Einstein Metrics, Sasakian Geometry and Kahler Orbifolds

项目编号：dms -0504367首席研究员：Charles P. Boyer和Krzysztof Galicki Boyer和Galicki教授提议研究几个几何和拓扑方面的项目。所有项目的目标是研究黎曼几何中的基本问题，主要有两个焦点：Calabi-Yau和Fano变体上的轨道束的接触几何以及这些空间上某些特殊度量（即爱因斯坦，正Ricci曲率，横向Calabi-Yau）的存在性。这里提出的问题和问题深深植根于主要研究者早期的工作，该工作利用了sasaki - einstein空间的接触几何与两种Kaehler几何之间的基本关系，即具有正标量曲率的Kaehler- einstein轨道度量的q ! Fano变体和具有Kaehler- Ricci-flatmetrics的Calabi-Yau流形。最近，主要研究人员和J. kollara解决了黎曼几何中的一个开放问题。我们已经在一篇即将发表在《数学年鉴》上的论文中证明了奇异球体上爱因斯坦度规的存在。此外，我们还证明了有界可并行流形的奇维同伦球允许存在大量的爱因斯坦度规。事实上，变形类的数量以及sasaki - einstein度量的模的数量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理论和交点理论的代数几何，Calabi猜想的分析，最后是孤立超曲面奇点链路的经典微分拓扑。这些方法可以进一步向各个方向推广。更一般地说，主要研究者想要解决几个关于5维和7维的紧凑sasaki - einstein流形的分类问题。这两个维度之所以重要，有两个不同的原因。鉴于早期的工作，高维的例子可以使用连接构造来构造。同时，这两个奇维似乎在超弦理论中起着特殊的作用。在弦理论和m理论最新发展的背景下，主要研究人员还提出了有关4维自对偶爱因斯坦度量的一些相关问题。数学是现代技术的基础，对数学的理解和发展必须先于技术进步。然而，我们对特定几何类型的研究与现代理论物理中的一些重要问题密切相关，应该为他们的理解提供重要的数学基础。例如，我们正在研究的数学模型目前被用于超对称弦理论，这是引力与其他自然基本力统一的模型。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.