PomeDefinition 5.19.1.DF4FCB370781 Let $Y$ be a complex manifold of dimension $N$ and $m\in \mathbb {N}$. We say an $(m,m)$-current $T$ on $Y$ is positive1 if either $m>N$ or for any smooth $(1,0)$-forms $\beta _1,\ldots ,\beta _{N-m}$ on $Y$, the measure
\[ T\wedge \mathrm{i}\beta _1\wedge \overline{\beta _1}\wedge \cdots \wedge \mathrm{i}\beta _{N-m}\wedge \overline{\beta _{N-m}} \]
is positive.
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