Stability of extremal Kahler manifolds
Stability of extremal Kahler manifolds
复制标题
DOI：
10.18910/9178
复制
发表时间：
2004-04
影响因子：
0.4
通讯作者：
T. Mabuchi
中科院分区：
文献类型：
--
作者：
T. Mabuchi
在Donaldson关于极化代数流形(M，L)渐近稳定性的研究中，由张[39](另见[22])定义的临界度量被称为平衡度量，当极化代数流形允许常数量曲率的Kähler度量时，临界度量起着中心作用。设T∼=(C∗)k是M的全纯自同构群的单位分支Aut(M)中的代数环面.本文定义了关于T的临界度量的概念，作为应用，选择适当的T，我们将证明[26]中关于临界度量的渐近逼近的一个结果(见[10]，[39])可以推广到(M，L)在极化类中允许极值Kähler度量的情况。然后，在我们即将发表的论文[27]中，我们将证明对稳定性概念稍作修改(见下面的定理A)，即使在像[26]中那样的障碍没有消失的情况下，我们也可以获得极值Kähler流形的渐近稳定性。特别地，通过类似于[10]的论证，将证明射影代数流形M上的固定积分Kähler类中的极值Kähler度量在群Aut(M)的作用下是唯一的†。
In Donaldson’s study [10] of asymptotic stability for polarized algebraic manifolds (M, L), critical metrics originally defined by Zhang [39] (see also [22]) are referred to as balanced metrics and play a central role when the polarized algebraic manifolds admit Kähler metrics of constant scalar curvature. Let T ∼= (C∗)k be an algebraic torus in the identity component Aut(M) of the group of holomorphic automorphisms of M . In this paper, we define the concept of critical metrics relative to T , and as an application, choosing a suitable T , we shall show that a result in [26] on the asymptotic approximation of critical metrics (see [10], [39]) can be generalized to the case where (M, L) admits an extremal Kähler metric in the polarization class. Then in our forthcoming paper [27], we shall show that a slight modification of the concept of stability (see Theorem A below) allows us to obtain the asymptotic stability of extremal Kähler manifolds even when the obstruction as in [26] does not vanish. In particular, by an argument similar to [10], an extremal Kähler metric in a fixed integral Kähler class on a projective algebraic manifold M will be shown to be unique† up to the action of the group Aut(M).