Keep $\pi :X\to B$, $(A,h)$, $\theta $, $\varphi $ and $V_ m$ as in subsection 18.39.6. A singularity of higher codimension on the base can become divisorial after a modification of the base. By lemma 5.43.22, the integrability test upstairs then applies to the pulled-back section multiplied by the Jacobian. This fixed factor shifts the divisorial orders, while the restriction estimate still bounds the number of lost sections.
Let $\rho :Y\to B$ be a modification of connected compact complex manifolds, and form the fiber product and its projections
\[ X'=X\times _ B Y,\qquad \pi ':X'\to Y,\qquad \mu :X'\to X. \]
In local submersion coordinates one sees that $X'$ is smooth, that $\pi '$ is a submersion with the same compact connected fibers as $\pi $, and that $\mu $ is a modification. Let $R_\rho $ and $J_\rho $ be the relative canonical bundle and the Jacobian section of $\rho $, as in subsection 5.43.21. The coordinate Jacobian matrix of $\mu $ is that of $\rho $, extended by the identity in the fiber variables, so
\[ R_\mu =\pi '^*R_\rho ,\qquad J_\mu =\pi '^*J_\rho . \]
Put $\varphi '=\varphi \circ \mu $ and $\theta '=\rho ^*\theta $. Choose distinct nonempty smooth connected reduced divisors $D_1,\ldots ,D_ N$ on $Y$ whose union has simple normal crossings, together with smooth metrics $H_ i$ on $\mathcal O_ Y(D_ i)$, canonical sections $s_ i$ and curvatures $\eta _ i$.
Theorem 18.39.16.1FC6712A00F0 Suppose that $c_ i>0$ and $u\in C^\infty (Y,\mathbb R)$ satisfy
\[ \beta =\theta '+\mathrm{dd}^{\mathrm{c}}u-\sum _ i c_ i\eta _ i>0, \qquad \varphi '\geq \sum _ i c_ i\log \| \pi '^*s_ i\| _{H_ i}^2-C. \]
Suppose that for each $i$ the function $\varphi '$ satisfies an anisotropic upper bound, with respect to $\pi '$, on a submersion polydisk over an open subset of $D_ i$ away from the other divisors, as in theorem 18.39.13, with fixed $p_ i,b_{ij}$. Define $a_ i$, $L_{m,i}$ and $K_{m,i}$ by the formulas of that theorem, and put
\[ \psi '=u+\sum _ i c_ i\log \| s_ i\| _{H_ i}^2, \qquad M_ m=\rho ^*A^ m\otimes R_\rho . \]
Then there is a unique $\psi \in \mathrm{PSH}(B,\theta )$ with $\psi \circ \rho =\psi '$, and
\[ \mathrm{MA}_\theta (\psi )=\rho _*(\beta ^ q),\qquad \int _ B\mathrm{MA}_\theta (\psi )=\int _ Y\beta ^ q\in (0,\infty ). \]
The map $T_ m(s)=J_\rho \rho ^*s$ identifies $H^0(B,A^ m)$ with $H^0(Y,M_ m)$, and for $W_ m=H^0(B,A^ m\otimes \mathcal I(m\psi ))$,
\[ \begin{split} T_ m(W_ m)& =H^0\left(Y,M_ m\left(-\sum _ iL_{m,i}D_ i\right)\right),\\ T_ m(W_ m)& \subseteq T_ m(V_ m) \subseteq H^0\left(Y,M_ m\left(-\sum _ iK_{m,i}D_ i\right)\right). \end{split} \]
Consequently $W_ m\subseteq V_ m$ and $\dim V_ m-\dim W_ m=O(m^{q-1})$. If $q=1$, the difference is at most $\sum _ i\lceil a_ i\rceil $ for every $m$.
Proof.
