The optimal symplectic connection conjecture for fibrations

Let $(X,H)\to(B,L)$ be a polarised fibration, with each fibre equipped with its fibrewise cscK geometry. An optimal symplectic connection is a fibrewise constant scalar curvature Kähler metric satisfying the associated geometric partial differential equation. Optimal symplectic connection conjecture. A fibration admits an optimal symplectic connection if and only if it is polystable. The paper proves that existence implies semistability; the conjectured converse and the upgrade from semistability to polystability remain open.