Automorphisms and connections on Higgs bundles over compact Kähler manifolds

Let (E,φ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of E by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by G the direct sum of these stable quotients, and let the singular set of G be called S ⊂ X. We construct a 1-parameter family of filtration preserving ^{C ∞} isomorphisms Φt:G|XS→E|XS, and a Hermitian metric h on G|X\S, such that as t → + ∞ , the Chern connection for the Hermitian Higgs bundle (Φt*E,Φt*φ,h)|X\S converges, in the ^{C ∞} Fréchet topology over any relatively compact open subset of X \ S, to the direct sum of the Yang-Mills-Higgs connections on the direct summands in G.We also prove an analogous result for principal Higgs G-bundles on X.