The geometry and stability of fibrations

This thesis concerns two topics. First is the existence of optimal symplectic connections on holomorphic submersions, and links to algebro-geometric stability of fibrations. The main result is that a fibration admitting an optimal symplectic connection is stable with respect to a large class of fibration degenerations. The second topic is the deformation theory and families of Kähler metrics with weighted constant scalar curvature. Here we study the existence of such metrics when either the complex structures or the weight functions vary, and also prove the existence of a weighted Weil--Petersson metric associated to a family of these metrics. The results are new even in the fibrewise extremal and Kähler--Ricci soliton settings.