THE GEOMETRY OF A MODULI SPACE OF BUNDLES

Let X be a class VII surface with b(2)(X) > 0. Following ideas developed in previous articles, we study the moduli space M-X := M-kappa X(pst )(E), where E is a differentiable rank 2 bundle on X with c(2)(E) = 0, and det(E) = K-X, the underlying differentiable line bundle of the canonical line bundle kappa(X). In this article we are interested in the non-minimal case: assuming that the minimal model of X is a primary Hopf surface, we prove that any point in the moduli space is a line bundle extension, and we give explicit geometric descriptions of M-X for b(2) (X) is an element of {1, 2}. Our motivation comes from the classification theory of class VII surfaces. Let X-0 be a minimal class VII surface with positive b(2) which is the deformation in large of a family of blown up primary Hopf surfaces. In other words X-0 is the central fiber of a holomorphic family (X-z)(z is an element of D), where X-z is a blown up primary Hopf surface for any z not equal 0. The classification of minimal class VII surfaces with this property is still an open problem. The moduli space M-X0 associated with an 'unknown such surface X-0 will be "the limit" of the family (M-Xz)(z is an element of D). of moduli spaces associated with blown up primary Hopf surfaces.