Invariant Metrics on Complex Manifolds

作为数学中最古老的分支之一，几何学关注与物体的大小，形状和距离有关的空间属性。现代数学利用微积分等强有力的分析工具来研究几何，创造了微分几何这一领域。微分几何一直是数学中最活跃的领域之一，与其他数学领域如拓扑学和微分方程以及物理学有着相互联系和应用-在爱因斯坦的相对论中，引力是度量的曲率。该项目研究微分几何的复杂方面，特别是称为不变度量的复杂空间上的重要度量，使用复分析和微分方程的工具。一个主要的目标是了解不变度量的形状或曲率，以及它们的基本数学结构。这种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，表示论和数论，并应用于物理学，如弦理论和科学计算通过共形映射。复流形上有四个经典的不变度量：Bergman度量、Caratheodory-Reiffen度量、Kobayashi-Royden度量和具有负标量曲率的完备Kähler-Einstein度量。它们在双全纯下是不变的，因此只依赖于复流形的基本复结构。对不变度量的研究在微分几何、多复变、拓扑学、非线性偏微分方程和代数几何之间产生了有趣的联系。例如，不变度量与几个长期存在的命题密切相关，例如一个单连通的、负弯曲的、完备的Kähler流形必须双全纯于一个有界复欧氏域。Kähler-Einstein度规和Kobayashi-Royden度规在理解正则丛的正性中起着关键作用。结合偏微分方程和多个复变量的方法，该项目旨在为一大类复流形提供对不变度量几何的更深入理解，包括完整的非紧Kähler流形，紧致复流形和拟投射流形。该研究还将导致代数几何和数论的应用。该奖项反映了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。

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.