It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric [6, 22, 38]. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page-Bérard-Bergery's ansatz [41, 14]. 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 [26, 27].
"Hermitian metrics of constant Chern scalar curvature on ruled surfaces." Kodai Math. J. 43 (3) 409 - 430, October 2020. https://doi.org/10.2996/kmj/1605063622