In Demailly's book p.272, Theorem 13.9, there is:

Let $X$ be an arbitrary complex manifold. (b) Let $\omega$ be a $\mathcal C^∞$ closed real (1, 1)-form such that ${ω}\in H^2_{dR}(X,\mathbb R)$ is the image of an integral class. Then there exists a hermitian line bundle $E\to X$ such that $\frac{i}{2π}Θ(E) = ω$.

It should be noted that the author dose not assume $X$ to be a compact Kähler manifold, while in Voisin's book *Hodge theory and complex algebraic geometry. I*, p.163-164, $X$ is assumed to be a compact Kähler manifold, which is used to get the $\partial\bar\partial$-lemma, and use it to deduce that $\omega-\omega_{L,h}=\frac{1}{2\pi i}\partial\bar\partial\phi$, where $\omega_{L,h}$ is the curvature form of a Hermitian holomorphic line bundle $L$.

So, my question is that: is the compact and Kähler (or $\partial\bar\partial$-) assumptions necessary to make that there is a holomorphic line bundle satisfying $\frac{i}{2π}Θ(E) = ω$?

By the way, I find Demailly'proof a bit hard to understand since there are some typos, for example, I can't find Th. I-3.35 in his book, so can anybody help me figure out whether his statement is right or wrong?