Holomorphic functions and some geometric problems on certain Kahler manifolds

获奖：DMS 1406593，首席研究员：Gang LiuKahler流形是宇宙弦理论模型的基本组成部分。本课题研究Kahler流形的全局结构。特别地，关于Kahler流形的均匀化的猜想将被处理。这些猜想推广了单复变量的经典均匀化定理。在PI的项目中，有许多数学分支之间的相互作用，例如代数几何，分析，拓扑和微分几何。PI领域的更广泛应用包括分子结构、宇宙大尺度结构和液气边界。PI计划开展三个项目，涉及卡勒流形的函数理论和几何。第一个课题是研究与丘氏均匀化猜想密切相关的问题。这些问题包括多项式生长的全纯函数环的有限生成，非负Ricci曲率流形上多项式生长的全纯函数的尖锐维数估计，以及Ni关于平均曲率衰减、最大体积增长和多项式生长的全纯函数存在性之间的等价性的猜想。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物。PI还计划证明具有非正对分曲率的紧凑Kahler流形的Kodaira维是一个同伦不变量。主要的工具是这些流形的PI结构定理。

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.