Hermitian holomorphic line bundle and curvature Chern form in Demailly's book

Question

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?

Answer 1

Thanks to the aid of @Gunnar Þór Magnússon, I will write down my understanding of Demailly's proof, if there is anything unclear, please comment below.

Let $\cup_{i\in I}U_i$ be a covering of $X$ such that $U_i\cap U_j$ is simply connected. Since $\omega$ is a closed real (1,1) form on $X$, by local $\partial\bar\partial$-lemma (see Demailly's book mentioned in the question, p.135. Prop 1.19), in $U_i$, there is a real-valued $\mathcal C^{\infty}$ function $\phi_i$ such that $\frac{i}{2\pi}\partial\bar\partial\phi_i=\omega$.

Note that $\phi_i-\phi_j\in \mathcal C^{\infty}(U_i\cap U_j)$ is $\partial\bar\partial$-closed, by Th. I-5.16 ( in Demailly's book p.42 ), there exists a holomorphic function $f_{ij}\in \mathcal O(U_i\cap U_j)$ satisfying $\phi_i-\phi_j=2\text{Re}f_{ij}$.

Let $g_{ij}=e^{2\pi if_{ij}}\in\mathcal O^*(U_i\cap U_j)$, it is easy to check $g_{ij}\cdot g_{jk}\cdot g_{ki}=1$ and $g_{ij}\cdot g_{ji}=1$. So the transition functions $\{g_{ij}\}$ determin a holomorphic line bundle $L$ over $X$. And $e^{\phi_i}$ is the Hermitian metric of $L$ which satsfies $\frac{i}{2\pi}\partial\bar\partial\phi_i=\omega$.