  Consider a holomorphic submersion between compact K\"ahler manifolds, such
that each fibre admits a constant scalar curvature K\"ahler metric. When the
fibres admit continuous automorphisms, a choice of fibrewise constant scalar
curvature K\"ahler metric is not unique. An optimal symplectic connection is
choice of fibrewise constant scalar curvature K\"ahler metric satisfying a
geometric partial differential equation. The condition generalises the
Hermite-Einstein condition for a holomorphic vector bundle, through the induced
fibrewise Fubini-Study metric on the associated projectivisation.
  We prove various foundational analytic results concerning optimal symplectic
connections. Our main result proves that optimal symplectic connections are
unique, up to the action of the automorphism group of the submersion, when they
exist. Thus optimal symplectic connections are canonical relatively K\"ahler
metrics when they exist. In addition, we show that the existence of an optimal
symplectic connection forces the automorphism group of the submersion to be
reductive, and that an optimal symplectic connection is automatically invariant
under a maximal compact subgroup of this automorphism group. We also prove that
when a submersion admits an optimal symplectic connection, it achieves the
absolute minimum of a natural log norm functional which we define.
