Resumen de Lefschetz type formulas for dg-categories

Alexander Polishchuk

• 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 wwww . We apply this formula to get some restrictions on the Betti numbers of a Z/2Z/2 -equivariant module over k[[x1,…,xn]]/(ww)k[[x1,…,xn]]/(ww) in the case when ww(−x)=ww(x)ww(−x)=ww(x) .