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

2025/05/15 by Allen Knutson, Knutson, Allen · 1 citation
Mathematics · #14M15 14H10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2505.09905

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Chow functions for partially ordered sets
• Cohomological and categorical concentration
• Richardson varieties, projected Richardson varieties and positroid varieties
• Virtual localization revisited
• Brick manifolds and toric varieties of brick polytopes
• Chow Quotients of Toric Varieties as Moduli of Stable Log Maps
• Real aspects of the moduli space of genus zero stable maps
• Higher algebraic K-theory for actions of diagonalizable groups
• One-skeleta, Betti numbers and equivariant cohomology

Cited by

• Relative Quasimaps and Tilting Module of U(\mathfrakgl_n)

Related

• Projections of Richardson Varieties
• Singularities of generalized Richardson varieties
• Richardson varieties, projected Richardson varieties and positroid varieties
• On the fundamental group of open Richardson varieties
• Derived categories of resolutions of cyclic quotient singularities