Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Abstract

Let $M$ be a Fano manifold, and $H^{\star }(M;\mathbb{C})$ be the quantum cohomology ring of $M$ with the quantum product $\star .$ For $\sigma \in H^{\ast }(M;\mathbb{C})$, denote by $[\sigma ]$ the quantum multiplication operator $\sigma \star$ on $H^{\ast }(M;\mathbb{C})$. 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 $M,$ ${\delta }_0\ge \dim M+1,$ 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.

10 pages. arXiv admin note: text overlap with arXiv:1704.00403