A Hilbert--Mumford criterion for polystability in Kaehler geometry

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dc.creatorMundet-i-Riera, Ignasi
dc.date2008-04-07
dc.date.accessioned2026-07-07T09:30:50Z
dc.date.available2026-07-07T09:30:50Z
dc.descriptionConsider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $μ:X\to\klie^*$. We characterize which orbits of the complexified action of $G=K^{\CC}$ in $X$ intersect $μ^{-1}(0)$ in terms of the maximal weights $\lim_{t\to\infty}\laμ(e^{\imag ts}\cdot x),s\ra$, where $s$ belongs to the Lie algebra of $K$. We do not impose any a priori restriction on the stabilizer of $x$. Assuming some mild growth conditions on the action of $K$ on $X$, we view the maximal weights as defining a maps $λ_x$ from the boundary at infinity of the symmetric space $K\backslash G$ to $\RR\cup\{\infty\}$. We prove that $G\cdot x$ meets $μ^{-1}(0)$ if: (1) $λ_x$ is everywhere nonnegative, (2) any boundary point $y$ such that $λ_x(y)=0$ can be connected with a geodesic in $K\backslash G$ to another boundary point $y'$ satisfying $λ_x(y')=0$. We also prove that $λ_{g\cdot x}(y)=λ_x(y\cdot g)$ for any $g\in G$ and $y\in \partial_{\infty}(K\backslash G)$.
dc.description20 pages, no figures
dc.identifierhttps://arxiv.org/abs/0804.1067
dc.identifierhttp://arxiv.org/abs/0804.1067
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/158252
dc.subjectSymplectic Geometry
dc.subjectComplex Variables
dc.subject53D20; 32M05
dc.titleA Hilbert--Mumford criterion for polystability in Kaehler geometry
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