Degenerations and Moduli Spaces of Kahler Manifolds

Kahler流形在现代几何和物理学的研究中具有重要意义。特别是，他们提供了框架，以寻找爱因斯坦方程的解决方案及其推广。在这个项目中，PI计划研究满足适当假设的Kahler流形的退化和模空间的各种观点之间的深层联系。这些问题是基础性的，这些问题的答案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。卡勒几何的最新发展，特别是卡勒-爱因斯坦关于法诺流形的结果的证明，涉及到从许多不同的角度，包括微分几何，代数几何和几个复变函数，理解卡勒流形在自然曲率假设下的极限。该项目旨在进一步扩展这一点，并在各学科之间建立更多的桥梁。主要重点将放在两个专题上。首先，PI将研究具有有界Ricci曲率的Kahler流形的Gromov-Hausdorff紧化的代数几何意义;其次，PI将研究抛物发展方程-Calabi流，并了解与代数退化的关系。技术工具将基于PI及其合作者以前的工作，但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.