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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

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

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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, for instance.

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  • $\begingroup$ Hi - do you remember which reference to $dd^c$ lemma for current you had in mind? The link is broken. Thanks! $\endgroup$
    – dummy
    Commented Dec 31, 2023 at 15:23

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