Lemma 5.38.5.65C35F77CB06 Let $Y$ be a connected compact complex manifold. Let $(D_ j)$ be a finite or countable family of distinct prime divisors on $Y$ and $c_ j>0$ real coefficients such that $S=\sum _ jc_ j[D_ j]$ is a positive current of finite mass, and let $\beta $ be a smooth real $(1,1)$-form on $Y$. Let $v\in \mathrm{QPSH}(Y)$ satisfy
\[ \mathrm{dd}^{\mathrm{c}}v=S-\beta . \]
Then, for $a\in \mathrm{QPSH}(Y)$, the following conditions are equivalent:
\[ a\preceq v,\qquad a\preceq _{\mathcal I}v,\qquad \nu (a,D_ j)\geq c_ j\quad \hbox{for every }j. \]
Proof.
We first note that $\nu (v,D_ j)=c_ j$. Indeed, the restriction of $S$ to $D_ j$ is $c_ j[D_ j]$: distinct divisors intersect in codimension at least two, and such sets carry no mass of any divisor current, also after countable summation. Smooth forms give every divisor zero mass. The local form of Siu's decomposition, applied after a smooth correction, therefore gives $\nu (v,D_ j)=c_ j$.
Ordinary domination implies ideal domination by proposition 5.33.2, and ideal domination implies the asserted inequalities for generic Lelong numbers by theorem 5.37.3. It remains to prove that these inequalities imply ordinary domination.
Choose a Hermitian form $\Lambda $ with $a\in \mathrm{PSH}(Y,\Lambda )$. By lemma 5.38.1 and the inequalities $\nu (a,D_ j)\geq c_ j$, each finite partial sum of $S$ can be subtracted from $\Lambda +\mathrm{dd}^{\mathrm{c}}a$ without losing positivity. Letting the partial sums increase to $S$ gives
\[ \Lambda +\mathrm{dd}^{\mathrm{c}}a-S\geq 0. \]
Consequently $\mathrm{dd}^{\mathrm{c}}(a-v)\geq \beta -\Lambda $. Hence the integrable function $a-v$ has a quasi-plurisubharmonic representative, which is bounded above on the compact manifold $Y$. This yields $a\leq v+C$ almost everywhere, and lemma 5.24.3 extends this inequality everywhere. Thus $a\preceq v$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.