On the Geometry of Instanton Moduli Spaces

The study of instanton moduli spaces has led to striking results in the geometry of four-manifolds, for they constitute the groundwork for the construction of Donaldson's polynomial invariants, which are capable of distinguishing non-diffeomorphic smooth structures. This construction fails when the base manifold has a negative definite intersection form, since in this case the moduli spaces contain reducible solutions and these are usually singular points. This book is therefore concerned with the study of the geometry of instanton moduli spaces for those manifolds which have negative definite intersection form. In the first part, the topology and the Riemannian geometry near the reducible locus are described. This yields a purely topological condition for one construction of differential topological invariants. In the second part, explicit examples of instanton moduli spaces are computed for all minimal complex surfaces of class VII with second Betti number one and with respect to any Gauduchon metric. This monograph contains the author's PhD thesis, which was written at the Université de Provence Aix-Marseille I (France) under the supervision of Professor Andrei Teleman.