Let $ M=\mathbb C^n/Δ$ be a complex torus, then $M$ is abelian if and only if there is an integral closed positive $(1,1)$-current $\omega$ on $M$ and a point $p\in M $ such that $\omega−\epsilon\Sigma ^n_{i=1}dz_i∧d\bar z_i\geq0$ on $U$ in the sense of currents for some neighborhood $U$ of $p$ and for some positive constant number $\epsilon$, where $(z_1,⋯,z_n)$ is a coordinate system on $U$.

See Smoothing of currents and Moišezon manifolds. Several complex variables and complex geometry, by Ji, Shanyu

Let $ M=\mathbb C^n/Δ$ be a complex torus, then $M$ is abelian if and only if there is an integral closed positive $(1,1)$-current $\omega$ on $M$ and a point $p\in M $ such that $\omega−\epsilon\Sigma ^n_{i=1}dz_i∧d\bar z_i\geq0$ on $U$ in the sense of currents for some neighborhood $U$ of $p$ and for some positive constant number $\epsilon$, where $(z_1,⋯,z_n)$ is a coordinate system on $U$.

Note that the definition of positivity of current(introduced by Lelong) is different with positivity of a form

See Smoothing of currents and Moišezon manifolds. Several complex variables and complex geometry, by Ji, Shanyu