2020/12/08 by Michael Hallam, Hallam, Michael · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2012.04416
We derive the geodesic equation for relatively Kähler 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.