The Kobayashi-Hitchin Correspondence

By the Kobayashi-Hitchin correspondence, the authors of this book mean the isomorphy of the moduli spaces M^st of stable holomorphic — resp. M^HE 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 M^HE to M^st 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 M^st 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 Kähler) case compared to the algebraic or Kähler 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.

• Introduction
• Hermitian–Einstein Connections and Metrics
• Existence of Hermitian–Einstein Metrics in Stable Bundles
• The Kobayashi–Hitchin Correspondence
• Applications
• Examples of Moduli Spaces
• Appendices