Existence of higher extremal Kähler metrics on a minimal ruled surface

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 Kähler class admits a certain kind of ‘higher extremal Kähler metric’, which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a ‘higher constant scalar curvature Kähler (hcscK) metric’, which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler 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 Kähler class on this surface.