Existence of approximate Hermitian–Einstein structures on semistable principal bundles

Abstract

Let E G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over C. We show that E G is semistable if and only if it admits approximate Hermitian-Einstein structures.

Kobayashi—Hitchin correspondence for twisted vector bundles

“…As π − (Z) has measure zero and X \ Z is isomorphic, via π, to X \ π − (Z), the same argument presented in section 6. 5 We now prove that there are two constants C , C > such that…”

“…By de nition we have has a smooth solution, denoted φϵ. We now let hϵ := e φϵ h, which is a Hermitian metric on S (since it is a conformal change of h. The evolution equation (5) has then a unique smooth solution hϵ : [ , +∞) −→ Herm + ( S) such that hϵ( ) = hϵ. We let R ϵ,t be the Chern curvature of ( S, h ϵ,t ), where h ϵ,t = hϵ(t), and K ϵ,t will be its mean curvature.…”

“…Several other versions of the (approximate) Kobayashi-Hitchin correspondence have since appeared in other contexts: with no pretention of completeness we may cite re exive sheaves ( [2]), Higgs bundles ( [22], [45]), parabolic bundles ( [31], [44]), parabolic Higgs bundles ( [46], [4], [37], [38]), principal G−bundles ( [1], [5], [6]), principal pairs (see [3], [40], [8]), analitically stable bundles ( [45], [39]), holomorphic pairs ( [7], [48], [24]), generalized holomorphic vector bundles ( [20]). See moreover [36] (and references therein) for a universal treatment of the Kobayashi-Hitchin correspondence in several contexts.…”

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

“…We can now prove Theorem 4.1. Following the argument in [6] we can reduce to the case of semisimple structure group. If G is semisimple, the center z of the Lie algebra g is trivial and the formula in the thesis becomes |K σ ξ ,φ | < ξ .…”

“…If the principal Higgs bundle E = (E, φ) admits an approximate Hermitian-Yang-Mills structure, one shows that the same holds for the adjoint Higgs bundle Ad(E); the latter then is semistable [8], so that E is semistable as well. The proof of the converse result relies on an analysis of the flow of the Donaldson functional defined on the space of Hermitian metrics on the Higgs bundle Ad(E), showing that the flow preserves the condition that an Hermitian metric on Ad(E) comes from a reduction of the structure group of E to a maximal compact subgroup K (this implements in the case of principal Higgs bundles the ideas used in [6] for principal bundles, but we provide here a more detailed description of this technique).…”