Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

Question

I am looking for a reference for the following fact, which I believe should be true.

Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$). If $\omega$ is a positive closed $(1,1)$-current over $X,$ then there exists a plurisubharmonic function $\varphi$ such that $\omega=i\partial\overline{\partial} \varphi.$ In practice, I am interested in cases when $X= \mathbb{C}^n$ and $X= (\mathbb{C}\setminus \{0\})^n$.

I found the local version of this statement in Demailly's Complex Analytic and Differential Geometry, Chapter III, Proposition 1.19.

Answer 1

The fact that you are looking for requires the current to be exact.On any Stein manifold for which the second cohomology group with integer coefficients has non trivial torsion free part,there are holomorphic line bundles with closed positive (1,1) forms as curvature and are not exact. Under this additional assumption that the current is exact, the proof in Demailly goes through for Stein manifolds .