On orthogonal decompositions of hermitian Higgs bundles

TL;DR

This paper extends classical results on orthogonal decompositions and second fundamental forms from holomorphic hermitian vector bundles to hermitian Higgs bundles, providing new insights and alternative proofs, but also highlighting limitations in extending certain theorems.

Contribution

It generalizes key propositions about orthogonal decompositions and second fundamental forms to hermitian Higgs bundles, with new proofs and applications.

Findings

Classical propositions extend to hermitian Higgs bundles.

Alternative proofs without local computations are provided.

Some classical theorems do not extend straightforwardly to Higgs bundles.

Abstract

A hermitian Higgs bundle is a triple $({\mathfrak E},h) = (E,\Phi, h)$, where ${\mathfrak E}=(E,\Phi)$ is a Higgs bundle and $(E,h)$ is a holomorphic hermitian vector bundle. It is well-known that several results on holomorphic vector bundles extend to the Higgs bundles setting, although this is not always the case. In this article we show that some classical propositions, involving orthogonal decompositions of holomorphic hermitian vector bundles and the second fundamental form of its holomorphic subbundles, can be extended to hermitian Higgs bundles. The extended propositions concerning orthogonal decompositions have immediate applications in Higgs bundles, and we mention some of these throughout the article. Moreover, the extended propositions concerning the second fundamental form are generalizations of previously known results on Higgs bundles. In particular, here we include alternative proofs of these extended propositions without using local computations. Finally, as an application of the above results and due to the lack of a certain parallelism condition, we show that a classical theorem concerning the Kobayashi functional for holomorphic vector bundles does not admit a straightforward extension to Higgs bundles.