Algebraic Varieties and Valuation Theory

这个代数几何领域的项目涉及估值理论的各个方面。估值在数学的许多分支中自然出现。作为一个整体，赋值空间捕获了重要的信息，例如与奇点求解问题有关的信息。有许多自然的方法来观察估值并将它们打包到空间中，每种方法都为研究代数变量提供了有用的工具。本文研究的重点是两个评价空间：圆弧空间和代数变量的Berkovich分析。这些空间配备了有趣的结构，可以用来回答重要的数学问题。纳什研究了与奇点有关的弧空间；该空间作为动机积分的基础空间，已被应用于极小模型规划中奇异不变量的研究。Berkovich的非阿基米德几何已被广泛应用于各种领域，从p进几何到动力学、几何群论、镜像对称、热带几何，以及最近的双相几何。关于这些空间结构的许多有趣的问题仍然是开放的，更好地理解它们的几何形状将导致新的应用。本课题的一个具体目标是研究各种弧空间上的局部环，这个问题是由Shokurov关于最小对数差的半连续性猜想引起的，因此，间接地，由关于翻转终止的猜想引起的，这是最小模型规划中缺失的步骤之一。另一个目标提出了一种动机整合的新观点，即用Berkovich空间代替弧空间。该项目还解决了一些关于孤立奇点的链接、它们的接触结构和它们的CR结构的问题。虽然这些显然是不相关的问题，但链接的接触结构与估值理论之间存在潜在的联系。

This project in the field of algebraic geometry concerns aspects of valuation theory. Valuations appear naturally in many branches of mathematics. Spaces of valuations, as a whole, capture important information, for example in connection with the problem of resolution of singularities. There are many natural ways of looking at valuations and packaging them into spaces, and each provides useful tools for studying algebraic varieties. The focus of the proposed research is on two valuation spaces: the space of arcs and the Berkovich analytification of an algebraic variety. These spaces come equipped with interesting structure and can be used to answer important mathematical questions.Nash investigated the space of arcs in connection to singularities; the same space serves as the underlying space in motivic integration, and has been applied to study invariants of singularities in the minimal model program. Berkovich's non-Archimedean geometry has been used in a variety of contexts, from p-adic geometry to dynamics, geometric group theory, mirror symmetry, and tropical geometry, and most recently in birational geometry. Many interesting questions about the structure of these spaces remain open, and a better understanding of their geometry will lead to new applications. One specific goal of this project is to study the local rings on the arc space of a variety, a problem motivated by Shokurov's semicontinuity conjecture on minimal log discrepancies and therefore, indirectly, by the conjecture on termination of flips, one of the missing steps in the minimal model program. Another objective proposes a new point of view on motivic integration where Berkovich spaces are used in place of arc spaces. The project also addresses some questions about links of isolated singularities, their contact structure, and their CR structure. While these are apparently unrelated questions, there is an underlying connection between the contact structure of a link and valuation theory.