I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature.

There are of course the Fano (del Pezzo) Kähler-Einstein surfaces, all of which are obtained by blowing up $\mathbb C\mathbb P^2$ at finitely many distinct points, or $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$. In fact, $\mathbb C\mathbb P^1 \times C$ where $C$ is a compact Riemann surface admits cscK metrics of any sign by taking product metrics. *What are some other examples of non-KE positive cscK surfaces?*

*I would be particularly interested in examples which have holomorphic vector fields* since for example, by a result of Lebrun-Simanca, any scalar flat but not Ricci flat Kähler surface with no holomorphic vector fields also admits a positive cscK metric. Once one has a positive cscK surface with no holomorphic vector fields, one can apply results of Arezzo-Pacard to get a positive cscK metric on the blow up of any finite set of points.

Note that the existence of a positive cscK metric would imply that the surface has Kodaira dimension $-\infty$ (for a proof see the paper by Lebrun-Simanca linked above). Thus by the Kodaira-Enriques classification any such surface would be obtained by a sequence of blowups of $\mathbb C\mathbb P^2$ or a geometrically ruled surface (a holomorphic $\mathbb C\mathbb P^1$ bundle over a compact Riemann surface, or equivalently the projectivization of any rank 2 holomorphic vector bundle on a compact Riemann surface). Any ideas on how to construct the requested examples or direction to a resource containing such examples would be greatly appreciated!