Stable map quotients (and orbifold log resolutions) of Richardson varieties

Abstract

Let $X_\lambda^\mu := X_\lambda \cap X^\mu \subseteq G/P$ be a Richardson variety in a generalized partial flag manifold. We use equivariant stable map spaces to define a canonical resolution $\widetilde{X_\lambda^\mu}$ of singularities, albeit obtaining an orbifold not a manifold. The ``nodal curves'' boundary is an (orbifold) simple normal crossings divisor, and is conjecturally anticanonical. Its dual simplicial complex is the order complex of the open Bruhat interval $(\lambda,\mu) \subseteq W/W_P$, shown in [Bj\"orner-Wachs '82] to be a sphere or ball. In the case of $G/P$ a Grassmannian, the resolution $\widetilde{X_\lambda^\mu}$ is a GKM space, whose $T$-fixed points are indexed by rim-hook tableaux.