Conformally Kähler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functional

Abstract

We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge Kähler class a conformally Kähler, Einstein--Maxwell metric, or more generally, a Kähler metric of constant $(\xi ,a,p)$-scalar curvature, then this metric minimizes the $(\xi ,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson and developed by Li and Sano--Tipler, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--Tønnesen-Friedman, we describe the Kähler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally Kähler, Einstein-Maxwell metrics.

References added, presentation improved

References in corpus (2)

• Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons
• Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

Cited by in corpus (2)

• Kähler metrics with constant weighted scalar curvature and weighted K-stability
• Einstein--Maxwell equations on 4-dimensional Lie algebras