Examples of K-unstable Fano manifolds with the Picard number one

TL;DR

This paper demonstrates that certain four- and five-dimensional del Pezzo manifolds of degree five are K-unstable, providing counterexamples to a previous conjecture by Odaka and Okada.

Contribution

It presents explicit examples of K-unstable Fano manifolds with Picard number one, disproving a conjecture about their stability.

Findings

Del Pezzo manifolds of degree five in dimensions four and five are K-unstable.

Counterexamples to Odaka and Okada's conjecture.

The pair (X, -K_X) exhibits K-unstability in these cases.

Abstract

We show that the pair $(X, -K_X)$ is K-unstable for a del Pezzo manifold $X$ of degree five with dimension four or five. This disprove a conjecture of Odaka and Okada.