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.
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