Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles (Q2254172)

Let \(X\) be a compact Kähler manifold. A Higgs bundle is a holomorphic bundle \(E\) over \(X\) with a Higgs field \(\varphi : E \to E \otimes \Omega^1_X\), that is a holomoprphic \(\mathrm{End}(E)\)-valued 1-form with vanishing associated 2-form \(\varphi\wedge \varphi \). A Hermitian fibre metric \(h\) defines the canonical Chern connection \(\nabla\) in \(E\). It is called a Hermitian-Yang-Mills connection if the mean curvature endomorphism \(K_h \) of the connection (which is the contraction of the curvature of \(\nabla\) with the Kähler form) is a scalar endomorphism. It is said that a Higgs bundle \((E, \varphi)\) admits an approximate Hermitian-Yang-Mills structure if for any \(\epsilon >0\) there exists a metric with mean curvature \(K_h\) such that \(|K_h - c \mathrm{Id}|< \epsilon \) where \(c\) in an appropriate constant. The generalized Hitchin-Kobayashi correspondence states that a Higgs vector bundle is an approximate Hermitian-Yang-Mills structure if and only if it is semistable. The authors prove the similar result for principal Higgs bundle, which is defined as a principal \(G\)-bundle \(\pi : P \to X\) together with a holomorphic section \(\varphi \in \mathrm{Ad} E \otimes \Omega^1_X\) such that \([\varphi, \varphi] =0\) in \(\mathrm{Ad} E \otimes \Omega^2_X\). The authors define for principal Higgs bundle notions of approximate Hermitian-Yang-Mills structure and semistability and prove the following theorem: A principal Higgs \(G\)-bundle over a Kähler manifold \(X\) is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.