On the Geometry of Kahler-Einstein Manifolds

AbstractAward：DMS-9971506主要研究者：LuZhiqin该项目将以三种不同的方式分析紧致Kaehler流形上的Kaehler-Einstein度量。第一部分是研究极化紧Kaehler流形的Bergman度量的势函数。作者计算了Zelditch渐近展开式的前几个系数。 我使用的方法是复杂的几何。与利用酉圆丛的Szego核的参数矩阵的方法相比，我们的方法更为直接。 我们希望有一些统一的估计。即使在复维数为1的情况下，这个问题也很有趣。 在二维空间中，问题已经够复杂了。Tian证明了Kaehler-Einstein曲面上的多反正则丛存在一致下界。 在高维中，这个问题非常重要，指向流形的稳定性。第二个问题是Futakiinvariants的计算。这个问题与前一个问题有关，因为势函数被用于Futaki不变量的定义。Futaki不变量不仅可以作为Kaehler-Einstein度规存在的障碍，而且可以用来检验流形的稳定性。由于我们只对嵌入到复射影空间中的簇的Futaki不变量感兴趣，因此我们有理由认为我们可以利用一些“外在”信息（如定义簇的多项式）来计算不变量。实际上，在簇伊萨完全交的情况下，我们找到了一个计算Futaki不变量的公式。 第三部分是关于极化Calabi-Yau流形的模空间几何。本文定义了模空间上的Kaehler度量，并证明了这种度量的Ricciccurvature是负的。此外，如果能证明度规的标量曲率是下界的，就能用微分几何方法证明维韦格的紧化定理。爱因斯坦的广义相对论将引力的概念解释为时空的几何性质。研究这些空间几何的主要数学工具是微分几何。自从广义相对论被发现以来，微分几何对数学家和物理学家都至关重要。我们的项目是在微分几何的主要领域之一，并应增加我们对重力的理解，并最终我们生活的空间。

AbstractAward: DMS-9971506Principal Investigator: Zhiqin LuThis project will analyze Kaehler-Einstein metrics on a compactKaehler manifold in three different ways. The first one is tostudy the potential function of the Bergman metric of a polarizedcompact Kaehler manifold. The first several coefficients of theasymptotic expansion of Zelditch have been computed by theauthor. The method I used is complex geometric. Comparing to themethod using the paramatrix of the Szego kernel of the unitcircle bundle, our method is more straightforward. We wish tohave some uniform estimates. Even in the case of complexdimension one, this problem is interesting. In dimension two,the problem is already complicated enough. Tian proved that thereis a uniform lower bound for the multi-anticanonical bundle overa Kaehler-Einstein surfaces. In higher dimensions, the problemis very important, pointing towards the stability ofmanifolds. The second one is the computation of the Futakiinvariants. This problem is related to the previous one in thatthe potential function is used in the definition of the Futakiinvariants. Futaki invariants can be used not only as theobstruction towards the existence of the Kaehler-Einstein metric,but also can be used to check the stability of manifolds. Sincewe are only interested in the Futaki invariants for thosevarieties which are embedded into some complex projective space,it is reasonable to think that we can compute the invariantsusing some "extrinsic" information(Such as the polynomials indefining the varieties). In fact, in the case that the variety isa complete intersection, we found a formula to calculate theFutaki invariants. The last one is on the geometry of modulispace of polarized Calabi-Yau manifold. A Kaehler metric on themoduli space has been defined and was found that the Riccicurvature of such a metric is negative away from zero by theauthor. Furthermore, if one can prove that the scalar curvatureof the metric is lower bounded, one would be able to proveViehweg's compactification theorem using differential geometricmethods.Einstein's general theory of relativity interprets the concept ofgravity as a geometric property of space-time. The mainmathematical tool to study the geometry of these spaces isdifferential geometry. Since the discovery of general relativity,differential geometry has been crucial to both mathematicians andphysicists. Our project is in one of the main fields indifferential geometry, and should add to our understanding ofgravity and ultimately the space we are living in.