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

We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant \((\xi, a, p)\)-scalar curvature, then this metric minimizes the \((\xi,a,p)...

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Bibliographic Details
Published in:arXiv.org 2018-09
Main Author: Lahdili, Abdellah
Format: Article
Language:English
Subjects:
Curvature
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Summary:We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge K\"ahler class a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler 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 K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally K\"ahler, Einstein-Maxwell metrics.
ISSN:2331-8422