Invariants in Several Complex Variables and Complex Geometry

不变几何的研究在数学中具有重要的基础意义。这些几何图形自然地出现在数学领域中，例如几个复杂变量、偏微分方程（PDE）以及代数、复杂和微分几何。该项目研究了一种特殊的几何- CR几何-在研究几个复杂变量和复杂几何时出现的。它与数学物理的当前主题有很深的联系，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等方面的应用。例如，障碍物平坦度的研究与共形重力的运动方程有直接的联系。Kähler-Einstein度量的概念与这些调查高度相关；这些度量是在高维空间（流形）中对距离和角度的测量，它们在真空中根据爱因斯坦的相对论方程弯曲。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，在复杂方向上以良好的方式弯曲；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、PDE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。该项目也将为研究生和博士后提供有趣的研究课题。由该项目产生的研讨会活动将启发和激励学生和其他研究人员。这个项目的目标是研究复变实子流形的几何、解析和代数方面，以及它们的对称性。这些方面包括，例如，高阶局部不变量的紧态CR流形上的全局消失的几何后果，它在复杂蒙日-安培方程的研究中成为一个障碍。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相吻合。这个蒙日-安培方程的解产生负曲率的完整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及其关系来表征域和复杂流形。作为一个例子，一个目标是理解什么假设确保一个领域上的伯格曼度规是Kähler-Einstein当且仅当该领域是生物全纯等价于球。另一个要考虑的问题是紧致CR 3-流形上CR脐点的存在性。歧管上的脐带点是歧管显示出意想不到的高度对称性和平滑度的位置。在复二维空间中，这些点是否总是存在于有界严格伪凸域上是未知的；如果定义域与球是微分同构的，这个问题仍然没有解决。最后，研究了CR映射的存在性、唯一性和正则性问题。其中包括无限型一般子流形之间的CR映射的研究，以及最近发现的无限型超曲面的生物全纯、形式和光滑CR等价分类是不同的现象。这些研究有望揭示无限型流形之间CR映射的性质，并有助于更好地理解在这种情况下产生的Pfaffian系统。总之，该项目将显著提高对不变物体和对称在几个复杂变量和复杂几何中的作用的理解，这反过来将增加理论在物理模型和应用中的效用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.