论文标题
catégoriesdesSingularités，分解矩阵和周期evanescents
Catégories des singularités, factorisations matricielles et cycles évanescents
论文作者
论文摘要
本论文的目的是研究对（x，s）的奇异性唱片（x，s）的DG类别，其中X是一个方案，S是一个方案，S是X的某个矢量束的全球部分。Sing（x，x，s）被定义为sing（x_0）从sing（x_0）中引起的（x）的dg函数的核心（x）的dg函数（x），沿着（x）沿着（x）的范围（x）（x）的包含在x中（x）。 在第一部分中，我们将自己限制在矢量捆绑包很小的情况下。当x = spec（b）仿射时，我们证明了sing（x，s）的结构定理。粗略地，它告诉我们，sing（x，s）中的每个对象都由集中在n+1连续度（如果b^n中的s \）中的B模块的复合体表示。通过专门针对n = 1的情况，我们将Orlov的定理概括为sing（x，s）与矩阵因数的DG类别mf（x，s），将s \ in O_x（x）中的s \不平坦。 在第二部分中，我们研究了Sing（X，S）的L -ADIC共同体（如A.为此，我们介绍了单层不变的消失周期的L-ADIC捆。使用J.〜Burke和M.〜Walker概括的D. Orlov定理，我们计算了SING的L-ADIC实现（spec（b），（f_1，..，f_n）），用于（f_1，..，..，f_n）\ in B^n。 在上一章中，我们介绍了在等级2的离散估值环上迭代迭代消失的循环的L-ADIC滑轮。我们将这些L-ADIC或骨之一与基础封闭的亚chemeememememememememememepheme的L-ADIC Sheaves之一联系在一起。
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.~Toën 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.
