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Lefschetz type formulas for dg-categories

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arxiv 1111.0728 v7 pith:2QBEPUMV submitted 2011-11-03 math.AG

Alexander Polishchuk This is my paper · ORCID
classification math.AG
keywords formulalefschetzdg-categoriesendofunctorsholomorphicproveanalogapplication
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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).

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  1. Iterated traces in 2-categories and Lefschetz theorems

    math.AT 2019-08 conditional novelty 7.0 of 10

    Iterated traces in any 2-dualizable symmetric monoidal bicategory commute, recovering and extending a wide family of Lefschetz-type theorems.

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