Holomorphic functions and some geometric problems on certain Kahler manifolds

全纯函数和某些卡勒流形上的一些几何问题

• 批准号: 1406593
• 负责人: Gang Liu
• 金额: $ 11.8万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2016-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406593&HistoricalAwards=false
• 关键词: Holomorphic; functions; some; geometric; problems

AbstractAward: DMS 1406593, Principal Investigator: Gang LiuKahler manifolds are a basic building block in the string theory model of the universe. This project studies the global structure of Kahler manifolds. In particular, conjectures on the uniformization of Kahler manifolds will be addressed. These conjectures generalize the classical uniformization theorem in one complex variable. In the PI's project, there are many interactions among several branches of mathematics, e.g. algebraic geometry, analysis, topology and differential geometry. Wider applications of the PI's field include the structure of molecules, the large scale structure of the universe and the liquid gas boundary. The PI proposes to work on three projects which involve function theory and geometry on Kahler manifolds. The first project is to study problems which are closely related to the uniformization conjecture of Yau. These problems include the finite generation of the ring of holomorphic functions of polynomial growth, sharp dimension estimates for holomorphic functions with polynomial growth on manifolds with nonnegative Ricci curvature, and a conjecture of Ni on the equivalence between average curvature decay, maximal volume growth and the existence of holomorphic functions of polynomial growth. In the second project, the PI will seek obstructions to Kahler metrics with nonnegative scalar curvature. The PI also plans to show that the Kodaira dimension of compact Kahler manifolds with nonpositive bisectional curvature is a homotopy invariant. The main tool is the PI's structure theorem for these manifolds.

摘要奖：DMS 1406593，首席研究员：刘刚卡勒流形是宇宙弦理论模型的基本组成部分。该项目研究卡勒流形的整体结构。 特别是，将讨论关于卡勒流形均匀化的猜想。这些猜想将经典均匀化定理推广到一个复变量中。 在PI的项目中，数学的几个分支之间有很多相互作用，例如数学。代数几何、分析、拓扑和微分几何。 PI领域更广泛的应用包括分子结构、宇宙大尺度结构和液气边界。 PI 提议开展三个项目，涉及卡勒流形上的函数论和几何。第一个项目是研究与丘均匀化猜想密切相关的问题。这些问题包括多项式增长的全纯函数环的有限生成、非负里奇曲率流形上多项式增长的全纯函数的尖锐维数估计，以及关于平均曲率衰减、最大体积增长和多项式增长的全纯函数的存在性。在第二个项目中，PI 将寻找具有非负标量曲率的卡勒度量的障碍。 PI 还计划证明具有非正二等分曲率的紧卡勒流形的 Kodaira 维数是同伦不变量。主要工具是这些流形的 PI 结构定理。