### 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 one-parameter family {A_{t}}_{t∈ℝ} of ∞^{∞} automorphisms of E with the following property. Let ∇^{t} and φ^{t} be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism A_{t}. If θ^{t} denotes the extended connection form on E associated to the triple h, ∇^{t} and φ^{t}, then as t→ +infin;00, the connection form θ^{t} 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→ +infin;00, the curvature of θ^{t} converges in the ∞^{00} Frechet topology to the curvature of θ.

Original language | English (US) |
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Pages (from-to) | 737-754 |

Number of pages | 18 |

Journal | Communications in Analysis and Geometry |

Volume | 22 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty

### Cite this

*Communications in Analysis and Geometry*,

*22*(4), 737-754. https://doi.org/10.4310/CAG.2014.v22.n4.a5

}

*Communications in Analysis and Geometry*, vol. 22, no. 4, pp. 737-754. https://doi.org/10.4310/CAG.2014.v22.n4.a5

**Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold.** / Biswas, Indranil; Loftin, John; Stemmler, Matthias.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

AU - Biswas, Indranil

AU - Loftin, John

AU - Stemmler, Matthias

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 one-parameter family {At}t∈ℝ of ∞∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt 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→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.

AB - 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 one-parameter family {At}t∈ℝ of ∞∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt 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→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.

UR - http://www.scopus.com/inward/record.url?scp=84907528433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907528433&partnerID=8YFLogxK

U2 - 10.4310/CAG.2014.v22.n4.a5

DO - 10.4310/CAG.2014.v22.n4.a5

M3 - Article

AN - SCOPUS:84907528433

VL - 22

SP - 737

EP - 754

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 4

ER -