PSC K\"ahler cones of minimal K\"ahler surfaces and a birational obstruction

Abstract

We characterize the positive scalar curvature K\"ahler cone of every minimal compact K\"ahler surface with Kodaira dimension $-\infty$. By the Enriques--Kodaira classification, every such surface is either $\mathbb{P}^2$ or a geometrically ruled surface $\mathbb{P}(E)\to\Sigma_g$, where $E$ is a rank-two holomorphic vector bundle over a compact Riemann surface of genus $g$. On $\mathbb{P}^2$, every K\"ahler class admits a K\"ahler metric of positive scalar curvature, whereas on $\mathbb{P}(E)$, every K\"ahler class of positive total scalar curvature admits a metric of positive scalar curvature if and only if $g\le1$ or $E$ is slope-semistable. We further prove that if a smooth compact K\"ahler surface admits a birational morphism onto a ruled surface $\mathbb{P}(E)\to\Sigma_g$, where $g\ge2$ and $E$ is slope-unstable, then it contains a K\"ahler class of positive total scalar curvature admitting no positive scalar curvature K\"ahler metric.