Topological classification of complex surfaces

Question

The famous Enriques–Kodaira classification classfies the minimal complex surfaces by algebraic invariants, which give tight restrictions on the topology of the underlying manifolds. What about the opposite direction? In other words, does there exist a classification of the homeomorphy types of topological $4$-manifolds which admit a complex stucture that makes them into minimal complex surfaces? Some scattered results can be derived from the E-K classification directly: For example, $\Bbb{CP}^2$, $\Bbb{S}^2 \times \Bbb{S}^2$ and $\Bbb{CP}^2 \# \overline{\Bbb{CP}^2}$ are the only manifolds that can be given a structure of rational minimal surfaces, and $2\Bbb{E}_8 \# 3\Bbb{S}^2 \times \Bbb{S}^2$ is the only homeomorphy type of $K3$ surfaces. Therefore, I'm seeking for a generic solution (or information) of this question.

Answer 1

I believe there is no such classification. I have two reasons for this belief:

1. We still don't know what complex surfaces belong to Class VII, namely, $b_1$ odd and Kodaira dimension $-\infty$. Examples of class VII surfaces include Hopf surfaces and Inoue surfaces. Every other known minimal examples are deformations of blowups of primary Hopf surfaces. It is conjectured that all minimal class VII surfaces are of one of the three aforementioned types. The missing ingredient is the existence of a global spherical shell.

2. There are too many surfaces of general type, namely surfaces with Kodaira dimension $2$. There are many open questions about surfaces in this class, see this article for example.

Having said that, the classification does yield a lot of topological information. In the context of your question, it is probably better to think of such information as a tool to rule out the existence of a complex surface of a given homeomorphism type.