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Stability of homogeneous vector bundles. (English) Zbl 0885.14024
The paper discusses the connections between the geometric invariant theory, Hermite-Einstein metrics and the Kempf-Ness theory for the case of homogeneous vector bundles on a homogeneous projective algebraic variety \(X=G/P\) over \(\mathbb{C}\) where \(G\) is a reductive algebraic group and \(P\) its parabolic subgroup. The author defines the notion of stability for representations of \(P\) and proves that for such a representation \(\alpha\), \(\alpha\) is semistable iff the corresponding homogeneous vector bundle \(E_\alpha\) on \(X\) is “Mumford semistable” and iff \(E_\alpha\) admits an approximate Hermite-Einstein metric. For the exact statement one should take care of polarizations, Kähler classes as it is usual in this type of statements. The proof uses the “easy” part of Kobayashi-Hitchin correspondence but does not use the “hard” part based on the Donaldson theorem about the existence of Hermite-Einstein metrics. The introduction of the representation stability permits to make the bypass.
As an application there are computations that prove the stability of some bundles on \(\mathbb{P}^n\) and the Kapranov bundles on quadrics. The author remarks that many of the result are known to the specialists but the paper provides a systematic treatment of the topic.

14M17 Homogeneous spaces and generalizations
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L24 Geometric invariant theory
53C30 Differential geometry of homogeneous manifolds