Motivic realizations of singularity categories and vanishing cycles

Anthony Blanc; Marco Robalo; Bertrand Toën; Gabriele Vezzosi

doi:10.5802/jep.81

Motivic realizations of singularity categories and vanishing cycles
[Réalisations motiviques des catégories de singularités et cycles évanescents]

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

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 651-747.

Résumé
Abstract

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.

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 $ℓ$ -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 $ℓ$ -adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$ -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.

Reçu le : 2017-09-07
Accepté le : 2018-09-03
Publié le : 2018-10-09

MR Zbl

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, $\ell $-adic sheaves, algebraic K-theory
Mot clés : Modèles de Landau-Ginzburg, dg-catégories de singularités, factorisations matricielles, cycles évanescents, cycles proches, motifs, motifs non-commutatifs, théorie homotopique motivique des schémas, réalisations motiviques, faisceaux $\ell $-adiques, K-théorie algébrique

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Motivic realizations of singularity~categories and vanishing~cycles},
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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, Tome 5 (2018), pp. 651-747. doi : 10.5802/jep.81. https://jep.centre-mersenne.org/articles/10.5802/jep.81/

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