Theorem 18.39.3.AE701AEF7FDE The function $\psi $ is $\theta $-plurisubharmonic, with positive non-pluripolar mass
\[ \int _ B\mathrm{MA}_\theta (\psi )=\int _ B\alpha ^ q>0. \]
For every integer $m\geq 1$ put $\ell _ m=\lfloor mc\rfloor $ and $k_ m=\max \{ \ell _ m-r,0\} $. Under holomorphic descent, the spaces selected by $\varphi $ on $X$ and by $\psi $ on $B$ are
\[ \begin{split} V_ m& :=H^0\bigl (X,\pi ^*A^ m\otimes \mathcal I(m\varphi )\bigr ) =H^0\bigl (B,A^ m(-k_ m Z)\bigr ),\\ W_ m& :=H^0\bigl (B,A^ m\otimes \mathcal I(m\psi )\bigr ) =H^0\bigl (B,A^ m(-\ell _ m Z)\bigr )\subseteq V_ m. \end{split} \]
There is a constant $C$, depending only on the fixed data, such that
\[ 0\leq \dim V_ m-\dim W_ m\leq Cm^{q-1}. \]
For $q=1$ the difference is at most $r\, \# Z$ for every $m$.
Proof.
By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\alpha +c[Z]\geq 0$. The divisorial measure identity (11.41.2), applied with the residual form $\alpha >0$ and the bounded part zero, gives $\mathrm{MA}_\theta (\psi )=\alpha ^ q$, which proves the mass formula.
Every section of $\pi ^*A^ m$ descends by proposition 18.36.1. In the $r+1$ normal variables $(x,w)$, lemma 5.37.1 gives
\[ \mathcal I(m\varphi )=(x,w_1,\ldots ,w_ r)^{k_ m}. \]
The local coefficient $g(x,z')$ of a pulled-back section does not depend on $w$. Expanding $g$ in powers of $x$, we see that $g$ lies in this ideal exactly when the coefficients of $x^ j$ vanish for $j<k_ m$, that is, when $x^{k_ m}$ divides $g$. This is an identity of germs in all tangential variables, so it holds at every point of $Z$. Outside $Y$ the function $\varphi $ is locally bounded. On the base, the same lemma in the single normal variable $x$ gives $\mathcal I(m\psi )=\mathcal O_ B(-\ell _ mZ)$. These facts prove the two identities and the inclusion.
For $k_ m\leq j<\ell _ m$, the restriction sequence along $Z$,
\[ \begin{split} 0& \longrightarrow A^ m(-(j+1)Z)\longrightarrow A^ m(-jZ)\\ & \longrightarrow \bigl (A^ m(-jZ)\bigr )|_ Z\longrightarrow 0, \end{split} \]
shows that passing from vanishing order $j$ to order $j+1$ along $Z$ loses at most $h^0(Z,(A^ m(-jZ))|_ Z)$ dimensions. Apply lemma 18.39.1 on $Z$ with $E=A|_ Z$ and $F=\mathcal O_ B(-Z)|_ Z$; since $0\leq j\leq cm$, each loss is at most $Cm^{q-1}$, uniformly in $j$. There are at most $r$ steps, which proves the estimate. When $q=1$, the divisor $Z$ is a finite set and each restriction space has dimension $\# Z$, which gives the last assertion directly.
$\square$
Pome
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