Degenerations and Moduli Spaces of Kahler Manifolds

卡勒流形的退化和模空间

• 批准号: 1916520
• 负责人: Song Sun
• 金额: $ 13.16万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-05-31 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1916520&HistoricalAwards=false
• 关键词: Degenerations; Moduli; Spaces; Kahler; Manifolds

Kahler manifolds are of fundamental importance in the study of modern geometry and physics. In particular, they provide framework to finding solutions of Einstein's equation and its generalizations. In this project the PI plans to investigate the deep connection between various points of view on degenerations and moduli spaces of Kahler manifolds satisfying suitable assumptions. The problems are of foundational nature, and answers to these will lead to new techniques and interactions among different fields of mathematics, and will have potential applications in string theory. Recent development in Kahler geometry, in particular, the proof of the Kahler-Einstein result on Fano manifolds, involve the understanding of limits of Kahler manifolds under natural curvature assumptions, from many different angles, including differential geometry, algebraic geometry, and several complex variables. This project aims to extend this further and to build more bridges among various subjects. The main focus will be on two topics. Firstly the PI would like to investigate the algebro-geometric meaning of Gromov-Hausdorff compactifications of Kahler manifolds with bounded Ricci curvature; Secondly, the PI will study a parabolic evolution equation - the Calabi flow, and understand the relationship with algebraic degenerations. The technical tools will be based on previous work of the PI and his collaborators, but essentially new ideas will have to be explored to tackle these questions.

在现代几何和物理学的研究中，卡勒歧管至关重要。特别是，它们提供了寻找爱因斯坦方程及其概括的解决方案的框架。在该项目中，PI计划研究对脱位的各种观点与卡勒流形的模量空间之间的深厚联系，以满足适当的假设。这些问题是基本的性质，对这些问题的答案将导致不同数学领域之间的新技术和相互作用，并将在字符串理论中具有潜在的应用。尤其是卡勒几何形状的最新发展，Kahler-Einstein的证据导致了Fano歧管，涉及对自然曲率假设下的Kahler歧管的限制的理解，包括从许多不同的角度，包括差分几何，代数几何形状，代数几何形状和几种复杂的变量。该项目旨在进一步扩展这一点，并在各种主题之间建造更多的桥梁。主要重点将是两个主题。首先，PI希望研究具有有限的RICCI曲率的Kahler歧管的Gromov-Hausdorff压缩的代数几何含义。其次，PI将研究抛物线进化方程 - 卡拉比流，并了解与代数退化的关系。技术工具将基于PI及其合作者的先前工作，但基本上必须探索新的想法来解决这些问题。

项目成果

Algebraic Tangent Cones of Reflexive Sheaves

自反滑轮的代数切锥

• DOI: 10.1093/imrn/rny276
• 发表时间: 2018
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Chen, Xuemiao;Sun, Song
• 通讯作者: Sun, Song