Stability of Algebraic Manifold

摘要奖：DMS-9971387主要研究者：大卫A. Gieseker和Luo Huazhang在几何不变论中，为了研究模问题，引入了任何极化射影变分的稳定性概念。然而，稳定性的校核往往是一个难题. 它已经完成了芒福德的曲线与亏格大于1，由Gieseker的一般类型的表面，和Viehweg的高维品种半amplecanonical线丛。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，并且它的Ricci曲率满足一些非退化约束。这个度规不是Kahler-Einstein度规，而是与Kahler-Einstein度规密切相关。这个项目的一个动机来自于唐纳森，Uhlenbeck和Yau关于向量丛的稳定性与Yang-Mills联络存在性之间关系的工作。我们的计划是把数学中两个重要但不同的领域联系起来。一个是特殊度量的研究，它涉及到流形上微分方程的研究，通常与物理学有关。 另一个是模空间理论中稳定性的研究，主要涉及代数。我们的方法是在这两个领域之间建立一种对应关系，特别是用特殊度量的存在来描述稳定性的意义。这样做的好处是不仅为我们提供了关于这些特殊度量的新的视角，而且为我们提供了模空间理论的新的技术。例如，这种对应将使我们能够解决一些困难的数学问题，如检查Gieseker-Mumford稳定性的一些流形。这个问题是模空间理论中的基本问题之一，但至今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.