In their paper ``Nonalgebraic hyperkahler manifolds'' Campana, Oguiso and Peternell mention in Theorem 2.3 that if $Y$ is a smooth, Kähler, non-algebraic base of a fibration $f: X \dashrightarrow Y$ then $h^{2,0}(Y) > 0$, quoting Kodaira's paper about what is now known as Kodaira embedding theorem.
What particular consequence of Kodaira embedding, or a fact mentioned in Kodaira's paper, implies this statement?
Because $Y$ is Kahler, it has a symplectic form, which cannot be zero in $H^2(Y)$. If $h^{2,0}(Y) = 0$, we see that the Kahler cone (the set of cohomology classes for which there is a Kahler form given the complex structure $J$) is actually open in $H^2(Y)$, as it's of dimension $h^{1,1} \neq 0$. In this open set, therefore, there must be a rational (and hence there must be an integral) cohomology class. Then by this form of the Kodaira embedding theorem, $Y$, equipped with the new Kahler structure, admits a holomorphic embedding into some $\Bbb P^n$, and hence with this Kahler structure it's algebraic. But in changing the Kahler form, we never changed the complex structure, so we can honestly say that the original complex manifold was algebraic. This contradicts our assumption.