On orthogonal decompositions of hermitian Higgs bundles

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.

💡 Research Summary

The paper investigates how classical results concerning orthogonal decompositions of holomorphic Hermitian vector bundles extend to the setting of Hermitian Higgs bundles. A Hermitian Higgs bundle is a triple ((E,\Phi,h)) where ((E,\Phi)) is a Higgs bundle (i.e., a holomorphic vector bundle equipped with a holomorphic Higgs field (\Phi\in H^{0}(M,\Omega^{1,0}!\otimes!\operatorname{End}E))) and (h) is a Hermitian metric on the underlying holomorphic bundle. The authors begin by recalling the standard theory of Chern connections (D_h), curvature (R_h), and mean curvature operator (K_h) for a holomorphic Hermitian vector bundle ((E,h)). They restate two fundamental propositions from Kobayashi’s book: (1) if a (C^{\infty}) complex subbundle (E’\subset E) is invariant under the Chern connection, then its orthogonal complement (E’’) is also invariant, and the decomposition (E=E’\oplus E’’) is holomorphic; (4) if the second fundamental form of a holomorphic subbundle vanishes identically, then the orthogonal complement is holomorphic, yielding a holomorphic orthogonal splitting.

The core contribution is to lift these statements to the Higgs context. The appropriate connection is the Hitchin–Simpson connection (D_{h,\Phi}=D_h+