论文标题
稳定性，实际还原性谎言组的分析稳定性
Stability, analytic stability for real reductive Lie groups
论文作者
论文摘要
我们介绍了对希尔伯特稳定性理论标准的系统处理方法，用于对真正的submanifold $ x $ akähler歧管$ z $的真实还原组$ g $的动作。更确切地说，我们假设与Lie代数$ \ Mathfrak {U} $相连的连接Lie组$ U $的动作将全体态延伸至复杂的组$ u^{\ mathbb c} $的动作，并且$ u $ - $ u $ -Asaction in $ z $是hamiltonian。 If $G\subset U^{\mathbb C}$ is closed and compatible, there is a corresponding gradient map $μ_\mathfrak{p} : X\longrightarrow \mathfrak{p}$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of the Lie algebra of $G$.引入了$ g $在$ x $上的$ g $的概念。对于此类操作，可以使用与梯度映射相关的$ G $ equivariant函数（称为最大权重函数）来表征点的稳定性，可准性和多稳定性。我们还证明了经典的希尔伯特·姆福德（Hilbert-Mumford）的标准，用于半决赛和多稳定性条件。 我们感谢匿名裁判仔细阅读我们的论文，并给出了这种建设性的评论，从而实质上有助于提高论文的质量。
We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group $G$ on a real submanifold $X$ of a Kähler manifold $Z$. More precisely, we suppose the action of a compact connected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^{\mathbb C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^{\mathbb C}$ is closed and compatible, there is a corresponding gradient map $μ_\mathfrak{p} : X\longrightarrow \mathfrak{p}$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of the Lie algebra of $G$. The concept of energy complete action of $G$ on $X$ is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a $G$-equivariant function associated with a gradient map, called maximal weight function. We also prove the classical Hilbert-Mumford criteria for semistabilty and polystability conditions. We thank the anonymous referee for carefully reading our paper and for giving such constructive comments which substantially helped improving the quality of the paper.
