Class VII surface

Remarks

Example: a Hopf surface is a class VII surface. For the diagonal Hopf surface, $b_1 = 1$ because it is diffeomorphic to $S^1 \times S^3$, and no power of the canonical bundle has a nonzero holomorphic section. Inoue surfaces are class VII surfaces with $b_2 = 0$ and no curves.

Non-example: $\mathbb{CP}^2$ has $\kappa = -\infty$, but $b_1 = 0$, so clause 2 fails; it is a rational surface.

Since $b_1$ is odd, a class VII surface is not Kähler, and in particular not projective. Blowing up a point keeps $\kappa$ and $b_1$ and raises $b_2$ by $1$, so every class VII surface is a blow-up of a minimal one. A theorem of Bogomolov states that every class $\mathrm{VII}_0$ surface with $b_2 = 0$ is a Hopf surface or an Inoue surface. Li, Yau and Zheng, and later Teleman, gave complete proofs. For $b_2 > 0$ the classification is open; the global spherical shell conjecture predicts that every class $\mathrm{VII}_0$ surface with $b_2 > 0$ contains a global spherical shell.