Stability and canonical metrics on projective spaces blown up along a line

TL;DR

This paper investigates the existence of special Kähler metrics on blow-ups of projective spaces along a line, showing non-existence of constant scalar curvature metrics but existence of extremal metrics in nearby classes.

Contribution

It provides explicit formulas for extremal metrics on these blow-ups and clarifies the non-existence of constant scalar curvature Kähler metrics in all rational classes.

Findings

No constant scalar curvature Kähler metrics in any rational class.

Existence of extremal metrics close to the pullback of the Fubini--Study class.

Explicit formula for extremal metrics in action-angle coordinates.

Abstract

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a K\"ahler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature K\"ahler metrics in any rational K\"ahler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in K\"ahler classes that are close to the pullback of the Fubini--Study class.