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Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

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 \(C^\infty \) Hermitian structure on \(E_G\) together with a \(1\)-parameter family of \(C^\infty \) automorphisms \(\{F_t\}_{t\in \mathbb R }\) of the principal \(G\)-bundle \(E_G\) with the following property: Let \(\nabla ^t\) 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 \(t\rightarrow -\infty \), the connection \(\nabla ^t\) converges in \(C^\infty \) 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 \(t\rightarrow -\infty \), the curvature of \(\nabla ^t\) converges in \(C^\infty \) 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 \(\{F_t\}_{t\in \mathbb R }\) is constructed by generalizing the method of [6]. Given a holomorphic vector bundle \(E\) on \(X\), in [6] a \(1\)-parameter family of \(C^\infty \) automorphisms of \(E\) is constructed such that as \(t\rightarrow -\infty \), the curvature converges, in \(C^0\) topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.

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Correspondence to Indranil Biswas.

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Biswas, I., Bradlow, S.B., Jacob, A. et al. Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface. Ann Glob Anal Geom 44, 257–268 (2013). https://doi.org/10.1007/s10455-013-9365-1

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Keywords

  • Hermitian–Einstein connection
  • Principal bundle
  • Parabolic subgroup
  • Atiyah bundle
  • Automorphism

Mathematics Subject Classification (2000)

  • 53C07
  • 32L05