Hochschild Lefschetz class for $$\mathcal{D }$$ -modules

Abstract

We introduce a notion of Hochschild Lefschetz class for a good coherent D-module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann-Roch formula for a D-module on a compact complex orbifold. This is actually a slight abuse of terminology. In fact, HH( EX , EX) is an object in the derived category of sheaves of C-vector spaces on X.2 See Eq.(1) for the definition of HH( EX , E γ X ).

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].…”

Microlocal Euler classes and Hochschild homology

“…Although (9.4) is not a trace kernel in the sense of Definition 5.1, it should be possible to adapt the previous constructions to the case of D-modules and to elliptic pairs, and then to recover a theorem of [7], but we do not develop this point here (see [21] for related results).…”

Microlocal Euler classes and Hochschild homology

“…ω ∆ M ≃ i * ω L , we get the morphisms Here the middle arrow is derived from (9.4). Although (9.4) is not a trace kernel in the sense of Definition 5.1, it should be possible to adapt the previous constructions to the case of D-modules and to elliptic pairs, then to recover a theorem of [Gu96] but we do not develop this point here (see [RTT12] for related results).…”