Degenerations and Moduli Spaces of Kahler Manifolds

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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.