Invariants in Several Complex Variables and Complex Geometry

不变几何的研究在数学中具有基本的重要性。这样的几何自然地出现在数学领域，如多个复变量、偏微分方程(PDE)以及代数、复几何和微分几何。这个项目研究的是一种特殊的几何--CR几何--它是在研究几个复变量和复杂几何时出现的。它与当前数学物理中的主题有着深刻的联系，包括量子场论、广义相对论和弦理论，以及在系统工程和控制理论中的应用。例如，对障碍物平坦度的研究与共形重力中的运动方程有直接的联系。卡勒-爱因斯坦度规的概念与这些研究高度相关；这种度规是对高维空间(流形)中的距离和角度的测量，这些空间(流形)根据爱因斯坦的相对论在真空中弯曲。CR流形的一种产生方式是作为高维复空间中沿复数方向以良好方式弯曲的区域的边界；这样的区域被称为伪凸域。本项目中使用的技术来自各种数学领域：复杂分析/几何、偏微分方程组和微分几何。同时，本项目开发的技术和工具也将对这些领域产生影响。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。本项目的目标是研究复变型实子流形的几何、解析和代数方面，以及它们的对称性。这些方面包括，例如，在具有高阶局部不变量的紧致CR流形上全局消失的几何后果，这是作为研究复杂的Monge-Ampere方程的障碍而出现的。在三维空间中，这个不变量与Bergman核的渐近展开式中对数项边界上的迹重合。这个Monge-Ampere方程的解产生了具有负曲率的完备的Kähler-Einstein度量，该项目的一个主要目标是根据各种度量和核的性质以及它们之间的关系来刻画整环和复流形。作为一个例子，一个目的是理解什么假设确保域上的Bergman度量是Kähler-Einstein当且仅当域是双全纯等价于球。另一个要考虑的问题是紧致CR3-流形上的CR脐点的存在性。歧管上的脐带点是歧管表现出出人意料的高度对称性和光滑度的位置。这类点是否总是存在于复2维空间中有界的严格伪凸区域上尚不清楚；如果区域与球微分同胚，则这个问题仍然悬而未决。最后，该项目考虑了CR映射的存在唯一性和正则性问题。其中包括研究无限型子流形之间的CR映射，以及最近发现的现象，即对于无限型超曲面，双全纯、形式和光滑的CR等价分类是不同的。这些研究有望揭示无穷型流形之间的CR映射的性质，并有助于更好地理解在此背景下产生的Pfaffian系统。总而言之，这个项目将显著提高对几个复变量和复杂几何中不变物体和对称性作用的理解，这反过来将增加该理论在物理模型和应用中的实用性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.