Richardson Varieties Have Kawamata Log Terminal Singularities

TL;DR

This paper proves that Richardson varieties in flag varieties associated with Kac-Moody groups have Kawamata log terminal singularities and are log Fano, using explicit resolutions in finite cases and Frobenius splitting methods in the general case.

Contribution

It establishes the KLT singularities and log Fano property of Richardson varieties in the Kac-Moody setting, extending known results from finite-dimensional cases.

Findings

Richardson varieties have Kawamata log terminal singularities.

The pair (X^v_w, Δ) is log Fano.

Frobenius splitting implies log canonical singularities.

Abstract

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.