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

Indranil Biswas, John Loftin, Matthias Stemmler

Research output: Contribution to journalArticle

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 {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.

Original languageEnglish (US)
Pages (from-to)737-754
Number of pages18
JournalCommunications in Analysis and Geometry
Volume22
Issue number4
DOIs
StatePublished - Jan 1 2014

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All Science Journal Classification (ASJC) codes

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

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