Richardson Varieties Have Kawamata Log Terminal Singularities

Abstract

Let $X_w^v$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X_w^v$ 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 $\hat{I}”$ on $X_w^v$ and prove that the pair $(X_w^v,\hat{I}”)$ has Kawamata log terminal singularities. In fact, $-K_{X_w^v}-\hat{I}”$ is ample, which additionally proves that $(X_w^v,\hat{I}”)$ 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 $X_w^v$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X_w^v$ 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.

15 pages, improved exposition and explanation. To appear in the International Mathematics Research Notices