Abstract
We generalize the Hitchin–Kobayashi correspondence between semistability and the existence of approximate Hermitian–Yang–Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle \({\mathfrak {E}}\) on a compact Kähler manifold, with structure group a connected linear algebraic reductive group \(G\), is semistable if and only if it admits an approximate Hermitian–Yang–Mills structure.
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Notes
Note that if \(\tau \) is an element in the centre \({\mathfrak {z}}\) of \({\mathfrak {g}}\), the constant equivariant morphism \(f_\tau :E \rightarrow {\mathfrak {g}}\), \(f_\tau (u) =\tau \), defines a section of \(\mathrm{Ad }E\).
This is a general feature: if \(E\rightarrow X\) is a principal Higgs \(G\)-bundle, \(G'\subset G''\subset G\) are nested subgroups, and \({\mathfrak {E}}'\) is a reduction of the structure group of \(E\) to \(G'\), the latter corresponds to a section \(X \rightarrow E/G'\); composing with the morphism \(E/G'\rightarrow E/G''\), this gives a reduction \(E''\) of \(E\) to \(G''\). Note that \(E'' \simeq E' \times _{G'}G''\).
Actually this space should be suitably completed using Sobolev norms, but we shall omit these details here.
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This research was partly supported by GNSAGA-Istituto Nazionale per l’Alta Matematica and PRIN “Geometria delle varietà algebriche”.
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Bruzzo, U., Graña Otero, B. Approximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles. Ann Glob Anal Geom 47, 1–11 (2015). https://doi.org/10.1007/s10455-014-9433-1
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DOI: https://doi.org/10.1007/s10455-014-9433-1
Keywords
- Principal (Higgs) bundles
- Semistability
- Approximate Hermitian–Yang–Mills structures
- Hermitian–Yang–Mills metrics