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)

Abstract

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.
• Status: Published
• Refereed: Yes
• 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: 0024-6115
• 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