Existence of Higher Extremal K\"ahler Metrics on a Minimal Ruled Surface

TL;DR

This paper proves that on a special minimal ruled surface, every Kähler class admits a higher extremal Kähler metric, but no constant scalar curvature Kähler metrics exist, highlighting differences in geometric structures.

Contribution

It demonstrates the existence of higher extremal Kähler metrics on a pseudo-Hirzebruch surface and shows the non-existence of hcscK metrics in any Kähler class on this surface.

Findings

Higher extremal Kähler metrics exist on the surface.

No hcscK metrics exist in any Kähler class.

Computations involve the top Bando-Futaki invariant.

Abstract

In this paper we prove that on a special type of minimal ruled surface, which is an example of a `pseudo-Hirzebruch surface', every K\"ahler class admits a certain kind of `higher extremal K\"ahler metric', which is a K\"ahler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal K\"ahler metric. From an already proven result, it will follow that this specific higher extremal K\"ahler metric cannot be a `higher constant scalar curvature K\"ahler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature K\"ahler (cscK) metric, to be a K\"ahler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any K\"ahler class on this surface.