Jordan-Hölder reductions for principal Higgs bundles on curves
type: Item
instance of: scholarly article
title: Jordan-Hölder reductions for principal Higgs bundles on curves
zbMATH Open document ID: 1207.14026
author: Beatriz Graña Otero
published in: Journal of Geometry and Physics
publication date: 2010-09-24T00:00:00Z
review text: Let \(C\) be a smooth connected projective curve. It is well-know that, given a semistable vector bundle \(V\) over \(C\), it admits a filtration \[ 0=V_0\subset V_1\subset\dots\subset V_k=V \] of vector subbundles, such that each quotient \(V_{i+1}/V_i\) is a stable vector bundle of the same slope as \(V\). The associated graded object is \(\mathrm{gr}(V)=\bigoplus_iV_{i+1}/V_i\). In case \(V\) is polystable, then \(\mathrm{gr}(V)\cong V\). The is the so-called Jordan-Hölder reduction for the vector bundle \(V\). In [\textit{A. Ramanathan}, Proc. Indian Acad. Sci., Math. Sci. 106, No.3, 301-328 (1996; Zbl 0901.14007)] it is proved the existence of a Jordan-Hölder reduction for every semistable principal \(G\)-bundle over the curve \(C\), where \(G\) is a reductive complex Lie group. If \(E\) is the bundle of linear frames of a rank \(n\) vector bundle \(V\), then the structure group of \(E\) is \(\mathrm{GL}(n,\mathbb{C})\), and giving a filtration of \(V\) is the same as giving a reduction of structure group of \(E\) to a parabolic subgroup \(P\subset\mathrm{GL}(n,\mathbb{C})\). In this case, a Jordan-Hölder reduction is then a certain reduction of structure group to a parabolic subgroup \(P\), such that the principal bundle obtained by extending the structure group to the Levi factor \(L(P)\) is stable. Later, \textit{C. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003)] proved the existence of a Jordan-Hölder for semistable Higgs bundles. On the curve \(C\), a Higgs bundle is a pair \((V,\phi)\) where \(V\) is a vector bundle and \(\phi\) is a section of \(\mathrm{End}(V)\otimes\Omega^1\). In this case, a Jordan-Hölder reduction is a filtration of \(E\) as above but such that each \(V_i\) is \(\phi\)-invariant i.e. \(\phi(V_i)\subset V_i\otimes\Omega^1\). This main result of this paper is the proof of the existence of a Jordan-Hölder reduction for every semistable Higgs \(G\)-bundle over \(C\), where \(G\) is a complex reductive Lie…
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 5791219
zbMATH Keywords: principal Higgs bundles
zbMATH Keywords: semistability
zbMATH Keywords: Jordan-Hölder
reviewed by: André G. Oliveira
MaRDI profile type: Publication
full work available at URL: j.geomphys.2010.07.004
OpenAlex ID: W2009177757
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DOI: 10.1016/J.GEOMPHYS.2010.07.004
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description: scientific article; zbMATH DE number 5791219