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arXiv · 1607.03012

Motivic Realizations of Singularity Categories and Vanishing Cycles

A. Blanc·M. Robalo·B. Töen·G. Vezzosi
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Abstract

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over a complete discrete valuation ring. By using noncommutative motives, we first construct a motivic $\ell$-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 $\ell$-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$-categorical 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.

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A. Blanc, M. Robalo, B. Töen, G. Vezzosi. 2016-07-11. Motivic Realizations of Singularity Categories and Vanishing Cycles. https://arxiv.org/abs/1607.03012

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