Connections, metrics and Higgs fields on complex fiber bundles

We give a differential-geometric representation of the extension class associated to a holomorphic fibration, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat holomorphic bundles reductive of Kähler type to the category of nonlinear Higgs bundles over the same connected compact Kähler base. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.