Nonlinear Hodge correspondence for morphisms

We study Higgs sections, flat sections, and their higher-dimensional and functorial analogues in the setting of nonlinear harmonic bundles.

For a harmonic vector bundle over a compact Kähler manifold, a global section is flat if and only if it is a Higgs section.

We show that for general nonlinear harmonic bundles this equivalence requires the vanishing of a degree obstruction.

We then extend the result to sub-fibrations and to morphisms by interpreting morphisms as graph sub-fibrations in a fiber product.