环流形的相对代数几何稳定性
Relative Algebro-Geometric stabilities of Toric Manifolds
AI总结：
本文研究环流形的相对Chow与K-稳定性，给出环Fano流形判据，通过两种约化方法部分确定三维情形，并指出已发表版本错误。
摘要：
本文研究了环流形在环意义下的相对Chow稳定性和相对K-稳定性。首先，我们给出了环Fano流形在环意义下相对K-稳定与不稳定的判据。利用Hilbert-Mumford判据，我们以两种方式研究环流形上相对Chow稳定性的约化。一种方式是考虑极大环面作用及其权重多面体，我们通过Ono [Ono13]的策略获得约化，这符合Székelyhidi所检测的相对GIT稳定性。另一种方式依赖于$\mathbb{C}^*$-作用以及与环退化相关的Chow权重，遵循Donaldson和Ross-Thomas [D02, RT07]的方法。作为我们主要定理的应用，我们部分确定了环Fano三维流形的相对K-稳定性，并给出了在环意义下相对K-稳定但渐近相对Chow不稳定的反例。最后，我们解释了本文已发表版本（对应第1-5节）中的错误部分，这些错误导致表6中关于相对K-稳定性的某些结论不确定。
英文摘要：
In this paper we study the relative Chow and $K$-stability of toric manifolds in the toric sense. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [Ono13], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [D02, RT07]. As applications of our main theorem, we partially determine the relative $K$-stability of toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable. In the end, we explain the erroneous parts of the published version of this article (corresponding to Sections 1-5), which provides some inconclusive results for relative $K$-stability in Table 6.
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