Lefschetz type formulas for dg-categories.

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 Lunts (J Algebra 356:230-256, ) 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 Ngô (Bull Math Sci 1(1):129-199, ). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ . [ABSTRACT FROM AUTHOR]