The universal Kobayashi-Hitchin correspondence on Hermitian manifolds
Over the last few decades, a lot of attention has been paid to a description of many different moduli spaces connected with various holomorphic objects. A typical (classical) problem of that sort can be described as follows. Fix a compact complex manifold X and a smooth vector bundle E on it. We would like to classify all holomorphic structures on E inducing a given holomorphic structure on the determinant line bundle det E on X. To have a Hausdorff moduli space, a condition of stability (semistablity) should be introduced. The book is devoted to a description of a suitable (universal) moduli problem, which includes a significant number of special moduli problems studied recently as special cases. In particular, the authors introduce their universal moduli problem and describe suitable stability (polystability) conditions for it, and they characterize polystable pairs in a geometric way. They also study (using ideas from Donaldson theory) metric properties of a large class of moduli spaces and they apply the results to some interesting cases (Douady Quot spaces, moduli spaces of oriented connections and moduli spaces of non-Abelian monopoles on Gauduchon surfaces).