PomeC43B33AA4A18Proposition 12.35.1.C43B33AA4A18 The following are equivalent:
there is $u\in \mathrm{PSH}(X,\theta )$ with $\nu (u,D_ j)\geq a_ j$ for every $j$;
the residual background $\beta _ D$ is pseudoeffective;
for every Gauduchon metric $\gamma $,
The first condition does not require $u-\sigma _ D$ to be bounded, nor the Lelong numbers of $u$ to equal the prescribed ones.
Proof. Suppose first that $u$ satisfies the first condition. By lemma 5.38.1, these Lelong numbers can be subtracted: $\theta +\mathrm{dd}^{\mathrm{c}}u-[D]\geq 0$. Consequently
where the difference $u-\sigma _ D$ is first taken in $L^1$ and then replaced by its quasi-plurisubharmonic representative. This proves the second condition. Conversely, if $v\in \mathrm{PSH}(X,\beta _ D)$, then $u=\sigma _ D+v$ satisfies
so its generic Lelong numbers along the $D_ j$ are at least $a_ j$. For $n\geq 2$, Lamari's duality, theorem 11.6.1, says that a smooth background is pseudoeffective exactly when all its Gauduchon integrals are nonnegative; applied to $\beta _ D$, together with the degree identity, it gives the equivalence of the second and third conditions. For $n=1$ the same criterion reads $\int _ X\beta _ D\geq 0$: necessity follows from Stokes' theorem, and for sufficiency one solves $\mathrm{dd}^{\mathrm{c}}f=(\int _ X\beta _ D)\mu -\beta _ D$ for a smooth real function $f$ with the scalar Green operator. $\square$
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