Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold: Harder-Narasimhan reduction

Boudjemaa Anchouche, Hassan Azad, Indranil Biswas

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Abstract

Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle EG over a compact Kähler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where G = GL(n, ℂ), this reduction is the Harder-Narasimhan filtration of the vector bundle associated to EG by the standard representation of GL(n, ℂ). The reduction of EG in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and EP ⊂ EG the canonical reduction, then the first condition says that the principal Lbundle obtained by extending the structure group of the P-bundle EP using the natural projection of P to L is semistable. Denoting by u the Lie algebra of the uhipotent radical of P, the second condition says that for any irreducible P-module V occurring in u/[u, u], the associated vector bundle EP × P V is of positive degree; here u/[u, u] is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character χ of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to EP for χ is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here.

Original languageEnglish
Pages (from-to)693-712
Number of pages20
JournalMathematische Annalen
Volume323
Issue number4
DOIs
Publication statusPublished - 2002

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Principal Bundle
Compact Manifold
Vector Bundle
Parabolic Subgroup
Filtration
Bundle
Module
Reductive Group
Line Bundle
Algebraic Groups
Reformulation
Lie Algebra
Non-negative
Equivalence
Roots
Projection
Subgroup
Denote

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold : Harder-Narasimhan reduction. / Anchouche, Boudjemaa; Azad, Hassan; Biswas, Indranil.

In: Mathematische Annalen, Vol. 323, No. 4, 2002, p. 693-712.

Research output: Contribution to journalArticle

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