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



Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

John Loftin (Department of Mathematics and Computer Science, Rutgers University at Newark, New Jersey, U.S.A.)

Matthias Stemmler (Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany)


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 $\widehat{\theta}$.

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