Canonical metrics on four dimensional manifolds, and orbifold structures

摘要奖：DMS 1309029，首席研究员：Ioana Suvaina拟议的研究的主要目标是研究四维Orb olold空间的黎曼几何，以及它们如何表现为光滑流形上的正则度量空间的边界点。这位首席研究员建议分析一系列问题，以解决常数量曲率Kaehler曲面空间的研究。所提出的方案既关注奇点的局部发展问题，又关注对紧光滑空间的理解。主要研究人员的目标是在问题的代数几何方面与Kaehler几何中的对应方面建立联系。这些项目的解决将提供常数量曲率Kaehler曲面或常数量曲率几乎Kaehler 4-流形的新例子的构造。在另一个方向上，主要研究人员提出了一系列关于曲率分量的估计、四环形的拓扑和微分不变量以及爱因斯坦度规的存在或不存在之间的关系的项目。主要的研究方法是通过Seiberg-Witten理论。在时空模型的推动下，对四维空间的研究在数学和物理上都有很长的历史。在数学上，有强有力的证据表明，四维空间是非常特殊的，因为它们允许无限多的光滑结构。这项研究计划使用数学中不同领域的技术：微分拓扑学、黎曼和凯勒几何，以及代数几何来研究四维空间。特别是，首席研究员的目标是在这些不同领域之间建立新的联系。

AbstractAward: DMS 1309029, Principal Investigator: Ioana Suvaina The main goal of the proposed research is the study of Riemannian geometry of orbifold spaces in dimension four, and how they appear as boundary points of the space of canonical metrics on smooth manifolds. The principal investigator proposes to analyze a family of problems addressing the study of the space of constant scalar curvature Kaehler surfaces. The projects proposed concern both problems on the local development of singularities, but also the understanding of compact smooth spaces. The principal investigator aims to establish connections between the algebraic geometry aspects of the problems and their counterparts in Kaehler geometry. The resolution of these projects will provide constructions of new examples of constant scalar curvature Kaehler surfaces, or constant scalar curvature almost Kaehler 4-manifolds. In another direction, the principal investigator proposes a series of projects on the relations between estimates of the curvature components, the topological and differential invariants of a 4-orbifold, and existence or non-existence of Einstein metrics. The main approach is via the Seiberg-Witten theory.The study of the four dimensional spaces has a long history both in mathematics and physics being motivated by the study of the space-time model. In mathematics, there is strong evidence that the four dimensional spaces are very special as they admit infinitely many smooth structures. This research program plans to use techniques from different fields in mathematics: differential topology, Riemannian and Kaehler geometry, and algebraic geometry to study four dimensional spaces. In particular, the principal investigator aims to establish new connections among these various fields.