Hermitian metrics of constant Chern scalar curvature on ruled surfaces

TL;DR

This paper constructs Hermitian metrics with constant Chern scalar curvature on Hirzebruch and ruled surfaces, including positive and zero curvature cases, expanding understanding beyond known Kähler metrics.

Contribution

It introduces new methods to construct Hermitian metrics with constant Chern scalar curvature on ruled surfaces, including cases not admitting Kähler metrics.

Findings

Constructed Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces.

Built Hermitian metrics of zero Chern scalar curvature on some ruled surfaces.

Discussed existence of critical metrics of total Chern scalar curvature in conformal classes.

Abstract

It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature K\"ahler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page--B\'erard-Bergery's ansatz \cite{P78,B82}. We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by Gauduchon in \cite{G80,G84}.