Poisson--K\"ahler fibration I: curvature of the base manifold

We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula of a Weil--Petersson type metric for general relative K\"ahler fibrations. In particular, our curvature formula implies a Burns type negative curvature property of the base manifold for a special class of maximal variation K\"ahler fibrations (named Poisson--K\"ahler fibrations).Comment: One correction: stability in Theorem B should be polystabilit