#### 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 $\mathsc{𝒪}$ 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 $\mathsc{𝒪}$ is a conjecture that property $\mathsc{𝒪}$ holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture $\mathsc{𝒪}$ has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture $\mathsc{𝒪}$ 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