On the existence problem of Einstein-Maxwell K\"ahler metrics

TL;DR

This paper reviews the existence of Einstein-Maxwell K"ahler metrics, extends related concepts to more general settings, and provides numerical evidence for the existence of toric K-stable pairs in specific blow-up cases.

Contribution

It extends volume minimization and K-stability concepts to a broader setting and analyzes the toric case with numerical methods for specific blow-up manifolds.

Findings

Extended volume minimization principle and K-stability to general set-up.

Numerical analysis suggests existence of toric K-stable pairs in specific blow-up cases.

Identification of a Killing vector field K related to stability in the toric case.

Abstract

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the K\"ahler class $\Omega$ is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of self-intersection number 1. We observe by numerical analysis that there should be a Killing vector field $K$ which gives a toric K-stable pair $(\Omega, K)$ in the sense of Apostolov-Maschler.