Lefschetz type formulas for dg-categories

TL;DR

This paper develops a Lefschetz formula for endofunctors in smooth compact dg-categories, generalizing existing formulas and applying to matrix factorizations and equivariant modules, with implications for algebraic geometry and singularity theory.

Contribution

It introduces a Lefschetz formula for dg-categories, generalizes Lunts' formula, and applies it to matrix factorizations and equivariant modules, expanding the theoretical framework.

Findings

Derived a holomorphic Lefschetz formula for dg-categories.

Generalized Lunts' Lefschetz reciprocity law for commuting endofunctors.

Computed explicit ingredients for matrix factorizations of isolated singularities.

Abstract

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of V. Lunts that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngo. Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity w(x). We apply this formula to get some restrictions on the Betti numbers of a Z/2-equivariant module over k[[x_1,...,x_n]]/(w) in the case when w(-x)=w(x).