Abstract: 
The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $b_2>0$ has $b_2$ curves. By the results of [Ka1]--[Ka3], [Na1]--[Na3], [DOT], [OT] this conjecture (if true) would solve the classification problem completely. We explain a new approach (based on techniques from Donaldson theory) to prove existence of curves on class VII surfaces, and we present recent results obtained using this approach.

1. LATP, CMI
   Université de Provence
   39 Rue Frédéric Joliot-Curie
   13453 Marseille, France