The analytic moduli space of holomorphic submersions

Abstract We construct a moduli space that parametrises stable proper holomorphic submersions over a fixed compact Kähler base. Stability is described in terms of the existence of a canonical relatively Kähler metric on the submersion, called an optimal symplectic connection. The construction of the moduli space combines techniques from geometric invariant theory with the study of the geometric PDE defining an optimal symplectic connection. A special case of this moduli space is the moduli space of stable vector bundles over a compact Kähler manifold. We also show that the moduli space is a Hausdorff complex space equipped with a Weil-Petersson type Kähler metric.