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Lefschetz type formulas for dg-categories

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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 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 \({\varvec{w}}(-x)={\varvec{w}}(x)\).

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Notes

  1. The definition of the Mukai pairing in [10, Sec. 5] seems to contain a misprint.

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Acknowledgments

I am grateful to Luchezar Avramov and David Eisenbud for helpful discussions.

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  1. Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA

    Alexander Polishchuk

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  1. Alexander Polishchuk
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Correspondence to Alexander Polishchuk.

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Supported in part by NSF grant.

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Polishchuk, A. Lefschetz type formulas for dg-categories. Sel. Math. New Ser. 20, 885–928 (2014). https://doi.org/10.1007/s00029-013-0143-5

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