Moment maps and stability of holomorphic submersions

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.