Arc Spaces, Singularities, and Motivic Integration

本计画的研究探讨代数簇在其奇异（非流形）点的局部几何。研究奇点的一个经典方法是限制在一个小邻域内并分析其边界。在一篇有影响力的论文写在六十年代，约翰纳什提出了一种替代方法，它依赖于分析空间的芽的曲线通过奇异轨迹。20年后，弗拉基米尔·贝尔科维奇发展了一个非阿基米德几何的一般理论，它提供了一个研究奇点的新方法。这三种截然不同的观点其实是密切相关的。本项目设定的目标解决了这些领域中的不同开放问题，寻求新的联系和应用。该项目还为研究生提供了研究培训活动。代数簇的弧空间提供了motivic集成的基础空间，并已被用于研究最小模型程序中的奇点不变量。项目的第一部分设定了两个关于弧空间的不同目标。第一个涉及一个定理的德林费尔德，格林伯格，和Kazhdan的正式附近的弧空间在非退化的弧和地址的问题是否规定的分解全球化沿着一个适当的分层的弧空间。第二个目标涉及纳什问题的家庭弧通过奇异的各种：纳什问题最近已解决的表面特征为零，有几个结果，在更高的维度，但鲜为人知的是，积极的特点，即使在二维，和建议解决这种情况。第三个目标是继续正在进行的工作，致力于发展理论的动机整合的Berkovich分析的代数品种，旨在了解其与其他现有理论的整合，如Kontsevich的动机整合，p-adic整合，和Hrushovski-Kazhdan的整合。第四个也是最后一个目标的动机是李普曼-扎茨基猜想，并旨在利用以前的工作的孤立奇点的联系和复杂的高原问题的想法，建立一个新的方法来实现猜想。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

The research in this project explores the local geometry of algebraic varieties at their singular (non-manifold) points. A classical way of studying singularities it to restrict to a small neighborhood and analyze its boundary. In an influential paper written in the sixties, John Nash proposed an alternative approach which relies on the analysis of the space of germs of curves through the singular locus. Twenty years later, Vladimir Berkovich developed a general theory of non-Archimedean geometry which provides, among other things, a yet new approach to study singularities. These three very different points of view are in fact closely related to each other. The goals set in this project address different open problems in each of these areas, seeking new connections and applications. The project also provides research training activities for graduate students.The space of arcs of an algebraic variety provides the underlying space in motivic integration and has been used to study invariants of singularities in the minimal model program. The first part of the project sets two distinct objectives regarding arc spaces. The first pertains a theorem of Drinfeld, Grinberg, and Kazhdan of the formal neighborhood of the arc space at non-degenerate arcs and addresses the question whether the stated decomposition globalizes along a suitable stratification of the arc space. The second objective concerns the Nash problem on families of arcs through the singularities of a variety: the Nash problem has recently been settled for surfaces in characteristic zero and there are several results in higher dimensions, but little is known in positive characteristic, even in dimension two, and the proposal addresses this case. A third objective sees the continuation of ongoing work devoted to the development of a theory of motivic integration on the Berkovich analytification of an algebraic variety and aims to understand its connections with other existing theories of integration such as Kontsevich's motivic integration, p-adic integration, and Hrushovski-Kazhdan’s integration. The fourth and last objective is motivated by the Lipman–Zariski conjecture, and aims to use ideas from prior works on links of isolated singularities and the complex Plateau problem to set up a new approach toward the conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.