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Transactions of the American Mathematical Society

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A Hilbert-Mumford criterion for polystability in Kaehler geometry

Author: I. Mundet i Riera
Journal: Trans. Amer. Math. Soc. 362 (2010), 5169-5187
MSC (2010): Primary 53D20; Secondary 32M05
Published electronically: May 19, 2010
MathSciNet review: 2657676
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Abstract: Consider a Hamiltonian action of a compact Lie group $ K$ on a Kaehler manifold $ X$ with moment map $ \mu:X\to\mathfrak{k}^*$. Assume that the action of $ K$ extends to a holomorphic action of the complexification $ G$ of $ K$. We characterize which $ G$-orbits in $ X$ intersect $ \mu^{-1}(0)$ in terms of the maximal weights $ \lim_{t\to\infty}\langle\mu(e^{\mathbf{i} ts}\cdot x),s\rangle$, where $ s\in\mathfrak{k}$. We do not impose any a priori restriction on the stabilizer of $ x$. Under some mild restrictions on the action $ K\circlearrowright X$, we view the maximal weights as defining a collection of maps: for each $ x\in X$,

$\displaystyle \lambda_x:\partial_{\infty}(K\backslash G)\to\mathbb{R}\cup\{\infty\},$

where $ \partial_{\infty}(K\backslash G)$ is the boundary at infinity of the symmetric space $ K\backslash G$. We prove that $ G\cdot x\cap\mu^{-1}(0)\neq\emptyset$ if: (1) $ \lambda_x$ is everywhere nonnegative, (2) any boundary point $ y$ such that $ \lambda_x(y)=0$ can be connected with a geodesic in $ K\backslash G$ to another boundary point $ y'$ satisfying $ \lambda_x(y')=0$. We also prove that the maximal weight functions are $ G$-equivariant: for any $ g\in G$ and any $ y\in \partial_{\infty}(K\backslash G)$ we have $ \lambda_{g\cdot x}(y)=\lambda_x(y\cdot g)$.

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Additional Information

I. Mundet i Riera
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Keywords: Hamiltonian actions, Kaehler geometry, Hilbert--Mumford criterion
Received by editor(s): April 4, 2008
Received by editor(s) in revised form: May 20, 2008
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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