The singular support of l-adic sheaves; the relative continuous p-adic K-theory and cyclic homology

特征循环是复杂空间奇点的一个关键不变量；它所有已知的结构都具有超验的性质。该项目第一部分的目标是开发一种可用于特征 p 几何的纯几何方法。这个想法是使用几何氡变换来查看特征循环的隐藏组件，类似于计算机断层扫描（基于经典氡变换）对人体内部结构进行成像的方式。该项目第二部分的目标是解释如何从 p 进空间上的 p 进循环从它们的旋转数（微分形式的周期）中恢复出来。光滑代数簇上的可构造层 F 的特征循环提供了有关所有具有孤立奇点的函数（相对于 F）的消失循环空间的（总）维数信息。由于柏原和夏皮拉，它的构造因复杂的多样性而闻名，而且它纯粹是超验的。该项目第一部分的目标是找到可应用于特征 p 几何的特征循环的纯代数几何构造。关键工具是 Brylinski 的几何 Radon 变换，它是经典 Lefschetz 铅笔理论的有力推广。该项目的第二部分旨在利用 p-adic 环的相对连续 K 理论与其连续循环同源性之间的规范同源性（如主要研究者最近的预印本 (arXiv:1312.3299) 中所定义）来理解 p-adic 流形上的代数循环。

The characteristic cycle is a key invariant of singularities of complex spaces; all its known constructions have transcendental nature. The aim of the first part of the project is to develop a purely geometric approach to the theory that can be used in characteristic p geometry. The idea is to use a geometric Radon transform to see hidden components of the characteristic cycle, similar to the way that computer tomography (based on a classical Radon transform) allows imaging of the internal structure of the human body. The goal of the second part of the project is to explain how p-adic cycles on a p-adic space can be recovered from their rotation numbers (the periods of differential forms).The characteristic cycle of a constructible sheaf F on a smooth algebraic variety provides information about the (total) dimension of the spaces of vanishing cycles for all functions with isolated singularities (with respect to F). Its construction is known, due to Kashiwara and Shapira, for complex varieties, and it is purely transcendental. The goal of the first part of the project is to find a purely algebro-geometric construction of the characteristic cycle that can be applied in characteristic p geometry. The key instrument is Brylinski's geometric Radon transform which is a powerful generaliztion of the classical Lefschetz pencil theory. The second part of the project aims to use a canonical isogeny between the relative continuous K-theory of a p-adic ring and its continuous cyclic homology, as defined in a recent preprint (arXiv:1312.3299) of the principal investigator, to understand algebraic cycles on a p-adic manifold.