Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface
arXiv:2302.07127 · doi:10.1016/j.bulsci.2023.103345
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.
32 Pages, AMS Article Style. Revised Preprint, Submitted to Bulletin des Sciences Mathématiques. Includes corrections of some errata and typos, removal of one old Subsection, addition of one new Section and addition of some new material. Supersedes Version 1. Any comments are most welcome