Extremal metrics on fibrations

Dervan, R. and Sektnan, L. M. (2020) Extremal metrics on fibrations. Proceedings of the London Mathematical Society, 120(4), pp. 587-616. (doi: 10.1112/plms.12297)

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Consider a fibred compact Kähler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature Kähler metric and has discrete automorphism group. Assuming the base of the fibration admits a twisted extremal metric where the twisting form is a certain Weil-Petersson type metric, we prove that X admits an extremal metric for polarisations making the fibres small. Thus X admits a constant scalar curvature Kähler metric if and only if the Futaki invariant vanishes. This extends a result of Fine, who proved this result when the base admits no continuous automorphisms. As consequences of our techniques, we obtain analogues for maps of various fundamental results for varieties: if a map admits a twisted constant scalar curvature Kähler metric metric, then its automorphism group is reductive; a twisted extremal metric is invariant under a maximal compact subgroup of the automorphism group of the map; there is a geometric interpretation for uniqueness of twisted extremal metrics on maps.

Item Type:Articles
Additional Information:R. Dervan received funding from the ANR grant ‘GRACK’. L. M. Sektnan received funding from CIRGET.
Glasgow Author(s) Enlighten ID:Dervan, Dr Ruadhaí
Authors: Dervan, R., and Sektnan, L. M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Proceedings of the London Mathematical Society
Publisher:London Mathematical Society
ISSN (Online):1460-244X
Published Online:11 September 2019
Copyright Holders:Copyright © 2019 Authors(s)
First Published:First published in Proceedings of the London Mathematical Society 120(4):587-616
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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