K-stability and parabolic stability

TL;DR

This paper establishes the equivalence between parabolic polystability, K-polystability, and the existence of constant scalar curvature Kähler metrics on certain iterated blowups of ruled surfaces, under specific polarization conditions.

Contribution

It proves the equivalence of three stability notions and metric existence for iterated blowups of ruled surfaces with rational weights.

Findings

Parabolic polystability, K-polystability, and cscK metrics are equivalent in this setting.

The results hold for polarizations near the boundary of the Kähler cone.

Provides a link between algebraic stability and differential geometry in complex surfaces.

Abstract

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature K\"ahler metrics on the iterated blowup are equivalent, for certain polarizations close to the boundary of the K\"ahler cone.