Pome1FFB44AD196A1FFB44AD196A
In this section $\alpha $ is pseudoeffective. A current in $\alpha [-\varepsilon \omega ]$ may have singularities that the class does not force. To isolate the forced ones, take in each family a current with the mildest singularities and let the allowed negative error tend to zero. The Lelong numbers that persist define the non-nef locus.
For $\varepsilon >0$ put
where $V_{\theta +\varepsilon \omega }=P_{\theta +\varepsilon \omega }(0)$ is the normalized envelope of section 5.40. Since $\alpha $ is pseudoeffective, $\mathrm{PSH}(X,\theta +\varepsilon \omega )$ is nonempty, so $V_\varepsilon $ is $(\theta +\varepsilon \omega )$-plurisubharmonic and $T_{\min ,\varepsilon }\in \alpha [-\varepsilon \omega ]$. Moreover, $V_\varepsilon $ has minimal singularities in this family: $u-\sup _ Xu\leq V_\varepsilon $ for every $u\in \mathrm{PSH}(X,\theta +\varepsilon \omega )$.
Definition 10.12.1.72E9AF170CBA For $x\in X$, the minimal multiplicity of $\alpha $ at $x$ is
The non-nef locus is
As $\varepsilon $ decreases, the family $\alpha [-\varepsilon \omega ]$ shrinks, so the Lelong numbers of its minimal current can only increase; the supremum is therefore a limit as $\varepsilon \downarrow 0$. Two Hermitian forms are comparable up to a constant factor, so the families defined by them are nested after rescaling $\varepsilon $; hence $\nu (\alpha ,x)$ does not depend on $\omega $. Replacing $\theta $ by $\theta +\mathrm{dd}^{\mathrm{c}}f$ changes the minimal potentials by $-f$ up to a bounded normalization error, which does not affect Lelong numbers.
The smooth regularization of 5.36.3 turns vanishing Lelong numbers into an arbitrarily small loss of positivity. This gives the converse direction in the following characterization.
Proposition 10.12.2.4075D8C83DE1 For a pseudoeffective class $\alpha $, the set $E_{\mathrm{nn}}(\alpha )$ is a countable union of analytic subsets. Moreover,
Proof. By the monotonicity just noted,
and each set in this union is analytic by Siu's analyticity theorem (5.34.2).
If $\alpha $ is nef, each family $\alpha [-\varepsilon \omega ]$ contains a smooth current $\theta +\mathrm{dd}^{\mathrm{c}}f$. By minimality, $V_\varepsilon \geq f-\sup _ Xf$, so $V_\varepsilon $ is bounded and all Lelong numbers of $T_{\min ,\varepsilon }$ vanish. Conversely, if the non-nef locus is empty, every $T_{\min ,\varepsilon }$ has zero Lelong numbers. The regularization just recalled then gives a smooth representative of $\alpha $ bounded below by $-2\varepsilon \omega $. Since $\varepsilon >0$ is arbitrary, $\alpha $ is nef. $\square$
The approximating families cannot be replaced by the single family of positive currents in $\alpha $: even for a nef class, a positive current with minimal singularities in $\alpha $ itself may have nonzero Lelong numbers; see the example discussed in section 25.18.
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