The classification problem for class VII surfaces

The classification of complex surfaces is not finished yet. The most important gap in the Kodaira-Enriques classification table concerns the Kodaira class VII, e.g. the class of surfaces $X$ having $\mathrm{kod}(X)=-\infty$, $b_1(X)=1$. These surfaces are interesting from a differential topological point of view, because they are non-simply connected 4-manifolds with definite intersection form. The main conjecture which (if true) would complete the classification of class VII surfaces, states that any minimal class VII surface with $b_2>0$ contains $b_2$ holomorphic curves. We explain a new approach, based on ideas from Donaldson theory, which gives existence of holomorphic curves on class VII surfaces with small $b_2$. In particular, for $b_2=1$ we obtain a proof of the conjecture, and for $b_2=2$ we prove the existence of a cycle of curves.