PomeDefinition 10.5.1.057A3C697CFD For $\alpha \in \mathcal E$, $x\in X$ and a prime divisor $D\subset X$ put
\[ \nu (\alpha ,x)=\sup _{\varepsilon >0}\nu (V_{\theta +\varepsilon \omega },x), \qquad \nu (\alpha ,D)=\sup _{\varepsilon >0}\nu (V_{\theta +\varepsilon \omega },D). \]
These are the minimal multiplicities of $\alpha $. The non-nef locus of $\alpha $ is
\[ E_{\mathrm{nn}}(\alpha )=\{ x\in X:\nu (\alpha ,x)>0\} . \]
If $\alpha $ is big, its non-Kähler locus is
\[ E_{\mathrm{nK}}(\alpha )=\bigcap _ T\{ \varphi _ T=-\infty \} , \]
where $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ T$ ranges over the Kähler currents in $\alpha $ for which $\varphi _ T$ has analytic singularities. For a class $\alpha \in H^{1,1}(X,\mathbb R)$ that is not big we put $E_{\mathrm{nK}}(\alpha )=X$, and for a class that is not pseudo-effective we put $E_{\mathrm{nn}}(\alpha )=X$.
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