheaves, including constructible sheaves on a real manifold $M$, $\mathcal D$-modules on a complex manifold and, more generally, elliptic pairs. For that purpose, we have to work "microlocally", that is, on the cotangent bundle $\pi : T^*M\to M$ and the role of Hochschild homology is played by $\pi^{-1}\omega_M$, the inverse image of the topological dualizing complex on $M$ (after having choosen a base ring $\mathbf k$). Then, to what we call a trace kernel we associate its microlocal Euler class, a class on $T^*M$ supported by the microsupport of the kernel. The main theorem asserts that this class is functorial with respect to the composition of kernels.
This construction gives a new approach to the Riemann-Roch or Atiyah-Singer theorems.