## Abstract

Let X be a compact connected Riemann surface and G a connected reductive complex affine algebraic group. Given a holomorphic principal G-bundle E_{G} over X, we construct a C^{∞} Hermitian structure on E_{G} together with a 1-parameter family of C_{∞} automorphisms {F_{t}}_{t∈ℝ} of the principal G-bundle E_{G} with the following property: Let ∇^{t} be the connection on E_{G} corresponding to the Hermitian structure and the new holomorphic structure on E_{G} constructed using F_{t} from the original holomorphic structure. As t → -∞, the connection ∇^{t} converges in C^{∞} Fréchet topology to the connection on E_{G} given by the Hermitian-Einstein connection on the polystable principal bundle associated to E_{G}. In particular, as t → -∞, the curvature of ∇^{t} converges in C^{∞} Fréchet topology to the curvature of the connection on E_{G} given by the Hermitian-Einstein connection on the polystable principal bundle associated to E_{G}. The family {F_{t}}_{t∈ℝ} is constructed by generalizing the method of [6]. Given a holomorphic vector bundle E on X, in [6] a 1-parameter family of C^{∞} automorphisms of E is constructed such that as t → -∞, the curvature converges, in C^{0} topology, to the curvature of the Hermitian-Einstein connection of the associated graded bundle.

Original language | English (US) |
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Pages (from-to) | 257-268 |

Number of pages | 12 |

Journal | Annals of Global Analysis and Geometry |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2013 |

## Keywords

- Atiyah bundle
- Automorphism
- Hermitian-Einstein connection
- Parabolic subgroup
- Principal bundle

## ASJC Scopus subject areas

- Analysis
- Political Science and International Relations
- Geometry and Topology