On Galkin's Lower Bound Conjecture

TL;DR

This paper investigates the spectral radius of a linear operator on quantum cohomology of certain toric Fano manifolds, providing a counterexample to Galkin's lower bound conjecture.

Contribution

It offers the first explicit estimate that disproves Galkin's lower bound conjecture for a specific class of toric Fano manifolds.

Findings

Spectral radius exceeds Galkin's proposed lower bound

Counterexample found in the case of $ ext{P}_{ ext{P}^n}( ext{O}igoplus ext{O}(3))$

Disproves the conjecture for this class of manifolds

Abstract

We estimate an upper bound of the spectral radius of a linear operator on the quantum cohomology of the toric Fano manifolds $\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(3))$. This provides a negative answer to Galkin's lower bound conjecture.