The geometry of holomorphic submersions, their deformations and moduli

Proper holomorphic submersions can be viewed as both generalising holomorphic vector bundles and as a way of studying families of smooth projective varieties. We consider submersions whose fibres are analytically K-semistable, thus they each admit a degeneration to a Kaehler manifold with constant scalar curvature. On such holomorphic submersions, we introduce and study certain canonical relatively Kaehler metrics, called optimal symplectic connections, which generalise Hermite-Einstein connections for vector bundles and are defined as solutions to a geometric partial differential equation. Using optimal symplectic connections, we first give a general construction of extremal metric on the total space, in adiabatic classes, generalising results of Dervan-Sektnan, Fine, Hong. We then construct an analytic moduli space of holomorphic submersions admitting an optimal symplectic connection. To do so, we develop a deformation theory of holomorphic submersions and we combine techniques from geometric invariant theory with the study of the analytic properties of the optimal symplectic connection equation. We also show that the moduli space is a Hausdorff complex space which admits a Weil-Petersson type Kaehler metric.