Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

TL;DR

This paper proves Galkin's lower bound conjecture for the eigenvalues of quantum multiplication operators on the quantum cohomology of Lagrangian and orthogonal Grassmannians, confirming the conjecture in these cases.

Contribution

The paper establishes Galkin's lower bound conjecture for Lagrangian and orthogonal Grassmannians, extending the class of Fano manifolds where the conjecture holds.

Findings

Galkin's lower bound conjecture verified for Lagrangian Grassmannians

Galkin's lower bound conjecture verified for Orthogonal Grassmannians

Conditions for equality cases discussed

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 $\sigma \in H^*(M;\mathbb{C})$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^*(M;\mathbb{C})$. 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 $M,$ $\delta_0\geq \mathrm{dim} \ M +1,$ 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.