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Extremal metrics on fibrations

Accepted version
Peer-reviewed

Type

Article

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Authors

Dervan, Ruadhai 
Sektnan, Lars Martin 

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.

Description

Keywords

Journal Title

Proceedings of the London Mathematical Society

Conference Name

Journal ISSN

0024-6115
1460-244X

Volume Title

120

Publisher

Wiley-Blackwell

Rights

All rights reserved
Sponsorship
CIRGET and ANR funding (Canadian and French governmental agencies).