arXiv subjects

Sergio A. H. Cardona

Publications and source records attributed to Sergio A. H. Cardona.

3 recordsLinked to original sources

arXiv2026

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.

math.DG↗

arXiv2020

Twistor sections of Dirac bundles

A Dirac bundle is a euclidean bundle over a riemannian manifold $M$ which is a compatible left $C\ell(M)$-module, together with a metric connection also compatible with the Clifford action in a natural way. We prove some vanishing theorems and introduce the twistor equation within this framework. In particular, we exhibit a characterization of solutions for this equation in terms of the Dirac operator $D$ and a suitable Weitzenböck-type curvature operator $\mathcal{R}$. Finally, we analyze the especial case of the Clifford bundle to prove existence of nontrivial solutions of the twistor equation on spheres.

math.DG↗

arXiv2019

On a functional of Kobayashi for Higgs bundles

We define a functional ${\cal J}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that ${\cal J}(h)$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating ${\cal J}(h)$ and another functional ${\cal I}(h)$, closely related to the Yang-Mills-Higgs functional \cite{Bradlow-Wilkin, Wentworth}, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of ${\cal J}(h)$, which is expressed as a certain $L^{2}$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of ${\cal J}(h)$ if and only if the corresponding Hitchin--Simpson mean curvature is parallel with respect to the Hitchin--Simpson connection.

math.DG↗