Conjecture $\mathcal{O}$ holds for some Horospherical Varieties of Picard Rank 1

TL;DR

This paper proves that Conjecture $ ext{O}$, related to eigenvalues of quantum multiplication, holds for certain non-homogeneous horospherical varieties of Picard rank 1, extending previous results.

Contribution

It demonstrates Conjecture $ ext{O}$ for additional classes of horospherical varieties, using graph-theoretic methods based on Perron-Frobenius theory.

Findings

Conjecture $ ext{O}$ holds for two classes of horospherical varieties.

Conjecture $ ext{O}$ holds for an example in a third class.

Graph-theoretic approach simplifies the proof process.

Abstract

Property $\mathcal{O}$ 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 $\mathcal{O}$ is a conjecture that Property $\mathcal{O}$ holds for any Fano variety. Pasquier listed the smooth non-homogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture $\mathcal{O}$ has already been shown to hold for the odd symplectic Grassmannians which is one of these classes. We will show that Conjecture $\mathcal{O}$ holds for two more classes and an example in a third class of Pasquier's list. The theory of Perron-Frobenius reduces our proofs to be graph-theoretic in nature.