Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: Galkin's conjecture, property $\mathcal{O}$, Gamma conjectures

MSC numbers: 14N35, 05E15, 14J33, 47N50

Supported by: The first author was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07045594)