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Microlocal Euler classes and Hochschild homology

Published online by Cambridge University Press:  18 July 2013

Masaki Kashiwara
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Japan Department of Mathematical Sciences, Seoul National University, Republic of Korea (masaki@kurims.kyoto-u.ac.jp)
Pierre Schapira
Affiliation:
Institut de Mathématiques, Université Pierre et Marie Curie, France Mathematics Research Unit, University of Luxemburg, Luxemburg (schapira@math.jussieu.fr)
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Abstract

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We define the notion of a trace kernel on a manifold $M$. Roughly speaking, it is a sheaf on $M\times M$ for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle ${T}^{\ast } M$ over $M$, and we prove that this class is functorial with respect to the composition of kernels.

This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, $\mathscr{D}$-modules and elliptic pairs.

Type
Research Article
Copyright
©Cambridge University Press 2013 

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