权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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，如多复变、偏微分方程、代数几何、复几何和微分几何。这个研究项目调查了一个特定的几何形状，它出现在几个复变量的研究中，并与数学物理学中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，以及应用，例如，系统工程与控制理论本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，本项目中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了有趣的研究课题。这个数学研究项目的目标是研究几何，分析和代数方面的一般真实的子流形在复杂的品种（更一般地说，CR结构）和他们的映射。这项工作将调查的几何后果的整体消失在一个紧凑的CR流形的高阶局部不变量，出现的障碍，以顺利扩展到边界的Cheng-Yau解决方案February man的复杂的蒙格-安培方程。在三维空间中，这个不变量与伯格曼核的渐近展开中对数项边界上的迹相吻合;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的一个问题的修正版本，该问题询问在复2-空间中有界严格伪凸域上是否总是存在这样的点;虽然答案通常是“否”，但如果域与球同构，这个问题仍然是开放的。PI还将考虑关于CR映射的存在性、唯一性和正则性的问题以及相关问题。他将考虑的背景下CR映射的列维非退化超曲面到一个非退化超二次的高维。本文的研究将加深我们对Levi非退化超曲面理论中构成平坦模型的超二次曲面的CR子流形结构的理解。这项工作也应该提供洞察力如何局部CR几何的超曲面影响各种属性，如概念的非退化和刚性，他们的CR地图。PI还将继续研究无限型的一般子流形之间的CR映射，通过研究定义CR映射的系统到喷丛上的奇异Pfiran系统的延伸。他将专注于理解最近发现的现象，无限型超曲面的双全纯，正式和光滑CR等价分类是不同的。这些调查预计将揭示新的光无限型流形之间的CR映射的性质，并导致更好地了解Pfidian系统在这种情况下产生的。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。