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Let $X$ be a complex manifold and $\theta$ a smooth $(1,1)$-form on $X$.

(1) If $\theta>0$ in the sense of currents, then can we deduce that $\theta>0$ in the sense of forms?

(2) If $\theta>0$ in the sense of forms, then can we deduce that $\theta>0$ in the sense of currents?

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    $\begingroup$ You should be more specific about what you mean by "> 0" for currents and forms. Do you mean > 0 as in strictly positive? In that case I would interpret "> 0 in the sense of currents" as \theta >= \epsilon \omega for some smooth strictly positive (1,1)-form \omega and \epsilon > 0, and if \theta is smooth, then then it is trivial that these notions of positivity are equal. If you mean "> 0" as in non-negative, then there are positive and strongly positive (k,k) forms and currents, but for a smooth (1,1)-form, all these notions of positivity coincide. $\endgroup$
    – Richard Lärkäng
    Commented Jun 23, 2020 at 8:39

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