Angle deformation of K\"ahler-Einstein edge metrics on Hirzebruch surfaces

TL;DR

This paper constructs and analyzes a family of K"ahler-Einstein edge metrics on Hirzebruch surfaces, exploring their angle deformation and confirming a conjecture about convergence in certain cases, while also providing a rigid example.

Contribution

It introduces a new family of K"ahler-Einstein edge metrics on Hirzebruch surfaces and investigates their angle deformation properties, confirming a conjecture and answering a posed question.

Findings

Verification of Cheltsov-Rubinstein conjecture in special cases

Construction of a rigid K"ahler-Einstein edge metric example

Demonstration of convergence towards Calabi-Yau fibrations

Abstract

We construct a family of K\"ahler-Einstein edge metrics on all Hirzebruch surfaces using the Calabi ansatz and study their angle deformation. This allows us to verify in some special cases a conjecture of Cheltsov-Rubinstein that predicts convergence towards a non-compact Calabi-Yau fibration in the small angle limit. We also give an example of a K\"ahler-Einstein edge metric whose edge singularity is rigid, answering a question posed by Cheltsov.