Arc Spaces, Singularities, and Motivic Integration

这个项目的研究探索了代数簇在奇异点(非流形)的局部几何。研究奇点的一种经典方法是将其限制在一个较小的邻域内并分析其边界。在60年代写的一篇有影响力的论文中，约翰·纳什提出了一种替代方法，该方法依赖于通过奇异轨迹分析曲线芽空间。二十年后，弗拉基米尔·伯克维奇发展了非阿基米德几何的一般理论，其中提供了一种研究奇点的新方法。这三种截然不同的观点实际上是密切相关的。这个项目设定的目标是解决这些领域中不同的开放问题，寻求新的联系和应用。该项目还为研究生提供了研究性培训活动。代数族的弧空间为动机整合提供了潜在的空间，并已用于研究最小模型程序中的奇点不变量。该项目的第一部分设定了关于弧形空间的两个不同目标。第一个定理涉及Drinfeld，Grinberg和Kazhdan关于非退化弧线上的弧空间的形式邻域的一个定理，并解决了所述分解是否沿着弧空间的适当分层而球化的问题。第二个目标涉及通过各种奇点的弧族上的Nash问题：Nash问题最近已经解决了特征为零的曲面的Nash问题，在更高的维度上也有一些结果，但关于正特征的Nash问题很少，甚至在维度2中也是如此，该建议解决了这个问题。第三个目标是继续致力于发展关于代数簇的Berkovich分析的动机整合理论，并旨在了解其与其他现有整合理论的联系，如康采维奇的动机整合、p-进整合和Hrushovski-Kazhdan的整合。第四个也是最后一个目标是由利普曼-扎里斯基猜想推动的，旨在利用之前关于孤立奇点联系和复杂高原问题的工作中的想法来建立一种新的方法来处理这个猜想。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.