A note on blow-ups of toric surfaces and cscK metrics

TL;DR

This paper demonstrates that by performing a specific sequence of torus-equivariant blow-ups on a compact toric surface, the resulting surface can admit a constant scalar curvature Kähler (cscK) metric, expanding the understanding of metric existence on toric surfaces.

Contribution

It introduces a method to obtain cscK metrics on blown-up toric surfaces through a sequence of equivariant blow-ups, providing new insights into the metric geometry of toric surfaces.

Findings

Existence of cscK metrics on certain blown-up toric surfaces

Construction of a sequence of equivariant blow-ups leading to cscK metrics

Extension of known results on Kähler metrics to toric surface blow-ups

Abstract

Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.