Stability of Algebraic Manifold

摘要奖：DMS-9971387主要研究人员：David A.Gieseker和罗华章在几何不变理论中，为了研究模问题，引入了任何极化射影的稳定性的概念。然而，稳定性的校核通常是一个难题。Mumford对于亏格大于1的曲线，Gieseker对于一般类型的曲面，Viehweg对于具有半扩幅正则线丛的高维变种已经做了这一点。我们将从微分几何的观点研究Gieseker和Mumford所使用的非极化光滑射影簇的稳定性。更具体地说，我们试图将Gieseker-Mumford稳定性与簇上存在一个特殊的Kahler度量联系起来，使得它的标量曲率是常数的，并且它的Ricci曲率满足一些非退化约束。这个度量不是卡勒-爱因斯坦度量，而是与卡勒-爱因斯坦度量密切相关。这个项目的一个动机来自于Donaldson，Uhlenbeck和Yau在向量丛的稳定性和Yang-Mills联络的存在之间的关系所做的先前的工作。我们的项目是将数学中两个重要但不同的领域联系在一起。一个是特殊度量的研究，它涉及流形上的微分方程式的研究，通常与物理有关。另一个是模空间理论中稳定性的研究，主要涉及到代数。我们的方法是在这两个域之间建立对应关系，特别是用特殊度量的存在来描述稳定性的含义。这样做的好处是，它不仅为我们提供了关于那些特殊度量的新视角，也为我们提供了模空间理论的新技术。例如，这种对应将使我们能够解决一些困难的数学问题，如检验某些流形的Gieseker-Mumford稳定性。这个问题是模空间理论中的基本问题之一，但一直没有得到普遍的解决。

AbstractAward: DMS-9971387Principal Investigator: David A. Gieseker and Huazhang LuoIn Geometric Invariant Theory, in order to study the moduliproblems, the notion of stability for any polarized projectivevariety is introduced. However to check the stability is usuallya difficult problem. It has been done by Mumford for curves withgenus bigger than 1, by Gieseker for surfaces of general type,and by Viehweg for high dimensional varieties with semi-amplecanonical line bundles. We will study the stability of apolarized smooth projective variety as used by Gieseker andMumford from a differential geometric viewpoint. Morespecifically, we try to relate the Gieseker-Mumford stability tothe existence of a special Kahler metric on the variety such thatits scalar curvature is constant and its Ricci curvaturesatisfies some non-degenerate constraints. This metric is notKahler-Einstein, but is closely related to Kahler-Einsteinmetric. One motivation for this project comes from the previouswork done by Donaldson, Uhlenbeck and Yau on the relation betweenstability of vector bundle and the existence of Yang-Millsconnection. Our project is to relate two important but different fields inmathematics together. One is the study of special metrics, whichinvolves the study of differential equations on manifold andusually has connection to physics. Another is the study ofstability in moduli space theory which involves mainlyalgebra. Our approach is to establish a correspondence betweenthese two fields, in particular to describe the meaning ofstability in terms of the existence of special metrics. Theadvantage for doing is it will not only provide us with newinsights about those special metrics, but also provide us withnew technique to moduli space theory. For example, this kind ofcorrespondence will enable us to solve some difficultmathematical questions such as checking the Gieseker-Mumfordstability of some manifolds. This question is one of thefundamental questions in the moduli space theory but has not beensolved in general.