The Kobayashi-Hitchin Correspondence
By the Kobayashi-Hitchin correspondence, the authors of this book mean the isomorphy of the moduli spaces Mst of stable holomorphic ? resp. MHE of irreducible Hermitian-Einstein ? structures in a differentiable complex vector bundle on a compact complex manifold. They give a complete proof of this result in the most general setting, and treat several applications and some new examples.After discussing the stability concept on arbitrary compact complex manifolds in Chapter 1, the authors consider, in Chapter 2, Hermitian-Einstein structures and prove the stability of irreducible Hermitian-Einstein bundles. This implies the existence of a natural map I from MHE to Mst which is bijective by the result of (the rather technical) Chapter 3. In Chapter 4 the moduli spaces involved are studied in detail, in particular it is shown that their natural analytic structures are isomorphic via I. Also a comparison theorem for moduli spaces of instantons resp. stable bundles is proved; this is the form in which the Kobayashi-Hitchin has been used in Donaldson theory to study differentiable structures of complex surfaces. The fact that I is an isomorphism of real analytic spaces is applied in Chapter 5 to show the openness of the stability condition and the existence of a natural Hermitian metric in the moduli space, and to study, at least in some cases, the dependence of Mst on the base metric used to define stability. Another application is a rather simple proof of Bogomolov's theorem on surfaces of type VII0. In Chapter 6, some moduli spaces of stable bundles are calculated to illustrate what can happen in the general (i.e. not necessarily K hler) case compared to the algebraic or K hler one. Finally, appendices containing results, especially from Hermitian geometry and analysis, in the form they are used in the main part of the book are included.
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A°(EndE analytic space analytic subspace associated bijection Chern class Chern connection cohomology compact complex manifold complex manifold constant Corollary curvature defined definition deg9 degree map denote Donaldson theory Einstein factor elliptic complex endomorphisms equation equivalent exists fibre fixed flat follows function g-stable Gauduchon metric hence Hermitian manifold Hermitian metric Hermitian-Einstein connections Hermitian-Einstein metric Hilbert manifold holds holomorphic bundle holomorphic line bundle Hopf surface implies induced irreducible isomorphism classes Kahler manifold Kahler metric Kobayashi-Hitchin correspondence Lemma metric g moduli space morphism natural neighborhood non-Kahler nontrivial orthogonal particular Pic(X Pic(X)T principal bundle projection proof Proposition prove PU(r quotient real-analytic Remark resp section 1.1 semiconnections sequence sheaf simple holomorphic structures smooth Sobolev spaces of stable stable bundles SU(r subbundle submanifold subset subsheaf surjective tangent bundle tangent space Theorem topological invariant torsion trivial unitary connection vanishing locus vector bundle weakly