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Galkin's lower bound conjecure for Lagrangian and orthogonal Grassmannians
정대웅,Manwook Han 대한수학회 2020 대한수학회보 Vol.57 No.4
Let $M$ be a Fano manifold, and $H^\star(M;\C)$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in H^\star(M;\C)$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^\star(M;\C)$. It was conjectured several years ago \cite{GGI, GI} and has been proved for many Fano manifolds \cite{CL1, CH2, LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real valued eigenvalue $\delta_0$ which is maximal among eigenvalues of $[c_1(M)]$. Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$ $\delta_0\geq \mathrm{dim} \ M +1,$ and the equality holds if and only if $M$ is the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.
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GALKIN'S LOWER BOUND CONJECURE FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS
Cheong, Daewoong,Han, Manwook Korean Mathematical Society 2020 대한수학회보 Vol.57 No.4
Let M be a Fano manifold, and H<sup>🟉</sup>(M; ℂ) be the quantum cohomology ring of M with the quantum product 🟉. For 𝜎 ∈ H<sup>🟉</sup>(M; ℂ), denote by [𝜎] the quantum multiplication operator 𝜎🟉 on H<sup>🟉</sup>(M; ℂ). It was conjectured several years ago [7,8] and has been proved for many Fano manifolds [1,2,10,14], including our cases, that the operator [c<sub>1</sub>(M)] has a real valued eigenvalue 𝛿<sub>0</sub> which is maximal among eigenvalues of [c<sub>1</sub>(M)]. Galkin's lower bound conjecture [6] states that for a Fano manifold M, 𝛿<sub>0</sub> ≥ dim M + 1, and the equality holds if and only if M is the projective space ℙ<sup>n</sup>. In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.
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