Geometry of Invariants and Mappings in Several Complex Variables

几个复杂变量中不变量的几何和映射

基本信息

批准号：
1900955
负责人：
Peter Ebenfelt
金额：
$ 27万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900955&HistoricalAwards=false
关键词：
Geometry Invariants Mappings Several Complex

项目摘要

The study of invariant geometries is fundamental in mathematics. Invariant geometries arise in many different areas, such as several complex variables, partial differential equations, and algebraic, complex, and differential geometry. This research project investigates a particular geometry that arises in the study of several complex variables and 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. The ideas and techniques needed for this study are drawn from a variety of mathematical areas, including complex analysis and geometry, partial differential equations, and differential geometry; at the same time, the tools developed in this project will inform these areas as well. The project also provides interesting research topics for graduate students and postdocs. The goal of this mathematics research project is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of CR structures) and their mappings. The work will investigate the geometric consequences of global vanishing on a compact CR manifold of a higher order local invariant that arises as the obstruction to smooth extension to the boundary of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation. In three dimensions, this invariant coincides with the trace on the boundary of the log-term in the asymptotic expansion of the Bergman kernel; hence this problem is also connected with a strong form of the Ramadanov Conjecture. The PI will study the existence of CR umbilical points on compact CR 3-manifolds, especially a revised version of a question by Chern-Moser, which asks if such points always exist on bounded strictly pseudoconvex domains in complex 2-space; although the answer is 'no' in general, the question is still open if the domain is diffeomorphic to the ball. The PI will also consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions. He will consider the context of CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. The PI will also continue study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds and lead to a better understanding of the Pfaffian systems arising in this context.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.

不变几何的研究是数学的基础。不变几何出现在许多不同的领域，例如多个复变量、偏微分方程以及代数、复数和微分几何。该研究项目研究了在研究多个复杂变量时出现的特定几何形状，并与当前数学物理主题（包括量子场论、广义相对论和弦理论）以及系统工程和数学领域的应用有着深刻的联系。控制理论。这项研究所需的思想和技术来自各种数学领域，包括复分析和几何、偏微分方程和微分几何；同时，该项目开发的工具也将为这些领域提供信息。该项目还为研究生和博士后提供有趣的研究课题。该数学研究项目的目标是研究复杂变体（更一般地说，CR 结构）中的通用实子流形及其映射的几何、解析和代数方面。这项工作将研究全局消失在高阶局部不变量的紧致 CR 流形上的几何后果，该流形是由于阻碍 Fefferman 复数 Monge-Ampere 方程的 Cheng-Yau 解的边界平滑扩展而出现的。在三维空间中，这个不变量与伯格曼核的渐进展开中对数项边界上的迹重合；因此这个问题也与拉马达诺夫猜想的强形式有关。 PI 将研究紧致 CR 3 流形上 CR 脐点的存在性，特别是 Chern-Moser 问题的修订版本，该问题询问这些点是否始终存在于复 2 空间中的有界严格伪凸域上；尽管答案通常是“否”，但如果域与球微分同胚，问题仍然存在。 PI 还将考虑有关 CR 地图的存在性、唯一性和规律性的问题，以及相关问题。他将考虑 Levi 非简并超曲面到更高维度的非简并超二次曲面的 CR 映射的背景。这项研究将增强我们对超二次曲面 CR 子流形结构的理解，超二次曲面构成了 Levi 非简并超曲面理论中的平面模型。这项工作还应该深入了解此类超曲面的局部 CR 几何形状如何影响其 CR 贴图的各种属性，例如非简并性和刚性的概念。 PI 还将继续研究无限类型通用子流形之间的 CR 映射，方法是研究定义 CR 映射的系统到射流束上的奇异普法夫系统的延伸。他将重点了解最近发现的现象，即对于无限型超曲面，双全纯、形式和光滑 CR 等价分类是不同的。这些研究预计将为无限类型流形之间的 CR 映射的性质提供新的线索，并有助于更好地理解在此背景下出现的普法夫系统。该奖项反映了 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。