2019/06/26 by Daewoong Cheong, Cheong, Daewoong, Manwook Han +1
Computer Science · Mathematics · #05E05 #14J33 #14N35 #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1906.11646
openalex publication_date 2019/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a Fano manifold, and H^⋆(M;ℂ) be the quantum cohomology ring of M with the quantum product ⋆. For σ∈ H^*(M;ℂ), denote by [σ] the quantum multiplication operator σ⋆ on H^*(M;ℂ). It was conjectured several years ago \citeGGI, GI and has been proved for many Fano manifols \citeCL1, CH2, LiMiSh, Ke, including our cases, that the operator [c1(M)] has a real valued eigenvalue δ0 which is maximal among eigenvaules of [c1(M)]. Galkin's lower bound conjecture \citeGa states that for a Fano manifold M, δ0≥ dim M +1, and the equlity holds if and only if M is the projective space ℙn. In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.