Positive scalar curvature Kähler cones on compact complex surfaces

Abstract

We determine the positive scalar curvature Kähler cone of every minimal compact Kähler surface of 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 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 $g\le 1$ or $E$ is slope-semistable. We further prove that if a smooth compact Kähler surface admits a birational morphism onto a ruled surface $\mathbb{P}(E)\to {\Sigma }_g$, where $g\ge 2$ and $E$ is slope-unstable, then it contains a Kähler class of positive total scalar curvature admitting no positive scalar curvature Kähler metric. Finally, for every smooth compact Kähler surface $X$ birational to $Y={\mathbb{P}}^2$ or $Y=\mathbb{P}(E)$, we show that Kähler classes admitting positive scalar curvature metrics on $Y$ are exactly the pushforwards of classes admitting such metrics on $X$.

arXiv comment25 pages; improved organization and notation; added Theorem 3