An investigation of stability on certain toric surfaces

TL;DR

This paper explores the connection between stability conditions and the existence of extremal Kähler metrics on certain toric surfaces, providing computable criteria for stability in quadrilaterals and implications for metric existence.

Contribution

It introduces a new framework for analyzing stability on toric surfaces with quadrilateral moment polytopes, including a computable stability criterion and stable region characterization.

Findings

Derived a stability criterion for quadrilaterals with specific weights.

Identified a stable region for generic quadrilaterals.

Connected stability conditions to the existence of extremal Kähler metrics.

Abstract

We investigate the relationship between stability and the existence of extremal K\"ahler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. We introduce a space of symplectic potentials for toric manifolds, which induces metrics with mixed Poincar\'e type and cone angle singularities. For quadrilaterals, we give a computable criterion for stability with 0 weights along two of the edges of the quadrilateral. This in turn implies the existence of a definite log-stable region for generic quadrilaterals. This uses constructions due to Apostolov-Calderbank-Gauduchon and Legendre.