Equivariant rigidity of Richardson varieties

Anders S. Buch, Pierre-Emmanuel Chaput and Nicolas Perrin

Vol. 338 (2025), No. 2, 209–229

DOI: 10.2140/pjm.2025.338.209

Abstract

We prove that Schubert and Richardson varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Białynicki-Birula cells under suitable conditions. This is used to prove a conjecture of Buch, Chaput, and Perrin, stating that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety. We pose a stronger conjecture which implies a Seidel multiplication formula in equivariant quantum $K$-theory, and prove this conjecture for cominuscule flag varieties.

Keywords

rigidity, Schubert varieties, equivariant cohomology, Białynicki-Birula decomposition, curve neighborhoods, Seidel representation, quantum $K$-theory, horospherical varieties

Mathematical Subject Classification

Primary: 14M15

Secondary: 14C25, 14L30, 14N35

Milestones

Received: 10 October 2024

Revised: 29 April 2025

Accepted: 14 June 2025

Published: 24 August 2025