Let Xwv be a Richardson variety in the full flag variety X associated
to a symmetrizable Kac-Moody group G. Recall that Xwv is the intersection
of the finite dimensional Schubert variety Xw with the finite codimensional
opposite Schubert variety Xv. We give an explicit \bQ-divisor Δ on
Xwv and prove that the pair (Xwv,Δ) has Kawamata log terminal
singularities. In fact, −KXwv−Δ is ample, which additionally
proves that (Xwv,Δ) 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 Xwv (similar to the BSDH resolution
of the Schubert varieties). In the general Kac-Moody case, in the absence of an
explicit resolution of Xwv 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.Comment: 15 pages, improved exposition and explanation. To appear in the
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