CHAPTER 2 The finite dimensional Kobayashi-Hitchin correspondence In this section we give a brief presentation of the stability theory in the finitely dimensional Kahlerian non-algebraic framework and explain the analogue of the Hilbert criterion in this framework. The Hilbert criterion we prove here gives a numerical characterization of the polystable orbits, i. e. of the orbits which intersect the vanishing locus of the moment map. Our proof - based on the continuity method - is a very good preparation for understanding the proof of our universal Kobayashi-Hitchin correspondence on Hermitian manifolds in section 5, which will follow the same strategy but will require involved analytical techniques. 2.1. Analytic Stability, Symplectic stability For complete proofs and more details about the notions and the results in- troduced in this section we refer to [Te3]. The analogous results in the algebraic geometric framework are well known ([Ki], [MFK], [KN]), but the non-algebraic framework raises specific difficulties. Let G be a complex reductive group. We denote by H{G) the subset H(G) C g defined by H(G) := {£ G g\ exp(zRf) is compact} = | J it . K CG maximal compact H(G) is a locally closed cone in g. The elements in H{G) will be called vectors of Hermitian type (see [Te3]). There is an obvious non-surjective injective map Hom(C*, G) -» H(G) given by dt ( R 3 * H - 0 ( 1 + *)) = d19{\) . t=o Therefore, compared to classical GIT, in non-algebraic complex geometry, one has to consider more general (non-algebraic!) one-parameter subgroups, which do not exponentiate to morphisms C* — G. To any £ G H(G) one can associate a parabolic subgroup G(£) of G by G(£) := {g G G\ lim exp(t£)#exp(—1£) exists in G} The subgroup G(£) fits in a short exact sequence (2.1) {1} -^ U(0 -^ G(0 -^ Z(0 -^ {1} , where U(0 := {g € G\ lim exp(^)ffexp(-^) = e} t—OC 11

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