Approximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles

Abstract

Abstract. We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle E on a compact Kähler manifold, with structure group a connected linear algebraic reductive group G, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

Non-existence of Higgs fields on Calabi-Yau Manifolds

“…where F A h is the curvature of Chern connection A h , θ * ,h is the adjoint of θ with respect to the metric h. In [9] and [18], it is proved that a Higgs bundle admits the Hermitian-Einstein if only if it is polystable. In [4,5,6], it shows that a Higgs bundle is semistable if only if it is admits approximate Hermitian-Einstein structure, i.e., for every positive ε, there is a Hermitian metric h on E such that…”