The Einstein-Maxwell Equations, Kaehler Metrics, and Hermitian Geometry

TL;DR

This paper explores the relationship between constant-scalar-curvature Kähler metrics and Einstein-Maxwell solutions on complex surfaces, revealing new solutions via conformally Kähler geometry and highlighting limitations of existing correspondences.

Contribution

It demonstrates how cscK metrics relate to Einstein-Maxwell solutions and introduces new solutions through conformally Kähler geometry, expanding the known solution space.

Findings

cscK metrics correspond to Einstein-Maxwell solutions on complex surfaces

not all Einstein-Maxwell solutions arise from cscK metrics

new solutions are constructed using conformally Kähler geometry

Abstract

Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kaehler geometry.