Automorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal K\"ahler cases. Combined with previous result...

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Bibliographic Details
Published in:arXiv.org 2017-10
Main Author: Lahdili, Abdellah
Format: Article
Language:English
Subjects:
Automorphisms
Deformation
Fields (mathematics)
Online Access:Get full text
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Summary:We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal K\"ahler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally K\"ahler, Einstein--Maxwell metrics on \(\mathbb{{CP}}^1 \times \mathbb{{CP}}^1\). We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of \((K, q, a)\)-extremal K\"ahler metrics in a given K\"ahler class, and show that the existence of \((K, q, a)\)-extremal K\"ahler metrics is stable under small deformation of the K\"ahler class, the Killing vector field \(K\) and the normalization constant \(a\).
ISSN:2331-8422