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, 2012) 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, 2011). 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)) .