Constant scalar curvature K\"ahler metrics on rational surfaces

TL;DR

This paper classifies rational surfaces that admit constant scalar curvature K"ahler metrics, showing only the projective plane and quadric surface qualify, and identifies destabilizing configurations on others.

Contribution

It proves that only the projective plane and quadric surface admit such metrics among rational surfaces, and demonstrates destabilizing configurations on others.

Findings

Only the projective plane and quadric surface admit constant scalar curvature K"ahler metrics.

All other rational surfaces admit destabilizing slope test configurations.

Hirzebruch surfaces other than the quadric do not admit such metrics.

Abstract

We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature K\"ahler metric for every K\"ahler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rational surfaces other than those two admit a destabilising slope test configuration for some polarization, as introduced by Ross and Thomas. We further show that all Hirzebruch surfaces other than the quadric surface and all rational surfaces with Picard rank $3$ do not admit a constant scalar curvature K\"ahler metric in any K\"ahler class.