Communications in Number Theory and Physics

Volume 17 (2023)

Number 1

On the fundamental group of open Richardson varieties

Pages: 77 – 101



Changzheng Li (School of Mathematics, Sun Yat-sen University, Guangzhou, China)

Frank Sottile (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Chi Zhang (Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)


We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these $\log$ Calabi–Yau varieties underlie a Landau–Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is $\log$ Calabi–Yau as it isomorphic to the complement of the Knutson–Lam–Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.


flag variety, open Richardson variety, fundamental group, mirror symmetry

2010 Mathematics Subject Classification

Primary 14J33, 14M15. Secondary 57M05.

C. Li is supported by NSFC Grants 11831017, 11822113, and 11771455.

F. Sottile is supported by grant 636314 from the Simons Foundation.

Received 20 August 2020

Accepted 1 October 2022

Published 23 February 2023