Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

Abstract

Given a flat Higgs vector bundle (E ,∇ , φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder–Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth 1–parameter family {At}t∈R of C∞ automorphisms of E with the following property. Let ∇ and φ be the flat connection and flat Higgs field respectively on E constructed from ∇ and φ using the automorphism At. If θ denotes the extended connection form on E associated to the triple h, ∇ and φ, then as t −→ +∞, the connection form θ converges in the C∞ Frechet topology to the extended connection form θ on E given by the affine Yang–Mills–Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t −→ +∞, the curvature of θ converges in the C∞ Frechet topology to the curvature of θ.