weak convergence of positive currents vs. $L^1$ convergence of normalized potentials

Question

I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I presume $L^1_{loc}$ in the non-compact case) convergence of their $dd^c$-potentials which are normialized (I presume, having a fixed mean). I tried to search the book of Demailly for it but couldn't quite find such a statement. What is the precise statement and how can it be proved?

Answer 1

Let $X$ be a compact complex manifold and let $T_n=\theta+dd^c \varphi_n$ be a sequence of positive currents where $\theta$ is a fixed smooth $(1,1)$-form on $X$. Assume that $\varphi_n$ are normalized such that $\sup_X \varphi_n=0$ (one could have an analogous statement choosing the normalization $\int_X \varphi_n dV=0$ for some fixed volume form $dV$ on $X$).

If $\varphi_n$ converges weakly to $\varphi$ in $L^1$, then we get immediately $dd^c \varphi_n \to dd^c \varphi$ in the sense of currents, hence $T_n \to \theta+dd^c \varphi$.

Conversely, assume that there exists a positive current $T:=\theta+dd^c \varphi$ such that $T_n \to T$ and $\sup_X \varphi=0$. Given that the functions $\varphi_n$ are $\theta$-psh and sup-normalized, the set $\{\varphi_n,n\in \mathbb N\}$ is precompact in $L^1$. Let $\psi=\lim \varphi_{\sigma(n)}$ be any cluster value. It satisfies $dd^c \psi=\lim dd^c \varphi_{\sigma(n)}=\lim T_{\sigma(n)}-\theta=T-\theta=dd^c \varphi$. Hence $\psi-\varphi$ is pluriharmonic on the compact manifold $X$ and it is constant. By the normalization choice, $\psi=\varphi$. Therefore, $(\varphi_n)$ converges to $\varphi$.

Side remark. I assumed that $\theta$ does not depend on $n$ for simplicity. One can adapt the statements to a slightly more general setting if necessary.