Algebro-geometric semistability of polarized toric manifolds
Algebro-geometric semistability of polarized toric manifolds
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DOI：
10.4310/ajm.2013.v17.n4.a3
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发表时间：
2010-09
期刊：
影响因子：
--
通讯作者：
Hajime Ono
中科院分区：
文献类型：
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作者：
Hajime Ono
设$\Delta\subset \mathbb{R}^n$是一个n维积分Delzant多面体.众所周知，在$X_\Delta$上存在$n$维紧环面流形$X_\Delta$和与$\Delta$相关的非常丰富的$（\mathbb{C}^\times）^n$-等变线丛$L_\Delta$。本文给出了极大环面作用$（X\Delta，L\Delta^i）$Chow半稳定的一个充要条件.然后我们看到，渐近（相对）Chow半稳定性蕴涵着复曲面退化的（相对）K-半稳定性，这是Ross和托马斯在没有Riemann-Roch定理和测试配置的情况下证明的.
Let $\Delta\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $(X_\Delta,L_\Delta^i)$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas, without any knowledge of Riemann-Roch theorem and test configurations.