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

特征环是复数空间奇异性的关键不变量；它所有已知的构造都具有先验性。该项目第一部分的目的是开发一种纯几何方法来研究可用于特征几何的理论。这个想法是使用几何Radon变换来查看特征周期的隐藏组件，类似于计算机断层扫描（基于经典Radon变换）允许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进空间的旋转数（微分形式的周期）中恢复p进循环。光滑代数变换上可构造轴F的特征环提供了所有具有孤立奇点的函数（关于F）的消失环空间的（总）维数信息。它的构造是已知的，由于柏原和夏皮拉，复杂的品种，它是纯粹先验的。项目第一部分的目标是找到一个可以应用于特征p几何的特征循环的纯代数几何构造。关键的工具是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使用p进环的相对连续k理论与其连续循环同调之间的正则等同性，如主要研究者最近的预印本（arXiv:1312.3299）中定义的那样，来理解p进流形上的代数循环。

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.