Conformally Kähler, Einstein–Maxwell geometry

  • Vestislav Apostolov

    UQAM, Montréal, Canada
  • Gideon Maschler

    Clark University, Worcester, USA
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On a given compact complex manifold or orbifold (M,J)(M,J), we study the existence of Hermitian metrics g~\tilde g in the conformal classes of Kähler metrics on (M,J)(M,J), such that the Ricci tensor of g~\tilde g is of type (1, 1) with respect to the complex structure, and the scalar curvature of g~\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)(M, J), and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of KK-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, KK-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].

Cite this article

Vestislav Apostolov, Gideon Maschler, Conformally Kähler, Einstein–Maxwell geometry. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1319–1360

DOI 10.4171/JEMS/862