In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic surface of algebraic dimension $a(X)=1$. Since $a(x)=1$, there is a non-constant meromorphic function $h$ on X. If $h$ had indeterminacy points, we would have $D^2>0$, which contradicts the fact that on non-algebraic surfaces $D^2\leq 0$.
However, if there were a meromorphic function $h$ on $X$, then the divisor $D=(h)$ would be principal and thus the first Chern class of the associated line bundle $L$ would vanish; i.e $c_1(L)=0$. Thus $D^2=(c_1(L)\cup c_1(L),[X])=0$. Thus, I do not understand what is the meaning of this phrase. Can someone give me a hand?
