Invariant Metrics on Complex Manifolds

作为数学中最古老的分支之一，几何学关注的是与物体的大小、形状和距离有关的空间特性。现代数学利用微积分等强大的分析工具来研究几何，从而产生了微分几何。微分几何一直是数学中最活跃的领域之一，它与拓扑学和微分方程等其他数学领域以及物理学有着相互联系和应用——在爱因斯坦的相对论中，引力是度规的曲率。该项目研究了微分几何的复杂方面，特别是在复杂空间的重要度量称为不变度量，从复杂分析和微分方程的工具。主要目标是理解不变度量的形状或曲率，以及它们的底层数学结构。复杂几何的研究与数学的代数方面相互作用，包括代数几何、表示理论和数论，并应用于物理学，如弦理论和通过保角映射的科学计算。该项目包括通过参与研究对研究生进行培训。复流形上有四种经典不变度规：Bergman度规、Caratheodory-Reiffen度规、Kobayashi-Royden度规和负标量曲率的完整度规Kähler-Einstein。它们在生物全纯下是不变的，因此只依赖于复流形的底层复结构。不变度量的研究引起了微分几何、几个复杂变量、拓扑、非线性偏微分方程和代数几何之间的有趣联系。例如，不变度量与几个长期存在的猜想密切相关，例如单连通的负弯曲的完全Kähler流形必须是有界复欧几里得域的生物全纯。Kähler-Einstein度规和Kobayashi-Royden度规在理解规范束的正性方面起着关键作用。结合偏微分方程和几个复变量的方法，该项目旨在为一大类复流形，包括完全非紧Kähler流形、紧复流形和拟射影流形，提供对不变度量几何的更深层次的理解。这项研究也将导致在代数几何和数论的应用。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。

As one of the oldest branches in mathematics, geometry concerns the properties of spaces related to sizes, shapes, and distances of objects. Modern mathematics makes use of powerful analytic tools such as calculus to study geometry, which creates the field differential geometry. Differential geometry has been one of the most active fields in mathematics, with interconnections and applications to other mathematical fields like topology and differential equations, and also with physics - in Einstein's theory of relativity, gravity is the curvature of a metric. This project investigates the complex aspects of differential geometry, especially the important metrics on complex spaces called invariant metrics, with tools from complex analysis and differential equations. A main goal is to understand the shape or curvature of the invariant metrics, and their underlying mathematical structures. This study of complex geometry interplays with the algebraic sides of mathematics, including algebraic geometry, representation theory, and number theory, and has applications to physics such as string theory and to scientific computing via conformal maps. The project includes training through research involvement for graduate students.There are four classical invariant metrics on complex manifolds, the Bergman metric, the Caratheodory-Reiffen metric, the Kobayashi-Royden metric, and the complete Kähler-Einstein metric with negative scalar curvature. They are invariant under biholomorphisms, and hence depend only on the underlying complex structure of the complex manifold. The study of invariant metrics give rise to intriguing connections between differential geometry, several complex variables, topology, nonlinear partial differential equations, and algebraic geometry. For instance, the invariant metrics are closely related to several long-standing conjectures such as that a simply-connected, negatively curved, complete Kähler manifold must be biholomorphic to a bounded complex Euclidean domain. The Kähler-Einstein metric and the Kobayashi-Royden metric play key roles in understanding the positivity of canonical bundle. Combining methods from partial differential equations and several complex variables, the project aims to provide deeper understanding on the geometry of the invariant metrics, for a large class of complex manifolds, including complete noncompact Kähler manifolds, compact complex manifolds, and quasi-projective manifolds. The research will also lead to applications in algebraic geometry and number theory.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.