A Hilbert-Mumford criterion for polystability in Kaehler geometry - Mathematics > Symplectic GeometryReport as inadecuate




A Hilbert-Mumford criterion for polystability in Kaehler geometry - Mathematics > Symplectic Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$on a Kaehler manifold $X$, with moment map $\mu:X\to\klie^*$. We characterizewhich orbits of the complexified action of $G=K^{\CC}$ in $X$ intersect$\mu^{-1}0$ in terms of the maximal weights $\lim {t\to\infty}\la\mue^{\imagts}\cdot x,s a$, where $s$ belongs to the Lie algebra of $K$. We do notimpose any a priori restriction on the stabilizer of $x$. Assuming some mildgrowth conditions on the action of $K$ on $X$, we view the maximal weights asdefining a maps $\lambda x$ from the boundary at infinity of the symmetricspace $K\backslash G$ to $\RR\cup\{\infty\}$. We prove that $G\cdot x$ meets$\mu^{-1}0$ if: 1 $\lambda x$ is everywhere nonnegative, 2 any boundarypoint $y$ such that $\lambda xy=0$ can be connected with a geodesic in$K\backslash G$ to another boundary point $y-$ satisfying $\lambda xy-=0$. Wealso prove that $\lambda {g\cdot x}y=\lambda xy\cdot g$ for any $g\in G$and $y\in \partial {\infty}K\backslash G$.



Author: Ignasi Mundet-i-Riera

Source: https://arxiv.org/







Related documents