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arXiv · 1910.09353

Hermitian metrics of constant Chern scalar curvature on ruled surfaces

Caner Koca·Mehdi Lejmi
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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}.

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Caner Koca, Mehdi Lejmi. 2019-10-21. Hermitian metrics of constant Chern scalar curvature on ruled surfaces. https://arxiv.org/abs/1910.09353

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