Richardson Varieties Have Kawamata Log Terminal Singularities
Richardson Varieties Have Kawamata Log Terminal Singularities
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DOI：
10.1093/imrn/rns241
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发表时间：
2012-03
影响因子：
1
通讯作者：
Shrawan Kumar;Karl Schwede
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文献类型：
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作者：
Shrawan Kumar;Karl Schwede
设$X^v_w$是与可对称化的Kac-Moody群$G$相关联的全旗簇$X$中的Richardson簇。回想一下，$X^v_w$是有限维舒伯特簇$X_w$与有限余维相反舒伯特簇$X^v$的交集。我们给出了X^v_w$上的一个显式$\bQ$-除数$\Delta$，并证明了对$（X^v_w，\Delta）$具有Kawamata对数终端奇点.事实上，$-K_{X^v_w} - \Delta$是充分的，这额外地证明了$（X^v_w，\Delta）$是log Fano。我们首先在有限情形下证明我们的结果（即，当$G$是一个有限维半单群的情况下）通过仔细分析的显式解决的奇异性$X^v_w$（类似于BSDH决议的舒伯特品种）。在一般的Kac-Moody情形下，在缺少上述X^v_w$的显式分解的情况下，我们给出了依赖于Frobenius分裂方法的证明。特别地，我们利用Mathieu的结果断言Richardson簇是Frobenius分裂的，并将它与N的一个结果联合收割机结合起来。Hara和K.- I. Watanabe将Frobenius分裂与对数正则奇点联系起来。
Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities.