On orthogonal decompositions of hermitian Higgs bundles

TL;DR

The paper develops Higgs-category extensions of fundamental holomorphic-bundle results, focusing on orthogonal decompositions and the second fundamental form for Hermitian Higgs bundles ${\mathfrak E}=(E,\Phi,h)$. It establishes Higgs versions of Gauss-Codazzi relations and shows that, under suitable invariance conditions, Higgs subbundles yield holomorphic or Higgs orthogonal decompositions with appropriately defined connections ${\mathcal{D}}_h$, ${\mathcal{D}}_{h,S}$, and ${\mathcal{D}}_{h,Q}$. It also analyzes the Kobayashi functional in the Higgs setting, highlighting that the lack of parallelism between the Hitchin-Simpson connection and the metric obstructs a direct extension of the classical decomposition theorem to Higgs bundles. Overall, the work provides computation-free Higgs-generalizations of classical results and clarifies structural differences from the non-Higgs case, with implications for both complex geometry and gauge-theoretic physics.

Abstract

A hermitian Higgs bundle is a triple $({\mathfrak E},h) = (E,Φ, h)$, where ${\mathfrak E}=(E,Φ)$ 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.

Theorems & Definitions (19)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Proposition 5
• Proposition 6