Hyperdiscriminants and Canonical Kahler metrics on algebraic manifolds

代数流形上的超判别式和规范卡勒度量

• 批准号: 1104448
• 负责人: Sean Paul
• 金额: $ 15.37万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104448&HistoricalAwards=false
• 关键词: Hyperdiscriminants; Canonical; Kahler; metrics; algebraic

One of the main problems in complex geometric analysis is to detect the existence of "canonical" Kahler metrics in a given class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. The presence of positive curvature makes this extremely difficult to answer and has led to a striking series of conjectures which relate the existence of these special metrics (solutions to the complex Monge-Ampere equation, a fully non-linear p.d.e ) to the algebraic geometry of the pluri-anticanonical images of the manifold. This geometry was suspected to be related to Mumford's deep "Geometric Invariant Theory". Recently this speculation has been completely justified by the PI (building upon work of Gang Tian) and it is the aim of this proposal to finish the proof of the "standard conjectures" in the Fano case and to develop and extend the entire Theory in the context of representations of algebraic groups. Here the PI hopes to make contact with Mikio Sato's beautiful theory of prehomogeneous vector spaces.Broadly speaking, there are two ways to mathematically approach, or model, a given problem: continuously, or discretely. These approaches are traditionally mutually exclusive. Analysis (differential equations in particular) is the time honored subject in the continuous domain, combinatorics and algebra (the study of enumerating a finite amount of data) is the hallmark of the discrete approach. In this proposal these two methods come together-the PI will explore the question of how the solution to an equation from analysis might be obtained by an infinite sequence of a purely finite (but large) collection of data. The equation arose in Einsteins' theory of Gravitation, whereas the finite set of data (hyperdiscriminants and resultants) arose in the work of Arthur Cayley a great Victorian era English mathematician.

复几何分析中的一个主要问题是在紧致复（Kahler）流形上检测给定类中“典范”Kahler度量的存在性。特别是一个寻求的必要和充分条件存在的卡勒爱因斯坦度量的法诺流形。正曲率的存在使得这个问题非常难以回答，并导致了一系列引人注目的成果，这些成果将这些特殊度量（复杂的蒙格-安培方程的解，一个完全非线性的p.d.e）的存在与流形的多反正则像的代数几何联系起来。这种几何被怀疑与芒福德的深刻的“几何不变理论”有关。最近这种猜测已完全合理的PI（建设工作的田刚），这是目的，这一建议，以完成证明的“标准结构”在法诺的情况下，并发展和扩大整个理论的背景下表示的代数群。在这里，PI希望与佐藤干雄（Mikio Sato）的漂亮的预齐次向量空间理论进行接触。广义地说，有两种方法可以在数学上接近或建模给定的问题：连续或离散。 这些方法传统上是相互排斥的。分析（特别是微分方程）是连续域中历史悠久的主题，组合学和代数（枚举有限数量数据的研究）是离散方法的标志。在这个提议中，这两种方法结合在一起-PI将探索如何通过纯粹有限（但大）数据集合的无限序列获得分析方程的解的问题。该方程出现在爱因斯坦的引力理论，而有限的数据集（超判别式和结果）出现在工作的亚瑟凯莱一个伟大的维多利亚时代的英国数学家。