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

TL;DR

This work constructs a canonical orbifold resolution of Richardson varieties $X_\lambda^\mu \subseteq G/P$ via stable map quotients by the circle action $\check{\rho}$, avoiding Bott–Samelson choices. The resulting space $\widetilde{X_\lambda^\mu}$ is smooth as an orbifold and admits a birational map to $X_\lambda^\mu$, with a boundary that forms a simple normal crossings divisor whose dual complex equals the order complex of the open Bruhat interval $(\lambda,\mu)$. For Grassmannians, the resolution is a $2$-GKM space with $T$-fixed data indexed by rim-hook tableaux, enabling explicit Betti-number computations via weight analysis and a Deodhar-style decomposition. The paper also relates the stable map quotient to the Chow quotient, develops a stratification framework, and investigates ample/anticanonical line bundles, anticanonical criteria, and the Björner–Wachs boundary topology, offering a cohesive approach to resolving singularities in a combinatorially controlled setting with potential broad applications in equivariant geometry and Schubert calculus.

Abstract

Let $X_λ^μ:= X_λ\cap X^μ\subseteq G/P$ be a Richardson variety in a generalized partial flag manifold. We use equivariant stable map spaces to define a canonical resolution $\widetilde{X_λ^μ}$ 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 $(λ,μ) \subseteq W/W_P$, shown in [Björner-Wachs '82] to be a sphere or ball. In the case of $G/P$ a Grassmannian, the resolution $\widetilde{X_λ^μ}$ is a GKM space, whose $T$-fixed points are indexed by rim-hook tableaux.

Theorems & Definitions (38)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Theorem 2