Lefschetz and Hirzebruch–Riemann–Roch formulas via noncommutative motives

Abstract

Abstract. V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemman-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.

Lefschetz type formulas for dg-categories

“…There are two important classes of examples in which the abstract formulas (0.2) and (0.4) can be made explicit: the categories of coherent sheaves on smooth projective varieties and the categories of matrix factorizations of a potential with an isolated singularity. Note that in [21] the formula (0.3) is rewritten in explicit terms for the case of coherent sheaves (see also [11]), while in [27] the case F = Id C of the formula (0.2) is made explicit for matrix factorizations. In Section 3.4 we extend this calculation to the case of an autoequivalence F induced by a diagonal linear map preserving the potential.…”

Iterated traces in 2-categories and Lefschetz theorems

“…• In the bicategory of spectral categories and their bimodules, Theorem 1.1 extends the main result of [Lun12,CT14] to the following new result.…”

“…In recent years, there has been a proliferation of Lefschetz-type theorems in noncommutative geometry [BZNb,BZNa,Pol14,Shk13,Lun12,CT14,Hoy14]. These results are comparisons of invariants and in their simplest form they compare the dimension of the Hochschild homology of a bimodule with the trace of the map induced by tensoring with that bimodule.…”

Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

“…One can also extend the classical Grothendieck-Riemann-Roch theorem in a non-commutative direction by studying traces of maps on Hochschild homology induced by functors between nice enough (e.g. smooth and proper) categories (see [Shk13], [Lun11], [Pol11], [CT14]). But while the Chern character makes perfect sense in the non-commutative context, it was already pointed out by Shklyarov that the Todd class seems to be of commutative nature and is missing in general non-commutative GRR-like theorems.…”