# Status of a conjecture of Hirzebruch

I was reading a paper from 1994 which claimed that the following statement was a conjecture of Hirzerbruch:

If a complex surface X is homeomorphic to either $$S^2 \times S^2$$ or $$\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$$ then it is biholomorphic to one of the Hirzerbruch surfaces.

I would like to know if this conjecture has been resolved or what the current status about it.

Also, I would like to know the state of the art about the variant of the conjecture in which "homeomorphic" is replaced by "diffeomorphic":

If a complex surface X is diffeomorphic to either $$S^2 \times S^2$$ or $$\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$$ then is it biholomorphic to one of the Hirzerbruch surfaces? If this has been proved, what is a reference?

Suppose $$X$$ is diffeomorphic to $$S^2\times S^2$$ or $$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$$. Then $$X$$ is biholomorphic to a Hirzebruch surface.

Note that $$b_1(X) = 0$$, so $$X$$ admits a Kähler metric. It follows from Seiberg-Witten theory that any compact Kähler surface with non-negative Kodaira dimension does not admit a metric of positive scalar curvature, but $$X$$ does, so $$X$$ has Kodaira dimension $$-\infty$$. By the classification of surfaces, $$X$$ is either rational (birational to $$\mathbb{CP}^2$$) or a ruled surface over a curve of genus $$g \geq 1$$. The latter possibility cannot occur as $$X$$ is simply connected, so $$X$$ is rational and hence biholomorphic to $$\mathbb{CP}^2$$ or a Hirzebruch surface; see Proposition VI.3.3 of Compact Complex Surfaces (second edition) by Barth, Hulek, Peters, and Van de Ven. As $$b_2(X) = 2 \neq 1 = b_2(\mathbb{CP}^2)$$, we conclude that $$X$$ is biholomorphic to a Hirzebruch surface.

If $$X$$ is only homeomorphic to $$S^2\times S^2$$ or $$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$$, then I believe it is still not known whether $$X$$ must be biholomorphic to a Hirzebruch surface.

As before, $$X$$ must admit a Kähler metric, but $$X$$ need not admit a metric of positive scalar curvature a priori, so we cannot immediately determine the Kodaira dimension of $$X$$ as we did above. Note, if we could show that $$X$$ had Kodaira dimension $$-\infty$$, we could conclude that $$X$$ is biholomorphic to a Hirzebruch surface by the exact same argument. We can however use classification to show that the Kodaira dimension of $$X$$ is not zero or one.

We can rule out Kodaira dimension zero because any Kähler surface of Kodaira dimension zero is finitely covered by a blownup torus or blownup $$K3$$ surface, which is not the case for $$X$$. The impossibility of Kodaira dimension one follows from the fact that $$c_1^2 = 0$$ for minimal such surfaces, so $$c_1^2(X) \in \{0, -1\}$$, and $$c_2(X) = \chi(X) = 4$$, so $$\operatorname{Td}(X) = \frac{1}{12}(c_1^2(X) + c_2(X)) \not\in \mathbb{Z}$$, which is impossible. Therefore, Hirzebruch's conjecture reduces to providing a negative answer to the following question:

Is there a general type surface homeomorphic to $$S^2\times S^2$$ or $$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$$?

It is natural to ask whether anything changes if $$X$$ is a compact complex surface which is only homotopy equivalent to $$S^2\times S^2$$ or $$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$$. As $$X$$ is simply connected, it follows from Freedman's Theorem that $$X$$ is actually homeomorphic to $$S^2\times S^2$$ or $$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$$, so we're again in the realm of Hirzebruch's conjecture.