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.
