# 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?

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.