Derived $V$-filtrations and the Kontsevich-Sabbah-Saito theorem

Abstract

Let $f: X \to \mathbb{A}^1$ be a regular function on a smooth complex algebraic variety $X$. We formulate and prove an equivalence between the algebraic formal twisted de Rham complex of $f$ and the vanishing cycles with respect to $f$ as objects in the category of sheaves valued in the derived $\infty$-category of modules over $\widehat{\mathscr{E}}_{\mathbb{C},0}^{\mathrm{alg}}$, the ring of germs of algebraic formal microdifferential operators. This is a direct generalization of Kontsevich's conjecture, proven in work by Sabbah and then Sabbah--Saito, of an algebraic formula computing vanishing cohomology. The novelty in our approach is the introduction of a canonical $V$-filtration on the derived $\infty$-category of regular holonomic $\mathscr{D}_{\mathbb{C},0}$-modules, and the use of various techniques from the theory of higher categories and higher algebra in the context of the subject of microdifferential calculus.