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

特征圈是复空间奇点的一个关键不变量，它的所有已知构造都具有超越性质。该项目的第一部分的目的是开发一个纯粹的几何方法的理论，可用于特征p几何。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。该项目的第二部分的目标是解释如何从p-adic空间上的p-adic循环的旋转数（微分形式的周期）中恢复p-adic循环。光滑代数簇上的可构造层F的特征循环提供了关于所有具有孤立奇点的函数（关于F）的消失循环空间的（总）维数的信息。它的建设是众所周知的，由于柏原和Shapira，复杂的品种，它是纯粹的先验。该项目的第一部分的目标是找到一个纯粹的代数几何结构的特征循环，可以应用于特征p几何。关键工具是Brylinski的几何Radon变换，它是经典Lefschetz束理论的一个强有力的推广。该项目的第二部分旨在使用p-adic环的相对连续K-理论与其连续循环同调之间的典范同构，如主要研究者最近的预印本（arXiv：1312.3299）中所定义的，以理解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.