Degenerations and Moduli Spaces of Kahler Manifolds

卡勒流形在现代几何和物理研究中具有重要的基础意义。特别是，它们为寻找爱因斯坦方程及其推广的解提供了框架。在这个项目中，PI计划研究各种关于退化的观点和满足适当假设的Kahler流形的模空间之间的深层联系。这些问题是基础性的，对这些问题的回答将导致新的技术和不同数学领域之间的相互作用，并将在弦理论中有潜在的应用。Kahler几何的最新发展，特别是关于法诺流形的Kahler- einstein结果的证明，涉及从许多不同的角度，包括微分几何、代数几何和几个复杂变量，在自然曲率假设下对Kahler流形极限的理解。这个项目的目的是进一步扩展这一点，并在各个学科之间建立更多的桥梁。主要的焦点将集中在两个主题上。首先，PI想要研究具有有界Ricci曲率的Kahler流形的Gromov-Hausdorff紧化的代数几何意义；其次，PI将研究抛物演化方程- Calabi流，并了解其与代数退化的关系。技术工具将基于PI和他的合作者以前的工作，但本质上必须探索新的想法来解决这些问题。

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.