Year: 2026
Type: preprint
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 $\mathbb P(E)\toΣ_g$, 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. 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 $X$.
Source: arXiv (Cornell University)
Author: Sha, Zehao
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214216633