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 ×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 T * 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 of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.

Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets

“…Note that in the more general setting of elliptic pairs a similar construction of microlocal Lefschetz classes was previously given in Guillermou [12]. The difference from his is that we explicitly realized them as Lagrangian cycles in T * M. For recent results on this subject, see also [14], [18] and [23] etc. Note also that if φ = id X , M = X and Φ = id F , our Lefschetz cycle LC(F, Φ) M coincides with the characteristic cycle CC(F ) of F introduced by Kashiwara [15].…”

Hyperbolic localization via shrinking subbundles

“…Note also that if φ = id X , M = X and Φ = id F , the Lagrangian cycle LC(F, Φ) coincides with the characteristic cycle CC(F ) of F introduced by Kashiwara [K1]. For recent results on this subject, see also [KS2], [I], etc.…”

Microlocal Lefschetz Classes of Graph Trace Kernels

“…We follow the notation of [KS14]. The results in this subsection are the same as in Section 3 of [KS14].…”

“…In [KS14], Kashiwara and Schapira introduced the notion of trace kernels and the method to associate a microlocal Euler class with a trace kernel.…”

“…In the last section of [KS14], the authors have remarked that trace kernels can be adapted to the Lefschetz fixed point formula for constructible sheaves. However, their construction is not suitable to prove the functoriality of the cohomology classes.…”