Connections, metrics and Higgs fields on complex fiber bundles

TL;DR

This paper extends classical concepts of connections and flat bundles to nonlinear settings on complex fiber bundles, establishing new correspondences and characterizations in the context of Kähler manifolds and Higgs bundles.

Contribution

It generalizes Atiyah's work on holomorphic principal bundles, introduces a nonlinear analogue of Weil's flat connection criterion, and constructs a functor linking nonlinear flat and Higgs bundles.

Findings

Representation of extension classes via curvature for holomorphic fibrations

Nonlinear analogue of Weil's characterization of flat connections

Functorial correspondence between nonlinear flat and Higgs bundles

Abstract

We give a representation of the extension class associated to a holomorphic fibration by curvature, 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 bundles reductive of K\"ahler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of K\"ahler type. 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.