Canonical metrics on four dimensional varieties

本文提出的两个主要研究主题是爱因斯坦度量的模空间和常数量曲率Kahler度量在分类空间边界上的模空间。该提案的第一部分是PI以前工作的自然发展，并计划推广Seiberg-Witten理论的一些方面，关于黎曼曲率的估计，以及4-orbilold上爱因斯坦度量存在的障碍。光滑流形上的障碍物对爱因斯坦度规存在的进一步影响是可以预料的。提案的另一部分涉及对卡勒结构的研究。关于Ricci平坦的渐近局部欧几里德Kahler流形(非超Kahler)的分类方案为研究常数量曲率Kahler流形的模空间的紧化奠定了基础。所使用的具体工具来自扭曲空间、几何分析、代数几何稳定性和Seiberg-Witten理论。尽管在过去的二十年中取得了巨大的发展，但关于低维流形的几何仍然有许多我们不了解的地方。这项研究旨在发展四维空间上不同几何结构之间的相互作用，如黎曼几何、微分拓扑和Kahler几何。PI计划探讨特殊几何(Ale Ricci Flat Kahler)的分类问题，并了解4-流形或具有奇点的4-流形上爱因斯坦或常标量曲率Kahler几何结构的存在和不存在。这些主题不仅在微分和代数几何中非常重要，而且在数学物理(弦理论)中也是如此。

The two main themes of the research proposed focus on the study of the moduli spaces of Einstein metrics and constant scalar curvature Kahler metrics at a point on the boundary of the classifying space. The first part of the proposal is a natural development of the PI's previous work, and plans to generalize aspects of Seiberg-Witten Theory, estimates on Riemannian curvatures, and obstructions to the existence of Einstein metrics on 4-orbifolds. Further implications of obstructions to the existence of Einstein metrics on smooth manifolds are expected. The other part the proposal addresses the study of Kahler structures. The project proposed on the classification of Ricci flat asymptotically locally euclidean Kahler manifolds (non-hyperkahler) establishes a starting point in the study of the compactification of the moduli space of constant scalar curvature Kahler manifolds. The specific tools used come from twistor spaces, geometric analysis, algebraic geometric stability, and Seiberg-Witten theory.In spite of the tremendous developments over the last twenty years, there is still a lot that eludes us about the geometry of manifolds in low dimensions. The proposed research aims to develop interactions between different geometric structures such as Riemannian geometry, differential topology and Kahler geometry on four dimensional spaces. The PI plans to approach classification problems of special geometries (ALE Ricci flat Kahler), and also to understand existence and non-existence of Einstein or constant scalar curvature Kahler geometric structures on 4-manifolds, or 4-manifolds with singularities. These topics are of great importance not only in differential and algebraic geometry, but also in mathematical physics (string theory).