Theorem 18.39.13.A7CF6A5A6A5C Suppose that $c_1,\ldots ,c_ N>0$ and $u\in C^\infty (B,\mathbb R)$ satisfy
\[ \beta =\theta +\mathrm{dd}^{\mathrm{c}}u-\sum _{i=1}^ Nc_ i\eta _ i>0, \qquad \varphi \geq \sum _{i=1}^ Nc_ i\log \| \pi ^*s_ i\| _{H_ i}^2-C \quad \hbox{on }X. \]
Suppose that for each $i$ there is a submersion polydisk over an open subset of $D_ i\setminus \bigcup _{j\ne i}D_ j$, with coordinates $(z_ i,t_ i,w_ i,v_ i)$ as in theorem 18.39.8, on which
\[ \varphi \leq \frac{c_ i}{p_ i}\log \left(|z_ i|^{2p_ i} +\sum _{j=1}^{r_ i}|w_{ij}|^{2b_{ij}}\right)+C_ i, \qquad 1\leq r_ i\leq f, \]
where $p_ i$ and $b_{ij}$ are positive integers and $C_ i$ is a constant. Put
\[ \begin{gathered} a_ i=p_ i\sum _{j=1}^{r_ i}b_{ij}^{-1},\quad L_{m,i}=\lfloor mc_ i\rfloor ,\quad K_{m,i}=\max \{ \lfloor mc_ i-a_ i\rfloor ,0\} ,\\ \psi =u+\sum _{i=1}^ Nc_ i\log \| s_ i\| _{H_ i}^2. \end{gathered} \]
Then $\psi \in \mathrm{PSH}(B,\theta )$ has non-pluripolar mass $\int _ B\beta ^ q\in (0,\infty )$, and for every $m\geq 1$,
\[ \begin{split} W_ m& :=H^0\bigl (B,A^ m\otimes \mathcal I(m\psi )\bigr ) =H^0\left(B,A^ m\left(-\sum _ iL_{m,i}D_ i\right)\right)\subseteq V_ m,\\ V_ m& \subseteq H^0\left(B,A^ m\left(-\sum _ iK_{m,i}D_ i\right)\right). \end{split} \]
Moreover $\dim V_ m-\dim W_ m=O(m^{q-1})$. For $q=1$ this difference is at most $\sum _ i\lceil a_ i\rceil $ for every $m$.
Proof.
By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\beta +\sum _ i c_ i[D_ i]$. Off the divisors the non-pluripolar measure equals $\beta ^ q$, and neither measure charges their union. Thus $\mathrm{MA}_\theta (\psi )=\beta ^ q$, with finite positive total mass.
Near a point of an intersection, the divisors through that point are distinct coordinate hyperplanes $\{ z_ i=0\} $. If a holomorphic germ is divisible by each corresponding $z_ i^{L_{m,i}}$, then it is divisible by their product; its residual coefficient is bounded, and since $L_{m,i}+1>mc_ i$, Fubini's theorem gives integrability. Conversely, integrability at generic nearby points of each coordinate hyperplane forces divisibility by $z_ i^{L_{m,i}}$, by the one-variable threshold, and the divisor of the germ then gives divisibility throughout the chart. This proves
\[ \mathcal I(m\psi )=\mathcal O_ B\left(-\sum _ iL_{m,i}D_ i\right). \]
Since $u$ is bounded, the global lower bound implies $\varphi \geq \psi \circ \pi -C'$, and integration in submersion coordinates gives $W_ m\subseteq V_ m$. For a nonzero section in $V_ m$, the polydisk over $D_ i$ on which the upper bound holds detects the generic vanishing order of the section along the irreducible divisor $D_ i$. The argument with a unit coefficient in the proof of theorem 18.39.8, together with the strict integrability threshold of lemma 18.39.7, shows that this order is at least $K_{m,i}$. Imposing all these orders proves the upper inclusion.
To compare dimensions, raise the orders from $(K_{m,i})_ i$ to $(L_{m,i})_ i$, one divisor and one unit at a time. At each step, the restriction sequence loses at most
\[ h^0\left(D_ i,\left.A^ m\left(-\sum _ j n_ jD_ j\right)\right|_{D_ i}\right), \qquad 0\leq n_ j\leq mc_ j, \]
dimensions. By lemma 18.39.12 with $E=A|_{D_ i}$ and $F_ j=\mathcal O_ B(-D_ j)|_{D_ i}$, this is $O(m^{q-1})$ uniformly over all steps, and there are at most $\sum _ i\lceil a_ i\rceil $ steps. When $q=1$, the distinct connected divisors are points, and each restriction space has dimension one.
$\square$
Pome
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