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

TL;DR

This paper constructs a canonical orbifold resolution of Richardson varieties in flag manifolds using equivariant stable maps, revealing topological and combinatorial properties related to Bruhat intervals and GKM spaces.

Contribution

It introduces a new orbifold resolution of Richardson varieties via stable maps, linking geometric, combinatorial, and topological aspects, especially in Grassmannians.

Findings

The resolution is an orbifold with a simple normal crossings boundary.

The boundary's dual complex is a sphere or ball, related to Bruhat intervals.

In Grassmannians, the resolution is a GKM space with fixed points indexed by rim-hook tableaux.

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.