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On Nahm’s equations and the Poisson structure of semi-simple complex Lie algebras. (Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes.) (French) Zbl 0843.53027
Using Nahm’s equations, we extend Kronheimer’s construction of hyperkähler metrics to all coadjoint orbits of a complex semisimple Lie group and more generally to symplectic leaves of holomorphic Poisson manifolds associated to the group.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Biquard, O.: Fibr?s Paraboliques Stables et Connexions Singuli?res Plates. Bull. Soc. Math. France119, 231-257 (1991) · Zbl 0769.53013
[2] Biquard, O.: Fibr?s paraboliques et connexions singuli?res sur une courbe ouverte. Th?se: Ecole Polytechnique 1991
[3] Bottacin, F.: G?om?trie symplectique sur l’espace des modules de paires stables. Th?se: Universit? d’Orsay 1993
[4] Donaldson, S.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys.96, 387-407 (1984) · Zbl 0603.58042 · doi:10.1007/BF01214583
[5] Donaldson, S.: Boundary value problems for Yang-Mills fields. J. Geom. Phys.8, 89-122 (1992) · Zbl 0747.53022 · doi:10.1016/0393-0440(92)90044-2
[6] Donaldson, S., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford: Clarendon Press 1990 · Zbl 0820.57002
[7] Kovalev, A.: Nahm’s equations and complex adjoint orbits. Oxford: preprint 1993 · Zbl 0852.53033
[8] Kronheimer, P.: A hyper-k?hlerian structure on coadjoint orbits of a semisimple complex group. J. London Math. Soc. (2)42, 193-208 (1990) · Zbl 0721.22006 · doi:10.1112/jlms/s2-42.2.193
[9] Kronheimer, P.: Instantons and the geometry of the nilpotent variety. J. Differential Geometry32, 473-490 (1990) · Zbl 0725.58007
[10] Lichnerowicz, A.: Les vari?t?s de Poisson et leurs alg?bres de Lie associ?es. J. Differential Geometry12, 253-300 (1977) · Zbl 0405.53024
[11] Morgan, J., Mrowka, T., Ruberman, D.: TheL 2-moduli space and a vanishing theorem for Donaldson invariants (Monographs in Geometry and Topology vol. 2) Cambridge: International Press, 1994 · Zbl 0830.58005
[12] Simpson, C.: Harmonic bundles on non compact curves. J. Amer. Math. Soc.3, 713-770 (1990) · Zbl 0713.58012 · doi:10.1090/S0894-0347-1990-1040197-8
[13] Uhlenbeck, K.: Connections withL p -bounds on curvature. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
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