Jordan-Hölder reductions for principal Higgs bundles on curves
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 group. Over \(C\), a Higgs \(G\)-bundle is a pair \((E,\phi)\) where \(E\) is a principal \(G\)-bundle and \(\phi\) is global section of \(\mathrm{Ad}(E)\otimes\Omega^1\), where \(\mathrm{Ad}(E)\) is the adjoint bundle. In the same spirit of Ramanathan's work, a Jordan-Hölder reduction of a Higgs \(G\)-bundle \((E,\phi)\) is a certain reduction of structure group of \((E,\phi)\) (hence compatible with \(\phi\)) to a parabolic subgroup \(P\subset G\), such that the Higgs \(G\)-bundle obtained by extending the structure group to the Levi factor \(L(P)\) of \(P\) is a stable principal Higgs \(L(P)\)-bundle.
scientific article; zbMATH DE number 5791219
scientific article