Motivic realizations of singularity categories and vanishing cycles

Abstract

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -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.