On the Existence Problem of Einstein–Maxwell Kähler Metrics

Abstract

In this expository paper we review on the existence problem of Einstein–Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein–Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric Kstability 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 Kähler class Ω is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of selfintersection number 1. We observe by numerical analysis that there should be a Killing vector field K which gives a toric K-stable pair (Ω,K) in the sense of Apostolov–Maschler.