Vol. 13, No. 4, 2020

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Conjecture $\mathcal{O}$ holds for some horospherical varieties of Picard rank 1

Lela Bones, Garrett Fowler, Lisa Schneider and Ryan M. Shifler

Vol. 13 (2020), No. 4, 551–558
Abstract

Property 𝒪 for an arbitrary complex, Fano manifold X is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of X. Conjecture 𝒪 is a conjecture that property 𝒪 holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture 𝒪 has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture 𝒪 holds for two more classes and an example in a third class of Pasquier’s list. Perron–Frobenius theory reduces our proofs to be graph-theoretic in nature.

Keywords
quantum cohomology, horospherical, conjecture $\mathcal{O}$
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14N15, 15B48
Milestones
Received: 10 July 2018
Revised: 12 July 2019
Accepted: 25 July 2020
Published: 20 November 2020

Communicated by Kenneth S. Berenhaut
Authors
Lela Bones
Department of Mathematics
Salisbury University
Salisbury, MD
United States
Garrett Fowler
Department of Mathematics
Salisbury University
Salisbury, MD
United States
Lisa Schneider
Department of Mathematics
Salisbury University
Salisbury, MD
United States
Ryan M. Shifler
Department of Mathematics
Salisbury University
Salisbury, MD
United States