Algebraicity of the metric tangent cones and equivariant K-stability
Algebraicity of the metric tangent cones and equivariant K-stability
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DOI：
10.1090/jams/974
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发表时间：
2018-05
影响因子：
3.9
通讯作者：
Chi Li;Xiaowei Wang;Chenyang Xu-
中科院分区：
文献类型：
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作者：
Chi Li;Xiaowei Wang;Chenyang Xu-
我们证明了两个新的结果的K K -多稳定性的Q \mathbb {Q} -Fano簇的基础上纯粹的代数几何参数。第一种是说任何K-K-半稳定的对数Fano锥都有一个特殊的退化，退化为唯一确定的K-K-多稳定的对数Fano锥。作为推论，我们将其与微分几何结果联合收割机相结合，完成了Donaldson-Sun猜想的证明，即出现在Kähler-Einstein Fano流形的Gromov-Hausdorff极限上的任何点的度量切锥只依赖于奇点的代数结构.第二个结果表明，对于任意具有环面作用的对数Fano簇，KK-多稳定性等价于等变的KK-多稳定性，也就是说，要检验KK-多稳定性，只需检验在环面作用下等变的特殊检验构形.
We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.