elliptic curve such that $X$ contains an irreducible curve $C$ with the following property: the class $\{C\}\in{H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ is nef, but contains only one positive current, which is of course the integration current $[C]$.

The following proposition gives a more “algebraic” characterization of ${\mathcal{MK}}$, which also explains the (seemingly dumb) terminology.

A class $\alpha$ lies in ${\mathcal{MK}}$ iff there exists a modification\
$\mu:{\widetilde}{X}\to X$ and a Kähler class ${\widetilde}{\alpha}$ on ${\widetilde}{X}$ such that $\alpha=\mu_{\star}{\widetilde}{\alpha}$.

$Proof$: the argument is adapted from \[DP01\], theorem 3.4. If ${\widetilde}{\omega}$ is a Kähler form on ${\widetilde}{X}$ and $\omega$ is our reference Hermitian form on $X$, then $\mu^{\star}\omega\leq C{\widetilde}{\omega}$ for some $C>0$, since ${\widetilde}{X}$ is compact. Since $\mu$ is a modification, we have $\mu_{\star}\mu^{\star}\omega=\omega$, so we get $T:=\mu_{\star}{\widetilde}{\omega}\geq C^{-1}\omega$, and $T$ is thus a Kähler current. Since the singular values of $\mu$ are in codimension at least 2, we immediately see that $\nu(T,D)=0$ for every prime divisor $D$ in $X$, and $\{T\}=\mu_{\star}\{\omega\}$ lies in ${\mathcal{MK}}$ as desired. Conversely, if $\alpha\in{\mathcal{MK}}$ is represented by a Kähler current $T$ with $\nu(T,D)=0$ for all $D$, there exists by (ii) of theorem 2.1 a Kähler current $T_k$ in $\alpha$ with analytic singularities along a subscheme $V_k$ with $\nu(T_k,D)\leq\nu(T,D)$, so that $V_k$ 