Example 10.5.9A7A7F69130A0

Example 10.5.9.A7A7F69130A0 The zero class $\alpha =0$ is nef, so $E_{\mathrm{nn}}(0)=\varnothing $ by theorem 10.5.4. It is not big: a Kähler current $T\geq \delta \omega $ would have $\int _ XT\wedge \omega ^{n-1}\geq \delta \int _ X\omega ^ n>0$, whereas this integral vanishes for every closed current in the zero class. Hence $E_{\mathrm{nK}}(0)=X$. More generally, $E_{\mathrm{nK}}(\alpha )=X$ for every class that is not big, while $E_{\mathrm{nn}}(\alpha )=\varnothing $ for every nef class; on the boundary of the pseudo-effective cone the two loci therefore detect different failures of positivity. This is also why the minimal multiplicities are defined with the allowances $\varepsilon \omega $ rather than through $V_\theta $: for a class that is not big, proposition 10.5.2 gives only the inequality $\nu (\alpha ,x)\leq \nu (V_\theta ,x)$, since the proof of equality uses a Kähler current in $\alpha $.