The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of the European Mathematical Society


Full-Text PDF (316 KB) | Metadata | Table of Contents | JEMS summary
Online access to the full text of Journal of the European Mathematical Society is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
subscriptions@ems-ph.org
Volume 21, Issue 5, 2019, pp. 1319–1360
DOI: 10.4171/JEMS/862

Published online: 2019-01-08

Conformally Kähler, Einstein–Maxwell geometry

Vestislav Apostolov[1] and Gideon Maschler[2]

(1) UQAM, Montréal, Canada
(2) Clark University, Worcester, USA

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of Kähler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type (1, 1) with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension 4, such Hermitian metrics provide a Riemannian counter-part 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].

Keywords: Einstein–Maxwell, Kähler metrics, conformally Kähler, Einstein metrics K-stability, Futaki invariant, toric geometry, ambitoric structures, orbifolds, ambikähler metrics, extremal metrics

Apostolov Vestislav, Maschler Gideon: Conformally Kähler, Einstein–Maxwell geometry. J. Eur. Math. Soc. 21 (2019), 1319-1360. doi: 10.4171/JEMS/862