By the Poincaré–Lelong formula, $\theta '+\mathrm{dd}^{\mathrm{c}}\psi '=\beta +\sum _ i c_ i[D_ i]$. Thus $\psi '\in \mathrm{PSH}(Y,\rho ^*\theta )$, and proposition 5.43.18 gives the unique potential $\psi $ on $B$ with $\psi \circ \rho =\psi '$. On $Y$, locality off the union of the divisors and the fact that the non-pluripolar measure does not charge this union give $\mathrm{MA}_{\theta '}(\psi ')=\beta ^ q$. The measure identity in proposition 18.35.1 then gives the pushforward formula and the mass.
Lemma 5.43.22, applied first with weight zero and then with weight $m\psi $, gives the identification of the full section spaces and the formula for $T_ m(W_ m)$; here $\mathcal I(m\psi ')=\mathcal O_ Y(-\sum _ iL_{m,i}D_ i)$ by the integrability computation for simple normal crossings in the proof of theorem 18.39.13. Since $u$ is bounded, the global lower bound gives $\varphi '\geq \psi '\circ \pi '-C'$. As $\rho \circ \pi '=\pi \circ \mu $ and $\mu $ is surjective, this descends to $\varphi \geq \psi \circ \pi -C'$, and integrability in bounded submersion charts proves $W_ m\subseteq V_ m$.
For $s\in V_ m$, the transformation of multiplier ideals under $\mu $ gives
\[ J_\mu \mu ^*s=\pi '^*T_ m(s) \in H^0\bigl (X',\pi '^*M_ m\otimes \mathcal I(m\varphi ')\bigr ). \]
Hence, in each polydisk where the upper bound holds, the descended coefficient of $T_ m(s)$ vanishes to order at least $K_{m,i}$ along $D_ i$, by lemma 18.39.7 and the argument with a unit coefficient in the proof of theorem 18.39.8. This is the vanishing order of $T_ m(s)$, including the Jacobian factor, and it proves the upper inclusion.
The chain of restrictions along the divisors on $Y$ has at most $\sum _ i\lceil a_ i\rceil $ steps. On $D_ i$, each target is $(\rho ^*A|_{D_ i})^ m$ twisted by powers of the bundles $\mathcal O_ Y(-D_ j)|_{D_ i}$ in a linear range and by the fixed bundle $R_\rho |_{D_ i}$. Including the latter in lemma 18.39.12 with exponent one, we see that every target has dimension $O(m^{q-1})$, uniformly. Summing gives the estimate. For $q=1$ each divisor is a point and every target has dimension one.
$\square$
The theorem requires the modification and the divisorial data to be given, with a positive residual; it does not say which selecting potentials admit such data. In the following model with a point pole on the base, both the residual and the Jacobian correction are explicit.
Example 18.39.17.17FFBB442DEB Let $q\geq 2$, $B=\mathbb P^ q$, $A=\mathcal O_ B(1)$, $p=[1:0:\cdots :0]$ and $X=B\times \mathbb P^1$. Use the normalized Fubini–Study forms $\omega _ B,\omega _ F$ and the corresponding metrics, and for a point $o\in \mathbb P^1$ put
\[ R=\frac{\sum _{j=1}^ q|Z_ j|^2}{\sum _{j=0}^ q|Z_ j|^2},\quad T=\| s_ o\| ^2,\quad \varphi =\tfrac 14\log (R^2+T^3),\quad \psi =\tfrac 12\log R. \]
Then $\varphi \in \mathrm{PSH}(X,\omega _ B+\omega _ F)$, and $\psi \in \mathrm{PSH}(B,\omega _ B)$ has mass $1-2^{-q}$. Let $\mathfrak m_ p$ be the ideal sheaf of $p$. The selected and reference spaces are exactly
\[ \begin{gathered} V_ m=H^0\bigl (B,A^ m\otimes \mathfrak m_ p^{k_ m}\bigr ),\qquad W_ m=H^0\bigl (B,A^ m\otimes \mathfrak m_ p^{\ell _ m}\bigr ),\\ k_ m=\max \{ \lfloor m/2-2/3\rfloor -q+1,0\} ,\qquad \ell _ m=\max \{ \lfloor m/2\rfloor -q+1,0\} . \end{gathered} \]
Their defect vanishes for $m<2q$, and for $m\geq 2q$ it is
\[ \dim V_ m-\dim W_ m=\binom {\lfloor m/2\rfloor -1}{q-1} \sim \frac{2^{1-q}}{(q-1)!}m^{q-1}. \]
Proof.
