Motivic realizations of singularity categories and vanishing cycles

Anthony Blanc (1) , Marco Robalo (2) , Bertrand Toën (3) , Gabriele Vezzosi (4)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 DMA - Dipartimento di Matematica Applicata [Firenze]

In this paper 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.

Domaines: Géométrie algébrique [math.AG]

Anthony Blanc, Marco Robalo, Bertrand Toën, Gabriele Vezzosi. Motivic realizations of singularity categories and vanishing cycles. Journal de l'École polytechnique — Mathématiques, 2018, Tome 5, pp.651-747. ⟨10.5802/jep.81⟩. ⟨hal-01891197v2⟩