would lead to a classification of the type VII surfaces. A class VII surface with positive second Betti number b2 has at most b2 rational curves, and has exactly this number if it has a global spherical shell. Conversely showed that if a minimal class VII surface with positive second Betti number b2 has exactly b2 rational curves then it has a global spherical shell. For type VII surfaces with vanishing second Betti number, the primary Hopf surfaces have a global spherical shell, but secondary Hopf surfaces and Inoue surfaces do not because their fundamental groups are not infinite cyclic. Blowing up points on the latter surfaces gives non-minimal class VII surfaces with positive second Betti number that do not have spherical shells. References Complex surfaces