For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} - 1$. Do we need to use the Enriques-Kodaira classification?

Question

In the Wikipedia article on the Enriques-Kodaira classification, before the classification itself, the following sentence appears:

For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} − 1$.

Of course, once the classification is given, one can verify this, but this fact is presented before the classification. As such, I am inclined to think that the result can be obtained without using the classification, and that it may even be a necessary observation to complete the classification itself. Is this inclination correct?

I did try to see if there was a statement of the above fact in Compact Complex Surfaces by Barth, Peters, and Van de Ven, but I wasn't able to find one. If there is a discussion of the above result in this book or any other reference, I would be happy to hear it.

Answer 1

This is contained in Theorem 2.6 in chapter IV in the book of Barth, Peters, Van de Ven (in the new edition with Hulek, it is Theorem 2.7, chapter IV).

It does not rely on classification, and it was first proved by Kodaira.