Bihermitian Gray surfaces (Q2496573)

From the introduction: Surfaces admitting two oppositely oriented complex structures are called bi-Hermitian surfaces. We give examples of bi-Hermitian compact surfaces \((M,g)\) whose Ricci tensor \(\rho\) satisfies \(\nabla_Z \rho(X,X)=\frac13 X\tau g(X,X)\). We construct one-parameter families of such metrics on all the Hirzebruch surfaces \(\Sigma_k\), for \(k\geq 0\).