» Luebke M., Teleman A. - The Kobayashi-Hitchin correspondence
Luebke M., Teleman A. - The Kobayashi-Hitchin correspondence
||The Kobayashi-Hitchin correspondence
||Luebke M., Teleman A.
||When in 1993, encouraged by several colleagues, we decided to write this book, we had two main reasons to do so.
On the one hand, we were able to give a complete proof of the Kobayashi-Hitchin correspondence, i.e. the natural isomorphy of the moduli spaces of stable holomor-phic structures respectively irreducible Hermitian-Einstein connections in a differ-entiable complex vector bundle over a compact complex manifold. In particular, we could give this proof in the most general Hermitian context, whereas in most of the existing literature on this subject only algebraic or Kahler manifolds were considered; even for these cases, there was no single reference containing a complete proof of the correspondence in detail.
On the other hand, the Kobayashi-Hitchin correspondence had found important applications. In Donaldson theory, it had been used in the algebraic context to compute moduli spaces of instantons by algebraic-geometric methods, and thus had been an important tool in proving spectacular results in 4-diinensional differential topology. (In fact, this was our main motivation.) Furthermore, it had been used in the non-Kahler case to give a new and comparatively simple proof of Bogomolov's theorem on surfaces of type Vila-
Therefore, we thought it might be useful to present a complete, as far as possible self-contained, and hopefully readable proof of the correspondence and some of its applications.
Although at the end of 1994 it became apparent that many results in Donaldson theory can be proved in a much simpler way, by means of the newly discovered Seiberg-Witten invariants, and using only a very simple variant of the Kobayashi-Hitchin correspondence, we still think that this fundamental result is important and interesting enough to justify the publication of this book.
First of all, we are most grateful to Christian Okonek for his steady encouragement and financial support, which made several visits of the first author to Zurich possible.
Furthermore, we would like to thank the Stieltjes Institute for Mathematics in Leiden, the EC Science project Geometry of Algebraic Varieties (no. SCI-0398-C(A)), and the HCM project AGE-Algebraic Geometry in Europe (contr. no. ER-