Extremal metrics on fibrations

TL;DR

This paper proves the existence of extremal metrics on fibred Kähler manifolds under certain conditions, extending previous results and providing new insights into automorphism groups and metric invariance.

Contribution

It extends Fine's result by establishing extremal metrics on fibrations with a twisted extremal base, linking scalar curvature and automorphism groups.

Findings

X admits an extremal metric when fibres are small and base has a twisted extremal metric.

X admits a constant scalar curvature Kähler metric iff the Futaki invariant vanishes.

Automorphism groups are shown to be reductive under certain metric conditions.

Abstract

Consider a fibred compact K\"ahler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature K\"ahler 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 K\"ahler 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 K\"ahler 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.