Geodesics in the space of relatively K\"ahler metrics

TL;DR

This paper develops the theory of geodesics in the space of relatively K"ahler metrics on fibrations, proving existence, uniqueness, and stability results, with applications to optimal symplectic connections and fibration degenerations.

Contribution

It derives the geodesic equation for relatively K"ahler metrics, proves uniqueness of geodesics, and establishes stability criteria related to optimal symplectic connections.

Findings

Unique smooth geodesics connect fiberwise constant scalar curvature metrics.

Convexity of the log-norm functional along geodesics is established.

Fibrations with optimal symplectic connections are shown to be polystable under certain degenerations.

Abstract

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm functional for this setting along geodesics, which yields simple proofs of Dervan and Sektnan's uniqueness result for optimal symplectic connections and a boundedness result for the log-norm functional. Next, we associate to a fibration degeneration a unique geodesic ray defined on a dense open subset. Calculating the limiting slope of the log-norm functional along a globally defined smooth geodesic ray, we prove that fibrations admitting optimal symplectic connections are polystable with respect to a large class of fibration degenerations that are smooth over the base. We give examples of such degenerations in the case of projectivised vector bundles and isotrivial fibrations.