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# Communications in Analysis and Geometry

## Volume 22 (2014)

### Number 4

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

Pages: 737 – 754

DOI: http://dx.doi.org/10.4310/CAG.2014.v22.n4.a5

#### Authors

#### Abstract

Given a flat Higgs vector bundle $(E, \nabla , \varphi)$ over a compact connected special affine manifold, we first construct a natural filtration of $E$, compatible with both $\nabla$ and $\varphi$, 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 one-parameter family $\{ A_t \}_{t \in \mathbb{R}}$ of $C^{\infty}$ automorphisms of $E$ with the following property. Let $\nabla^t$ and $\varphi^t$ be the flat connection and flat Higgs field, respectively, on $E$ constructed from $\nabla$ and $\varphi$ using the automorphism $A_t$. If $\theta^t$ denotes the extended connection form on $E$ associated to the triple $h$, $\nabla^t$ and $\varphi^t$, then as $t \to + \infty$, the connection form $\theta^t$ converges in the $C^{\infty}$ Fréchet topology to the extended connection form $\widehat{\theta}$ 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 $t \to + \infty$, the curvature of $\theta^t$ converges in the $C^{\infty}$ Fréchet topology to the curvature of $\widehat{\theta}$.