Cominuscule Richardson varieties

Abstract

A Richardson variety in a cominuscule flag variety is defined by a skew shape of boxes. We prove that any cominuscule Richardson variety has a unique decomposition as a product of indecomposable cominuscule Richardson varieties, which are given by the connected components of the skew shape. We also prove that the first Chow group and the Picard group of a cominuscule Richardson variety are dual to each other; both are free abelian groups of rank equal to the number of indecomposable factors. Along the way, we construct an explicit desingularization of any cominuscule Richardson variety of classical Lie type as a tower of (isotropic) Grassmannian bundles. Our results imply that the Hilbert polynomial of the minimal ample line bundle is a well-defined geometric invariant of any cominuscule Richardson variety. We give a positive formula for this Hilbert polynomial.