Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913718046031872 |
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| author | Windare, Oluwagbenga Joshua |
| author_facet | Windare, Oluwagbenga Joshua |
| contents | We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ We assume that the action of $U$, a maximal compact connected subgroup of $U^\mathbb{C}$ on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $μ_\mathfrak{p}: Z\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Our main results are the openness and connectedness of the set of semistable points associated with $G$-action on $Z$, a convexity theorem for the $G$-action on a $G$-invariant compact Lagrangian submanifold of $Z$, and a convexity result for two-orbit variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14725 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps Windare, Oluwagbenga Joshua Differential Geometry 53D20, 14L24 We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ We assume that the action of $U$, a maximal compact connected subgroup of $U^\mathbb{C}$ on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $μ_\mathfrak{p}: Z\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Our main results are the openness and connectedness of the set of semistable points associated with $G$-action on $Z$, a convexity theorem for the $G$-action on a $G$-invariant compact Lagrangian submanifold of $Z$, and a convexity result for two-orbit variety. |
| title | Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps |
| topic | Differential Geometry 53D20, 14L24 |
| url | https://arxiv.org/abs/2206.14725 |