Assume that $X$ is a Moishezon manifold, then there exists a modification $\pi:\tilde{X}\rightarrow X$, where $\tilde{X}$ is a projective algebraic manifold. Let $\tilde{w}$ be a Kahler metric on $\tilde{X}$, then we can construct a holomorphic line bundle $\tilde{L}$ whose first Chern class in terms of the curvature is $\tilde{w}$. Then we can find a divisor $\tilde{D}$ such that the line bundle associated to $\tilde{D}$ is $\tilde{L}$.
Now I consider the image of $\tilde{D}$ under the restriction of $\pi$ to $\tilde{X}\setminus \pi^{-1}(A)$, where $A$ is an analytic set of codimension$\ge 2$ such that $\pi:\tilde{X}\setminus \pi^{-1}(A)\rightarrow X\setminus A$ is a bihilomorphism. Then $\pi|_{\tilde{X}\setminus \pi^{-1}(A)}(\tilde{D})$ is a divisor in $X\setminus A$. By Remmert-Stein Theorem, $\pi|_{\tilde{X}\setminus \pi^{-1}(A)}(\tilde{D})$ can be extended to a divisor, say $D$, in $X$.
Now we denote $L$ the line bundle associated to $D$. Then I guess we can find a singular Hermitian metric $h$ such that the first Chern class of $L$ in terms of the curvature is $\pi_\ast\tilde{w}$?
I have tried to trace the correspondence among first Chern classes, line bundles and divisors, but I feel difficult solving this problem. Can you help me? Thanks in advance!
