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

Abstract

Let ${\text{Bl}}_{{\mathbb{P}}^1}{\mathbb{P}}^n$ be a Kähler 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 Kähler metrics in any rational Kähler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in Kähler classes that are close to the pullback of the Fubini--Study class.

29 pages, 2 figures

References in corpus (1)

• Blowing up extremal Kähler manifolds II