Extremal metrics on fibrations

Consider a fibred compact Kahler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature Kahler (cscK) 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 cscK 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 cscK 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.