Over a common denominator, $R^2+T^3$ is a sum of squared moduli of holomorphic coefficients of sections of $\mathcal O_ X(2,3)$, divided by the corresponding metric factor. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$, also across the common zero set by the extension theorem.
Let $\rho :Y\to B$ be the blowup of $p$, with exceptional divisor $E$, and let $\tau :Y\to \mathbb P^{q-1}$ be the extension to $Y$ of the projection $[Z_0:\cdots :Z_ q]\mapsto [Z_1:\cdots :Z_ q]$. Since $(\rho ,\tau )$ is an embedding, the forms
\[ \alpha =\rho ^*\omega _ B,\quad \gamma =\tau ^*\omega _{\mathbb P^{q-1}},\qquad \beta =\tfrac 12(\alpha +\gamma ) \]
satisfy $\beta >0$. The divisor bundle is $\mathcal O_ Y(E)=\rho ^*\mathcal O_ B(1)\otimes \tau ^*\mathcal O_{\mathbb P^{q-1}}(-1)$; its quotient metric has curvature $\alpha -\gamma $ and satisfies $R\circ \rho =\| s_ E\| ^2$. Thus the theorem applies with the single divisor $E$, $u=0$, $c=1/2$, $a=2/3$ and residual $\beta $. Along $E\times \{ o\} $ the pulled-back selecting potential has the exact local model $\tfrac 14\log (|z|^4+|w|^6)+O(1)$, and it is locally bounded elsewhere. If $T_ m(s)$ is divisible by $z^{K_ m}$, where $K_ m=\max \{ \lfloor m/2-2/3\rfloor ,0\} $, then lemma 18.39.7 gives integrability in every chart of this model; this gives the inclusion opposite to the upper inclusion of the theorem.
In the blowup coordinates $Z_1/Z_0=z$ and $Z_ j/Z_0=zt_ j$ for $j\geq 2$, the Jacobian of $\rho $ is $z^{q-1}$. Hence, if a section $s$ of $A^ m$ vanishes to order $d$ at $p$, then $T_ m(s)$ vanishes to order $d+q-1$ generically along $E$. The two divisorial thresholds $\lfloor m/2\rfloor $ and $K_ m$ therefore give exactly the orders $\ell _ m$ and $k_ m$ above. Homogeneous polynomials of degree $m$ realize every jet of total degree at most $m$ at $p$. For $m\geq 2q$ we have $k_ m=\ell _ m-1$, and the additional jets of degree $\ell _ m-1$ are spanned by $\binom {\ell _ m+q-2}{q-1}$ monomials, which gives the defect. For $m<2q$ both orders vanish.
Finally, in cohomology $[\beta ]=\rho ^*c_1(A)-\tfrac 12[E]$. The intersection numbers containing both $\rho ^*c_1(A)$ and $[E]$ vanish, because $\rho ^*c_1(A)$ restricts to zero on $E$. The normal bundle of $E\simeq \mathbb P^{q-1}$ is $\mathcal O_ E(-1)$, so $\int _ Y[E]^ q=(-1)^{q-1}$, while $\int _ Y(\rho ^*c_1(A))^ q=1$. Expanding gives $\int _ Y\beta ^ q=1-2^{-q}$, which is the mass of $\psi $ by the theorem.
$\square$
The selecting potential is finite at most points of the fiber over $p$ and equals $-\infty $ at the point $(p,o)$, so again it does not descend, even up to a bounded function. The reference potential $\psi $ has a pole of codimension $q$ at $p$, and the Jacobian factor is what converts the divisorial orders along $E$ into vanishing orders at $p$.
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