Moduli theory, stability of fibrations and optimal symplectic connections

K-polystability is, on the one hand, conjecturally equivalent to existence of certain canonical K\"ahler metrics polarised varieties, and, other gives correct notion form moduli. We introduce a stability for families K-polystable extending classical slope bundle, viewed as family varieties via associated projectivisation. conjecture that this is condition forming moduli fibrations. Our main result relates geometry: we prove an optimal symplectic connection implies semistability fibration. An choice fibrewise constant scalar curvature metric, satisfying geometric partial differential equation. such polystability finite dimensional analogue conjecture, by describing GIT problem fibrations embedded in fixed projective space, and showing zero moment map.