Bi-Hermitian Gray surfaces II (Q1001998)

A bi-Hermitian Gray surface is an irreducible Riemannian \(4\)-manifold \((M,g)\), with Ricci tensor \(\rho\), which satisfies \[ \nabla_X \rho(X,X)=\tfrac{1}{3}\, X(\text{Scal})\cdot g(X,X), \] for all vector fields \(X\), and which admits two oppositely oriented Hermitian complex structures commuting with \(\rho\). This paper is concerned with the classification of compact bi-Hermitian Gray surfaces with nonconstant scalar curvature and even first Betti number. The author first proves that any such manifold must be a ruled surface, and so they admit a \(\mathbb{CP}^1\) fibration over a curve of genus \(g\). The author then proceeds to classify them for \(g>0\).