Characters of automorphism groups associated with Kähler classes and functionals with cocycle conditions
Characters of automorphism groups associated with Kähler classes and functionals with cocycle conditions
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与凯勒类相关的自同构群和具有共循环条件的泛函的特征
DOI：
10.2996/kmj/1106157289
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发表时间：
2001
影响因子：
0.6
通讯作者：
Y. Nakagawa
中科院分区：
文献类型：
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作者：
A. Futaki;Y. Nakagawa
设M是连通的紧致Kaehler流形. M承认Kaehler-Einstein度量的一个明显的必要条件是，第一个陈类c1M是负的、零的或正的，其中一个真实的二维de Rham上同调类被称为是负的（分别是）。正的），如果它由负的（resp.阳性）de®nite 101; 1种β-型。相反，如果c1<$ M<$是负的或零，那么M通过卡拉比方程的解（奥宾[1]，Yau [21]）承认一个Kahler-Einstein度规。在剩下的情况下，其中c1<$ M<$是积极的，在这种情况下，M通常被称为法诺流形，有进一步的必要条件。首先，Kaehler-Einstein Fano流形M上所有全纯向量的李代数h∈M∈是约化的（Matsushima [14]）。其次，李代数特征标f：h<$ M<$！[10]中引入的C在Kaehler-Einstein Fano流形上必为零。Bando-Mabuchi [5]也证明了，如果M容许Kahler-Einstein度量，则M的某个泛函，称为K-能量，从下面有界。这一分析必要条件在后来的研究中发挥了重要的理论作用。事实上，丁和田[9]推广了[10]和[5]的结果，得到了一个适用于不携带任何非零全纯向量的流形的必要条件。Tian [20]进一步扩展了这些思想，定义了某些稳定性的概念，称为K-稳定性和CM-稳定性，并给出了一个没有非零全纯向量和没有Kaehler-Einstein度量的Fano流形的例子。另一方面，奥宾[2]，丁[8]，萧[17]，田[18]和纳德尔[15]也给出了正的Kaehler-Einstein度规存在的充分条件。现在人们希望有一个正的凯勒-爱因斯坦度规存在的必要和充分条件。为了陈述这样的条件，田[20]引入了K-能量的适当性概念和丁[8]引入的泛函。结合[20]和[4]，我们可以证明，至少当h 0时，Fano流形允许Kaehler-Einstein度量当且仅当
Let M be a connected compact KaÈhler manifold. An obvious necessary condition for M to admit a KaÈhler-Einstein metric is that the ®rst Chern class c1
M is either negative, zero or positive, where a real 2-dimensional de Rham cohomology class is said to be negative (resp. positive) if it is represented by a negative (resp. positive) de®nite
1; 1-form. Conversely, if c1
M is negative or zero then M admits a KaÈhler-Einstein metric by the solution to the Calabi conjectures (Aubin [1], Yau [21]). In the remaining case where c1
M is positive, in which case M is often called a Fano manifold, there are further necessary conditions. First of all the Lie algebra h
M of all holomorphic vector ®elds on a KaÈhler-Einstein Fano manifold M is reductive (Matsushima [14]). Secondly a Lie algebra character f : h
M ! C introduced in [10] must vanish on a KaÈhler-Einstein Fano manifold. It was also proven by Bando-Mabuchi [5] that if M admits a KaÈhlerEinstein metric then certain functional, called K-energy, of M is bounded from below. This analytic necessary condition played a theoretically important role in the later studies. In fact Ding and Tian [9] extended the results of [10] and [5] to obtain a necessary condition applicable to manifolds which do not carry any non-zero holomorphic vector ®elds. Tian [20] further extended these ideas to de®ne certain notions of stability, called K-stability and CM-stability, and presented an example of a Fano manifold with no non-zero holomorphic vector ®elds and no KaÈhler-Einstein metrics. On the other hand there are known su1⁄2cient conditions for the existence of positive KaÈhler-Einstein metrics by Aubin [2], Ding [8], Siu [17], Tian [18] and Nadel [15]. Now one would hope to have a necessary and su1⁄2cient condition for the existence of positive KaÈhler-Einstein metrics. To state such a condition, Tian [20] introduced a notion of properness for the K-energy and a functional introduced by Ding [8]. Combining [20] and [4] one can show, at least when h
M 0, that a Fano manifold admits a KaÈhler-Einstein metric if and only if
Bando-Calabi-Futaki 角色及其向群体角色的提升
DOI：
--
发表时间：
2003
期刊：
Math. Ann. 325
影响因子：
--
作者：
A.Kodama;S.Shimizu;A.Kodama;T.Mabuchi;S.Takeuchi;Y.Nakagawa
通讯作者：
Y.Nakagawa