This is probably fairly elementary, but does someone know how to prove the following or know a reference.


Let $X$ be a Kaehler manifold. Let $\theta$ be a closed $(1,1)$-form and $T$ be a closed positive $(1,1)$-current cohomologous to $\theta$. Then there is a quasi-plurisubharmonic function $\psi$ so that 


$$ T=\theta +dd^c \psi.$$



ddbar lemma for positive closed (1,1)-currents
If I remember correctly, all classic Kähler identities between operators $d, d^c \dots $ of differential forms are also satisfied by the corresponding operators of currents. A reference for this could be L. SCHWARTZ, Lectures on Complex Analytic Manifolds, Tata Inst. Fund. Res. Lectures on Math. and Phys. 4, Springer, Berlin, 1986. MR 0901469. You have in particular a $dd^c$ lemma for currents: see lemma 3.3.2 in [here](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1127831440), for instance.


