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arXiv · 2302.07127

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

Rajas Sandeep Sompurkar
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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.

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Rajas Sandeep Sompurkar. 2023-02-14. Existence of Higher Extremal K\"ahler Metrics on a Minimal Ruled Surface. https://doi.org/10.1016/j.bulsci.2023.103345

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