PSC Kähler cones of minimal Kähler surfaces and a birational obstruction

Abstract

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