Theorem 18.39.8.A79139E2A3C6 Suppose that $c>0$ and $u\in C^\infty (B,\mathbb R)$ satisfy
\[ \beta =\theta +\mathrm{dd}^{\mathrm{c}}u-c\eta >0, \qquad \varphi \geq c\log \| \pi ^*s_ D\| _{H_ D}^2-C \quad \hbox{on }X \]
for a constant $C$. Suppose further that there is one submersion polydisk over an open subset of $D$, with coordinates
\[ \pi (z,t,w,v)=(z,t),\quad D=\{ z=0\} ,\quad t\in \mathbb C^{q-1},\quad w\in \mathbb C^ r,\quad v\in \mathbb C^{f-r}, \]
where $1\leq r\leq f$, on which
\[ \varphi \leq \frac cp\log \left(|z|^{2p}+\sum _{j=1}^ r|w_ j|^{2b_ j}\right)+C' \]
for positive integers $p,b_1,\ldots ,b_ r$ and a constant $C'$. Put
\[ a=p\sum _{j=1}^ r b_ j^{-1},\quad L_ m=\lfloor mc\rfloor ,\quad K_ m=\max \{ \lfloor mc-a\rfloor ,0\} ,\quad \psi =u+c\log \| s_ D\| _{H_ D}^2. \]
Then $\psi \in \mathrm{PSH}(B,\theta )$, its non-pluripolar mass is $\int _ B\beta ^ q\in (0,\infty )$, and for every $m\geq 1$,
\[ W_ m:=H^0\bigl (B,A^ m\otimes \mathcal I(m\psi )\bigr ) =H^0(B,A^ m(-L_ mD))\subseteq V_ m \subseteq H^0(B,A^ m(-K_ mD)). \]
Moreover $\dim V_ m-\dim W_ m=O(m^{q-1})=o(m^ q)$. For $q=1$ this difference is at most $\lceil a\rceil $ for every $m$.
Proof.
By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\beta +c[D]$. Off $D$, locality identifies the non-pluripolar top power with $\beta ^ q$; the non-pluripolar product puts no mass on $D$, and neither does the smooth measure $\beta ^ q$. Hence $\mathrm{MA}_\theta (\psi )=\beta ^ q$, whose mass is finite and positive because $\beta >0$ and $B$ is compact. The one-variable integrability condition $\ell +1>mc$ gives $\mathcal I(m\psi )=\mathcal O_ B(-L_ mD)$, also when $mc$ is an integer. Since $u$ is bounded, the global lower bound implies $\varphi \geq \psi \circ \pi -C''$. Hence integrability on the base gives integrability of the pullback in every bounded submersion chart, which proves $W_ m\subseteq V_ m$.
For the upper inclusion, take a nonzero section in $V_ m$ and let $\ell $ be its vanishing order along $D$; note that $D$ is irreducible, being smooth and connected. In the polydisk of the upper bound, the coefficient of the section is $z^\ell g(z,t)$, where $g(0,t)$ does not vanish identically on the open subset of $D$, since otherwise the identity theorem would increase the order along $D$. Shrink the polydisk about a point with $w=0$ at which $g(0,t)\ne 0$, so that $g$ becomes a holomorphic unit. The upper bound on $\varphi $ and lemma 18.39.7 then force $\ell +1>mc-a$. For an integer $\ell \geq 0$ this is equivalent to $\ell \geq K_ m$, which proves the upper inclusion.
Since $\lfloor mc\rfloor -\lfloor mc-a\rfloor \leq \lceil a\rceil $, we have $0\leq L_ m-K_ m\leq \lceil a\rceil $. Use the restriction sequences along $D$ from the proof of theorem 18.39.3 for $K_ m\leq j<L_ m$. Each step loses at most $h^0(D,(A^ m(-jD))|_ D)$ dimensions, and by lemma 18.39.1 this is at most $Cm^{q-1}$, uniformly for $0\leq j\leq cm$. Since the number of steps is bounded, the estimate follows. For $q=1$ the connected divisor $D$ is a single point, and each restriction space has dimension one.
$\square$
Pome
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