Canonical metrics on holomorphic fibre bundles

TL;DR

This paper characterizes when optimal symplectic connections, a type of canonical metric, exist on isotrivial Kähler fibrations, linking their existence to polystability of associated principal bundles and generalizing Hermite--Einstein equations.

Contribution

It provides a complete description of the existence of optimal symplectic connections on isotrivial Kähler fibrations, connecting stability conditions to metric existence.

Findings

Optimal symplectic connections exist if and only if the principal bundle is polystable.

Many new examples of cscK metrics are generated on holomorphic fibre bundles.

The optimal symplectic connection equation generalizes the Hermite--Einstein equation for fibrations.

Abstract

In this article we completely describe the existence of canonical metrics, known as optimal symplectic connections, on isotrivial K\"ahler fibrations. In this setting an optimal symplectic connection is induced from a Hermite--Einstein connection on the holomorphic principal bundle of relative automorphisms, and the Hitchin--Kobayashi correspondence asserts the existence of such a connection precisely when the principal bundle is polystable. Combined with results of Dervan and Sektnan this generates many new examples of cscK metrics on the total space of holomorphic fibre bundles. Our results indicate that in general the optimal symplectic connection equation should be viewed as a generalisation of the Hermite--Einstein equation to holomorphic fibrations where the complex structure of the fibres varies.