发表机构
Università degli Studi di Parma; Università degli Studi di Firenze（帕尔马大学; 佛罗伦萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对紧致Kähler流形上实约化群作用建立Calabi-Matsushima型分解，将稳定子Lie代数按非负特征值分解，并证明紧致稳定子单位分支为极大紧子群。
AI 中文摘要
设$Z$为紧致Kähler流形，其上带有紧致连通Lie群$U$的Hamilton作用，并设$G\subset U^{\mathbb C}$为实相容子群。利用与动量映射相关联的梯度映射$\mu_{\mathfrak p}$，我们在$f\circ\mu_{\mathfrak p}$的临界点处建立了稳定子$G_z$的Lie代数的Calabi-Matsushima型分解，其中$f$是$i\mathfrak u$上适当的$Ad_U$-不变的严格凸函数在$\mathfrak p$上的限制。更精确地，我们证明迷向代数分解为伴随自同态的非负特征值对应的特征空间，零特征空间由其约化部分给出。这将对$U^{\mathbb C}$的复化作用的经典分解推广到实约化情形。作为应用，我们证明在$f\circ\mu_{\mathfrak p}$的临界点处紧致稳定子的单位连通分支是$G$-稳定子的单位连通分支的极大紧子群。
英文摘要
Let $Z$ be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $μ_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer $G_z$ at a critical point of $f\circμ_{\mathfrak p}$, where $f$ is the restriction to $\mathfrak p$ of a suitable $Ad_U$-invariant strictly convex function on $i\mathfrak u$. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of $U^{\mathbb C}$ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of $f\circμ_{\mathfrak p}$ is a maximal compact subgroup of the identity component of the $G$-stabilizer.
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