Einstein-Hermitian connections on principal bundles and stability.
Einstein-Hermitian connections on principal bundles and stability.
We show that a holomorphic principal G-bundle, where G is a complex reductive group, on a complex projective manifold is (quasi-)stable if and only if it carries an Einstein-Hermitian connection. We use a theorem of \textit{S. Ramanan} and the first author [Tôhoku Math. J., II. Ser. 36, 269-291 (1984; Zbl 0567.14027)] that if a principal G-bundle is quasi- stable, then any associated vector bundle is quasi-stable, and Donaldson's theorem for vector bundles in our proof. A special case is when the principal bundle does not have a reduction to any parabolic. Such a bundle is stable and hence carries an Einstein-Hermitian connection.
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510.mathematics, Special Riemannian manifolds (Einstein, Sasakian, etc.), Holomorphic bundles and generalizations, stable bundle, complex projective manifold, Einstein-Hermitian connection, holomorphic principal G-bundle, Global differential geometry of Hermitian and Kählerian manifolds, Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory, Article, Connections (general theory)
510.mathematics, Special Riemannian manifolds (Einstein, Sasakian, etc.), Holomorphic bundles and generalizations, stable bundle, complex projective manifold, Einstein-Hermitian connection, holomorphic principal G-bundle, Global differential geometry of Hermitian and Kählerian manifolds, Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory, Article, Connections (general theory)