Integral p_adic Hodge-theory and applications to p-adic deformation theory

我们将线性数据关联到几何对象，如曲线、曲面或三重几何对象，这些几何对象被定义为整数上或被素数的幂零化的数域上的代数方程的零集。人们可以问，通过这种方式可以捕获多少关于原始对象的信息。线性数据是由各种上同调理论给出的，这是纯数学中许多领域的标准技术，包括拓扑学、复数和算术几何。这个问题导致了所谓的p-进变形理论；该项目解决了晶体上同调或与代数环相关的等价de Rham-Witt上同调类的问题，以及它们是如何p-adad变形的。一个特别令人感兴趣的相关问题是识别定义在这些线性数据内的(p进)整数上的积分结构，这导致几何对象的更精细的不变量，并编码更多的信息。这种方法在算术几何中非常常见。

We associate linear data to geometric objects like curves, surfaces or threefolds defined as zero-sets of algebraic equations over the integers or over a number domain that is annihilated by a power of a prime number. One can ask how much information about the original object can be captured in this way. The linear data are given by various cohomology theories which is a standard technique in many areas in Pure Mathematics including topology, complex and arithmetic geometry. This problem leads to what is called p-adic deformation theory; the project addresses the question in the case of crystalline cohomology or equivalently de Rham-Witt cohomology classes associated to algebraic cycles and how they deform p-adically. A related problem of particular interest is to identify an integral structure defined over the (p-adic) integers inside these linear data which leads to finer invariants of the geometric objects and encode more information. This method is very common in arithmetic geometry.