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Fibrés stables et métriques d’Hermite-Einstein (d’après S. K. Donaldson, K. K. Uhlenbeck et S. T. Yau). (Stable bundles and Hermitian- Einstein metrics (following S. K. Donaldson, K. K. Uhlenbeck and S. T. Yau)). (French) Zbl 0637.53080

Sémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. No. 683, Astérisque 152/153, 263-283 (1987).
[For the entire collection see Zbl 0627.00006.]
This paper surveys recent work on a problem suggested by Hitchin and Kobayashi. Let (V,\(\omega)\) be a compact Kähler manifold and E a holomorphic vector bundle over E. Any Hermitian fibre metric on E induces a unique compatible connection, with curvature tensor F. The endomorphism \(F^ H\) of E is defined by the contraction with the metric form: \[ F^ H=F\cdot \omega =\sum F_{\alpha {\bar \beta}}g^{\alpha {\bar \beta}}. \] The metric is said to be “Hermitian-Einstein” or “Hermitian Yang- Mills” if \(F^ H\) (which is in some ways analogous to the Ricci tensor in Riemannian geometry) is a constant multiple of the identity endomorphism. Now we say that a holomorphic bundle \({\mathcal L}\) is [\(\omega\) ]-stable if for any coherent sub-sheaf \({\mathcal F}\subset E\) with \(0<rank {\mathcal F}<rank E\) we have: \[ c_ 1(F)\cdot \omega /rank(F) < c_ 1(E)\cdot \omega /rank(E). \] The main theorem asserts that a holomorphic bundle is stable if and only if it admits an indecomposable Hermitian-Einstein connection, and that this connection is then unique.
The “classical” case of this theorem is when the base manifold V is an algebraic curve; then we are looking for projectively flat unitary connections and the result is equivalent to a theorem of Narasimhan and Seshadri. That the existence of a Hermitian-Einstein metric implied stability was proved by Lübke and Kobayashi, and the converse, in general, was proved by Uhlenbeck and Yau. Their proof uses the method of continuity and a regularity theorem: a weakly holomorphic map into a complex projective variety is smooth.
An alternative proof in the case when V is a projective manifold and \(\omega\) is a Hodge metric was given by the reviewer. This proof avoids the regularity question but uses as input an algebro-geometrical theorem of Mehta and Ramanathan, characterizing stability by restriction to hypersurfaces. This is combined with a variational description of the equation and a solution scheme using a nonlinear heat equation.
Reviewer: S.K.Donaldson

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C05 Connections (general theory)
32L05 Holomorphic bundles and generalizations

Citations:

Zbl 0627.00006
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