Microlocal Euler classes and Hochschild homology

Abstract

We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle over M and we prove that this class is functorial with respect to the composition of kernels. This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.

The proof of Theorem 4.6 has been considerably simplified. To appear in the Journal of the Institute of Mathematics of Jussieu

References in corpus (5)

• Deformation quantization modules
• The relative Riemann-Roch theorem from Hochschild homology
• On a conjecture of Kashiwara relating Chern and Euler classes of O-modules
• Hochschild Lefschetz Class for $D$-modules
• The Coherent-Constructible Correspondence and Fourier-Mukai Transforms

Cited by in corpus (4)

• Calabi-Yau structures on topological Fukaya categories
• Microlocal Lefschetz classes of graph trace kernels
• Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets
• Action of micro-differential operators on quantized contact transformations