Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Bull. Korean Math. Soc. Published online December 11, 2019

Daewoong Cheong and Manwook Han
Chungbuk National University

Abstract : Let $M$ be a Fano manifold, and $H^\star(M;\C)$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in H^*(M;\C)$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^*(M;\C)$. It was conjectured several years ago \cite{GGI, GI}
and has been proved for many Fano manifols \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 eigenvaules 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 equlity 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.