Relative algebro-geometric stabilities of toric manifolds
Relative algebro-geometric stabilities of toric manifolds
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DOI：
10.2748/tmj/1576724790
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发表时间：
2016-02
影响因子：
0.5
通讯作者：
N. Yotsutani;Bin Zhou
中科院分区：
文献类型：
--
作者：
N. Yotsutani;Bin Zhou
本文研究了环面流形的相对Chow稳定性和$K$-稳定性。首先，我们给出了复曲面Fano流形的相对K$-稳定性和不稳定性的一个判据.在复曲面流形上的相对Chow稳定性的减少将通过Hibert-Mumford准则以两种方式进行研究。一个是考虑最大环面作用及其权多面体的准则。然后，我们通过Ono \cite{Ono 13}的策略获得约简，这符合Sz\'ekelyhidi检测到的相对GIT稳定性。另一种方法是使用与唐纳森和Ross-Thomas \cite{D 02，RT 07}之后的复曲面退化相关的$\mathbb{C}^{\times}$-作用和Chow权重的准则。最后，我们确定了所有复曲面Fano三重流形的相对K-稳定性，并给出了相对K$-稳定但渐近相对Chow不稳定的反例。
In this paper, we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds. The reduction of relative Chow stability on toric manifolds will be investigated by the Hibert-Mumford criterion in two ways. One is to consider the criterion for the maximal torus action and its weight polytope. Then we obtain a reduction by the strategy of Ono \cite{Ono13}, which fits into the relative GIT stability detected by Sz\'ekelyhidi. The other way is to use the criterion for $\mathbb{C}^{\times}$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas \cite{D02, RT07}. In the end, we determine the relative K-stability of all toric Fano threefolds and present counter-example for relatively $K$-stable manifold, but which is asymptotically relatively Chow unstable.