Cat\'egories des singularit\'es, factorisations matricielles et cycles \'evanescents

TL;DR

This thesis explores the structure of dg categories of singularities associated with schemes and sections, generalizing known theorems and introducing new l-adic sheaves to analyze their cohomological properties.

Contribution

It generalizes Orlov's theorem to non-flat sections, studies l-adic cohomology of singularity categories, and introduces l-adic sheaves of vanishing cycles for schemes over valuation rings.

Findings

Every object in Sing(X,s) is represented by a complex of modules in n+1 degrees.

Generalization of Orlov's theorem to non-flat sections.

Computed l-adic realization of Sing for specific schemes.

Abstract

The aim of this thesis is to study the dg categories of singularities Sing(X,s) of pairs (X,s), where X is a scheme and s is a global section of some vector bundle over X. Sing(X,s) is defined as the kernel of the dg functor from Sing(X_0) to Sing(X) induced by the pushforward along the inclusion of the (derived) zero locus X_0 of s in X. In the first part, we restrict ourselves to the case where the vector bundle is trivial. We prove a structure theorem for Sing(X,s) when X=Spec(B) is affine. Roughly, it tells us that every object in Sing(X,s) is represented by a complex of B-modules concentrated in n+1 consecutive degrees (if s \in B^n). By specializing to the case n=1, we generalize Orlov's theorem, which identifies Sing(X,s) with the dg category of matrix factorizations MF(X,s), to the case where s \in O_X(X) is not flat. In the second part, we study the l-adic cohomology of Sing(X,s) (as defined by A.~Blanc - M.~Robalo - B.~To\"en and G.~Vezzosi) when s is a global section of a line bundle. In order to do so, we introduce the l-adic sheaf of monodromy-invariant vanishing cycles. Using a theorem of D.~Orlov generalized by J.~Burke and M.~Walker, we compute the l-adic realization of Sing(Spec(B),(f_1,..,f_n)) for (f_1,..,f_n) \in B^n. In the last chapter, we introduce the l-adic sheaves of iterated vanishing cycles of a scheme over a discrete valuation ring of rank 2. We relate one of these l-adic sheaves to the l-adic realization of the dg category of singularities of the fiber over a closed subscheme of the base.