Hermitian metrics of constant Chern scalar curvature on ruled surfaces

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}.