Let $U \subset \mathbb{C}^n$ be an open set and let $f: U \to \mathbb{R}$ be a continuous function. Moreover, assume that $f$ is smooth on the complement of a complex hypersurface $Z \subset U$ and satisfies $i \partial \bar{\partial} f > 0$ on $U \setminus Z$.
Question: Is $f$ plurisubharmonic on $U$?
I would really appreciate a reference for this result (which I beleive is true).
I found a reference, Demailly, Complex Analytic and Differential Geometry, Theorem 5.24.
The point is that $Z$ is pluripolar, meaning that there is a psh function which identically $-\infty$ on $Z$, for example $\log |g|$ where $g$ is a defining holomorphic function for $Z$. One can check that $f_{\epsilon} =f + \epsilon g$ is psh for every $\epsilon>0$ and $f_{\epsilon} \to f$ in $L^1_{loc}$ as $\epsilon \to 0$.