Motivic realizations of singularity categories and vanishing cycles
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 651-747.

In this article we establish a precise comparison between vanishing cycles and the singularity category of Landau–Ginzburg models over an excellent Henselian discrete valuation ring. By using noncommutative motives, we first construct a motivic -adic realization functor for dg-categories. Our main result, then asserts that, given a Landau–Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the -adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and -categorical lax symmetric monoidal version of Orlov’s comparison theorem between the derived category of singularities and the derived category of matrix factorizations for a Landau–Ginzburg model over a Noetherian regular local ring.

Dans cet article, on démontre un théorème de comparaison entre la théorie des cycles évanescents à la SGA7 et la catégorie des singularités d’un modèle de Landau-Ginzburg définie sur un anneau de valuation discrète, complet. Dans une première partie, nous étendons au cadre infini-catégorique le théorème de comparaison d’Orlov entre catégories de singularités et catégories de factorisations matricielles. Dans une seconde partie nous démontrons l’énoncé de comparaison, à l’aide d’une notion de réalisations motiviques de catégories.

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DOI: 10.5802/jep.81
Classification: 14F42,  19E08,  32S30,  16S38
Keywords: Landau-Ginzburg model, dg-categories of singularities, matrix factorisations, vanishing cycles, nearby cycles, motives, noncommutative motives, motivic homotopy theory Morel-Voevodsky, motivic realisations, -adic sheaves, algebraic K-theory
Anthony Blanc 1; Marco Robalo 2; Bertrand Toën 3; Gabriele Vezzosi 4

1 Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265, 34136 Trieste, Italia
2 Sorbonne Université, Faculté des sciences et ingénierie Pierre et Marie Curie, Institut de Mathématiques de Jussieu-PRG 4 place Jussieu, Case 247, 75252 Paris Cedex 05, France
3 CNRS, Université de Toulouse, Institut de Mathématiques de Toulouse (UMR 5219) 118 route de Narbonne, 31062 Toulouse Cedex 9, France
4 Dipartimento di Matematica ed Informatica “Ulisse Dini”, Università degli Studi di Firenze Viale Morgagni 67/a, 50134 Firenze, Italy
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Anthony Blanc; Marco Robalo; Bertrand Toën; Gabriele Vezzosi. Motivic realizations of singularity categories and vanishing cycles. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 651-747. doi : 10.5802/jep.81. https://jep.centre-mersenne.org/articles/10.5802/jep.81/

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