Galkin’s lower bound conjecture holds for the Grassmannian

Abstract

Let $\text{Gr}(k,n)$ be the Grassmannian. The quantum multiplication by the first Chern class $c_1(\text{Gr}(k,n))$ induces an endomorphism ${\widehat{c}}_1$ of the finite-dimensional vector space ${\text{QH}}^{*}{(\text{Gr}(k,n))}_{|q=1}$ specialized at q = 1. Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of ${\widehat{c}}_1$ is greater than or equal to $\mathrm{dim\hspace{0.17em}}\text{Gr}(k,n)$+1 with equality if and only if $\text{Gr}(k,n)={\mathbb{P}}^{n-1}.$

Keywords:

• Algebraic geometry
• conjecture O
• quantum cohomology

2010 Mathematics Subject Classification:

• Primary 14N35
• Secondary 15B48
• 14N15
• 14M15

Acknowledgments

The third named author thanks Leonardo Mihalcea for useful discussions. We thank the anonymous referees for their comments and an alternative proof.