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In this paper we establish a precise comparison between vanishing cycles andthe singularity category of Landau-Ginzburg models over a complete discretevaluation ring. By using noncommutative motives, we first construct a motivic$\ell$-adic realization functor for dg-categories. Our main result, thenasserts that, given a Landau-Ginzburg model over a complete discrete valuationring with potential induced by a uniformizer, the $\ell$-adic realization ofits singularity category is given by the inertia-invariant part of vanishingcohomology. We also prove a functorial and $\infty$-categorical version ofOrlov's comparison theorem between the derived category of singularities andthe derived category of matrix factorizations for a Landau-Ginzburg model overa noetherian regular local ring.

Motivic Realizations of Singularity Categories and Vanishing Cycles