Einstein Metrics, Sasakian Geometry and Kahler Orbifolds

摘要奖：DMS-0504367首席研究员：Charles P.Boyer和Krzysztof Galicki教授Boyer和Galicki提议研究几何和拓扑学中的几个项目。所有项目的目的都是研究黎曼几何中的基本问题，主要集中在两个方面：Calabi-Yau和Fano簇上的ORBORBLOD丛的接触几何以及此类空间上某些特殊的(即，爱因斯坦度量、正Ricci曲率度量、横切Calabi-Yau度量)的存在性。这里提出的问题深深植根于主要研究者的早期工作，他们利用了Sasakian-Einstein空间的接触几何和两种Kaehler几何之间的基本关系，即具有正标量曲率的Kaehler-Einstein或双曲度量的q-阶乘Fano簇，以及具有Kaehler-Ricci-平坦度量的Calabi-Yau流形。最近，主要研究人员和J.科勒解决了黎曼几何中的一个公开问题。在即将发表在《数学年鉴》上的一篇论文中，我们证明了奇异球面上爱因斯坦度规的存在。此外，我们还证明了有界可平行流形的奇维同伦球面允许大量的爱因斯坦度量。事实上，Sasakian-Einstein度规的形变类数和模数随维度呈双指数增长。主要研究人员使用的技术借鉴了几个不同的领域：Mori理论和交集理论的代数几何，Calabi猜想的分析，最后是孤立超曲面奇异链环的经典微分拓扑。这些方法可以扩展得更远，方向是不变的。更广泛地说，主要研究人员想要解决几个关于5维和7维紧致Sasakian-Einstein流形的分类问题。这两个维度是重要的，有两个不同的原因。在前人的工作中，高维的例子可以使用连接结构来构造。同时，这两个奇数维在超弦理论中似乎扮演着特殊的角色。在弦和M理论的最新发展的背景下，首席研究人员还建议研究与4维自对偶爱因斯坦度规有关的一些相关问题。数学是我们现代技术的基础，它的许多理解和发展必须先于技术进步。然而，我们对特殊类型几何的研究与现代理论物理中的一些重要问题密切相关，应该为理解它们提供重要的数学基础。例如，我们正在研究的数学模型是目前在超对称弦理论中经常使用的，它是引力与自然其他基本力统一的模型。这也适用于黑洞物理学。

AbstractAward: DMS-0504367Principal Investigator: Charles P. Boyer and Krzysztof GalickiProfessors Boyer and Galicki propose to investigate severalprojects in geometry and topology. The objective of all theprojects is to study fundamental questions in Riemannian Geometrywith two main focal points: Contact Geometry of orbifold bundlesover Calabi-Yau and Fano varieties and the existence of somespecial (i.e., Einstein, positive Ricci curvature, transverselyCalabi-Yau) metrics on such spaces. The questions and problemsproposed here are deeply rooted in the principal investigators'earlier work which exploited a fundamental relationship betweencontact geometry of Sasakian-Einstein spaces and two kinds ofKaehler geometry, namely Q-factorial Fano varieties withKaehler-Einstein orbifold metrics with positive scalar curvature,and Calabi-Yau manifolds with their Kaehler Ricci-flatmetrics. Most recently the principal investigators and J. Kollarhave solved an open problem in Riemannian geometry. We haveproved the existence of Einstein metrics on exotic spheres in apaper to appear in the Annals of Mathematics. Furthermore, wehave shown that odd dimensional homotopy spheres that boundparallelizable manifolds admit an enormous number of Einsteinmetrics. In fact, the number of deformation classes as well asthe number of moduli of Sasakian-Einstein metrics grow doubleexponentially with dimension. The techniques used by theprincipal investigators borrow from several different fields; thealgebraic geometry of Mori theory and intersection theory, theanalysis of the Calabi Conjecture, and finally the classicaldifferential topology of links of isolated hypersurfacesingularities. These methods can be extended much further and invarious directions. More generally the principal investigatorswant to address several classification problems concerningcompact Sasakian-Einstein manifolds in dimensions 5 and 7. Thesetwo dimensions are important for two separate reasons. In viewof earlier work higher dimensional examples can be constructedusing the join construction. At the same time these two odddimensions appear to play special role in Superstring Theory. Inthe context of recent developements in String and M-Theory theprincipal investigators also propose to investigate some relatedproblems concerning self-dual Einstein metrics in dimension 4.Mathematics is the foundation upon which our modern technology isbuilt, and much of its understanding and development must preceedtechnological progress. Nevertheless, our research into aparticular type of geometry is closely linked to some importantproblems in modern Theoretical Physics and should provide animportant mathematical basis for their understanding. Forexample, the mathematical models that we are studying arecurrently being used in supersymmetric string theory which is amodel for the unification of gravity with the other fundamentalforces of nature. This also has applications to the Physics ofblack holes.