Class VII surface

Summary

A class VII surface is a compact complex surface with Kodaira dimension $-\infty$ and first Betti number $1$. The name comes from Kodaira's list of classes in the Enriques–Kodaira classification. These surfaces are never Kähler, since $b_1$ is odd. They form the part of the classification of compact complex surfaces that is still not complete. Hopf surfaces and Inoue surfaces are the minimal examples with second Betti number $0$.