Invariants in Several Complex Variables and Complex Geometry

多个复变量和复几何中的不变量

基本信息

批准号：
2154368
负责人：
Peter Ebenfelt
金额：
$ 30.79万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154368&HistoricalAwards=false
关键词：
Invariants Several Complex Variables Geometry

项目摘要

The study of invariant geometries is of fundamental importance in mathematics. Such geometries arise naturally in areas of mathematics such as several complex variables, partial differential equations (PDE), and algebraic, complex, and differential geometry. This project investigates a particular geometry – CR geometry -- that arises in the study of several complex variables and complex geometry. It has deep connections with current topics in mathematical physics, including quantum field theory, general relativity, and string theory, as well as applications in, e.g., systems engineering and control theory. For example, the study of obstruction flatness has a direct link to the equations of motion in conformal gravity. The notion of Kähler-Einstein metric is highly relevant for these investigations; such metrics are measurements of distance and angle in higher-dimensional spaces (manifolds), which curve according to Einstein’s equations of relativity in vacuum. One way in which CR manifolds arise is as the boundaries of regions in high-dimensional complex spaces which curve in a well-behaved fashion in complex directions; such regions are known as pseudoconvex domains. Techniques to be used in this project come from a variety of mathematical areas: complex analysis/geometry, PDE, and differential geometry. At the same time, techniques and tools developed in this project will influence these areas as well. The project will also provide interesting research topics for graduate students and postdocs. The seminar activity resulting from the project will inspire and stimulate both students and other researchers.The goal of this project is to study geometric, analytic, and algebraic aspects of real submanifolds in complex varieties, together with their symmetries. Such aspects include, for instance, the geometric consequences of global vanishing on a compact CR manifold of a higher order local invariant that arises as an obstruction in the study of a complex Monge--Ampere equation. In three dimensions, this invariant coincides with the trace on the boundary of the log-term in an asymptotic expansion of the Bergman kernel. Solutions to this Monge-Ampere equation give rise to complete Kähler-Einstein metrics with negative curvature, and a major goal of the project is to characterize domains and complex manifolds in terms of properties of various metrics and kernels, and their relations. As an example, one aim is to understand what assumptions ensure that the Bergman metric on a domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to the ball. Another topic to be considered is the existence of CR umbilical points on compact CR 3-manifolds. Umbilical points on a manifold are locations where the manifold exhibits an unexpectedly high degree of symmetry and smoothness. It is not known whether such points always exist on bounded strictly pseudoconvex domains in complex 2-dimensional space; this question remains open if the domain is diffeomorphic to the ball. Finally, the project considers existence, uniqueness, and regularity questions for CR maps. These include the study of CR maps between generic submanifolds of infinite type and the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are distinct. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds, and to lead to a better understanding of the Pfaffian systems arising in this context. In conclusion, this project will significantly enhance the understanding of the role of invariant objects and symmetries in several complex variables and complex geometry, which in turn will increase the utility of the theory in physical models and applications.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.

不变几何的研究在数学中具有根本性的重要性。这种几何自然出现在数学领域，例如多个复变量、偏微分方程（PDE）以及代数、复数和微分几何。该项目研究了一种特殊的几何——CR。几何——产生于对多个复杂变量和复杂几何的研究，它与数学物理学的当前主题有着深刻的联系，包括量子场论、广义相对论和弦理论，以及在系统工程和数学领域的应用。例如，障碍平面度的研究与共形引力运动方程有直接联系。凯勒-爱因斯坦度量的概念与这些研究高度相关；此类度量是高维中的距离和角度的测量。空间（流形），根据真空中的爱因斯坦相对论方程弯曲，CR 流形出现的一种方式是作为高维复杂空间中的区域边界，其以良好的方式弯曲。该项目中使用的技术来自多个数学领域：复数分析/几何、偏微分方程和微分几何。该项目还将为研究生和博士后提供有趣的研究主题，该项目的研讨会活动将激励和激励学生和其他研究人员。该项目的目标是研究几何、分析和数学。的代数方面复簇中的实子流形及其对称性包括，例如，高阶局部不变量的紧致 CR 流形上的全局消失的几何后果，这是研究复数 Monge-Ampere 的障碍。在三个维度中，该不变量与 Bergman 核的渐近展开式中对数项边界上的轨迹一致，该 Monge-Ampere 方程的解得出完整的解。具有负曲率的卡勒-爱因斯坦度量，该项目的一个主要目标是根据各种度量和核的属性及其关系来表征域和复杂流形，例如，一个目标是了解哪些假设可以确保域上的伯格曼度量是凯勒-爱因斯坦当且仅当该域双全纯等价于球时另一个要考虑的主题是紧凑 CR 上 CR 脐点的存在。 3-流形上的脐点是流形表现出意想不到的高度对称性和平滑性的位置，如果这个问题仍然存在，则不知道这些点是否总是存在于复杂的二维空间中。最后，该项目考虑了 CR 映射的存在性、唯一性和规律性问题，其中包括无限类型的通用子流形之间的 CR 映射的研究。最近发现的现象是，对于无限型超曲面，双全纯、形式和平滑的 CR 等价分类是不同的。这些研究有望为无限型流形之间的 CR 映射的性质提供新的启示，并导致更好地理解。总之，普法夫系统将显着增强对不变物体和对称性在多个复杂变量和复杂几何中的作用的理解，这反过来又会增加该理论在物理模型和应用中的实用性。授予 NSF 的法定使命，并通过评估反映使用基金会的智力优点和更广泛的影响审查标准，被认为值得支持。