Moment maps and stability of holomorphic submersions

TL;DR

This paper establishes a moment map framework for certain Kähler metrics on holomorphic fibrations, linking the existence of optimal symplectic connections to a stability condition akin to K-stability, and proves stability persists under deformations.

Contribution

It introduces a finite-dimensional moment map perspective for optimal symplectic connections and connects their existence to a fibration-specific K-stability, extending stability results to deformations.

Findings

Finite-dimensional moment map property for optimal symplectic connections.

Existence of zeroes of the moment map corresponds to stability of the fibration.

Stability of a fibration with an optimal symplectic connection persists under deformations.

Abstract

We prove a finite-dimensional moment map property for certain canonical relatively K\"ahler metrics on holomorphic fibrations, called optimal symplectic connections. We then relate the existence of zeroes of this moment map to the stability of the fibration, where the stability property we consider is a version of K-stability that takes into account the fibration structure, first introduced by Dervan--Sektnan. In particular, we prove that a stable deformation of a fibration admitting an optimal symplectic connection still admits an optimal symplectic connection, through a new approach using the finite-dimensional moment map properties and the moment map flow. We include an appendix with a proof of a result considered by Sz\'ekelyhidi that a K-polystable deformation of a constant scalar curvature K\"ahler manifold still admits a constant scalar curvature metric, using the same technique.