Arc Spaces, Singularities, and Motivic Integration

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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.