Mathematics
Conformally Kähler, Einstein-Maxwell geometry
Document Type
Article
Abstract
On a given compact complex manifold or orbifold (M, J), we study the existence of Hermitian metrics Q g in the conformal classes of Kähler metrics on (M, J), such that the Ricci tensor of g is of type (1, 1) with respect to the complex structure, and the scalar curvature of g is constant. In real dimension 4, such Hermitian metrics provide a Riemannian counterpart of the Einstein-Maxwell equations in general relativity, and have been recently studied in [3, 34, 35, 33]. We show how the existence problem of such Hermitian metrics (which we call in any dimension conformally Kähler, Einstein-Maxwell metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki [22, 25] in the constant scalar curvature Kähler case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally Kähler, Einstein-Maxwell metrics invariant under a certain group of automorphisms which are associated to a given Kähler class, a real holomorphic vector field on (M, J), and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of K-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally Kähler, Einstein-Maxwell metrics. We use the methods of [4] to show that on a compact symplectic toric 4-orbifold with second Betti number equal to 2, K-polystability is also a sufficient condition for the existence of (toric) conformally Kähler, Einstein-Maxwell metrics, and the latter are explicitly described as ambitoric in the sense of [3]. As an application, we exhibit many new examples of conformally Kähler, Einstein-Maxwell metrics defined on compact 4-orbifolds, and obtain a uniqueness result for the construction in [34].
Publication Title
Journal of the European Mathematical Society
Publication Date
2019
Volume
21
Issue
5
First Page
1319
Last Page
1360
ISSN
1435-9855
DOI
10.4171/JEMS/862
Keywords
Ambikähler metrics, ambitoric structures, conformally Kähler, Einstein metrics K-stability, Einstein-Maxwell, extremal metrics, Futaki invariant, Kähler metrics, orbifolds, toric geometry
Repository Citation
Apostolov, Vestislav and Maschler, Gideon, "Conformally Kähler, Einstein-Maxwell geometry" (2019). Mathematics. 32.
https://commons.clarku.edu/mathematics/32
