Cat\'egories des singularit\'es, factorisations matricielles et cycles \'evanescents
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(X0) to Sing(X) induced by the pushforward along the inclusion of the (derived) zero locus X0 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 ∈ Bn). 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 ∈ OX(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),(f1,..,fn)) for (f1,..,fn) ∈ Bn. 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.