Scalar Curvature and Stability of Toric Varieties
Scalar Curvature and Stability of Toric Varieties
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DOI：
10.4310/jdg/1090950195
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发表时间：
2002-10
影响因子：
2.5
通讯作者：
S. Donaldson
中科院分区：
文献类型：
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作者：
S. Donaldson
我们定义了一个极化代数变量的稳定性条件，并给出了一个关于常数标量曲率的Kahler度规存在性的猜想。本文的主要结果在某种程度上证实了这一猜想在环面情况下。在稳定性假设下，证明了Mabuchi泛函在不变度量上是有界的，并且证明了极小化序列具有一定的收敛性。在相反的方向上，我们给出了不允许常数标量曲率度量的偏振曲面的新例子。证明使用了由Guillemin和Abreu开发的一般框架，其中不变Kahler度量对应于环变矩多面体上的凸函数。本文是向寻找复射影变种允许常标量曲率的Kahler度规的条件这一一般问题的解决迈出了一步。人们所期望的答案模式是，这个微分几何条件应该等同于几何不变理论意义上的“稳定性”的某些概念。这种期望现在可能是民间传说的一部分：回到丘在KahlerEinstein度量中提出的建议，以及田和其他人在这种情况下的许多结果；用一张详细的形式化图加以加强，这张图清楚地说明了向量束的稳定性与Yang-Mills连接之间业已建立的关系[5]之间的类比。在这里，我们开始调查
We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety. This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of KahlerEinstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of