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Return to a fibration with compact connected fibers, and let the selecting potential have isolated logarithmic poles on the total space. Every selected section descends to the base, and in large tensor powers it must vanish to a certain order at the images of the poles. This order is computed from integrability in all variables of the total space, so it differs from the order imposed by a pole of the same coefficient on the base by at most the fiber dimension. The jets in between span $O(m^{q-1})$ dimensions on a $q$-dimensional base. This is negligible after normalization by $m^ q$, so the kernel measures converge to the same quotient equilibrium measure as for the corresponding poles on the base.
The comparison rests on the following elementary fact: for nested section spaces, the total variation of the difference of their kernel measures is the normalized codimension.
Lemma 18.37.1.22D696DB398E Let $B$ be a connected compact complex manifold of dimension $q\geq 1$, let $A\to B$ be a holomorphic line bundle with a smooth Hermitian metric, let $v\in C(B)$, and let $\nu $ be a smooth positive probability volume on $B$. For every integer $m\geq 1$ let $W_ m\subseteq V_ m\subseteq H^0(B,A^ m)$ be subspaces, both with the norm $\| \cdot \| _{m,v,\nu }$, and put $d_ m=\dim V_ m-\dim W_ m$. Write $B_{V_ m,v}$, $B_{W_ m,v}$ for their kernels and $\beta _{V_ m,v}^{(q)}$, $\beta _{W_ m,v}^{(q)}$ for their kernel measures, defined as in section 18.30 with the space indicated by the subscript. Then
\[ \begin{gathered} B_{V_ m,v}-B_{W_ m,v}\geq 0,\\ \| \beta _{V_ m,v}^{(q)}-\beta _{W_ m,v}^{(q)}\| _{\mathrm{TV}} =\frac{q!d_ m}{m^ q}. \end{gathered} \]
The determinant functionals of the two spaces, each normalized to vanish at the zero weight, satisfy
\[ |\mathcal F_{V_ m}^{(q)}(v)-\mathcal F_{W_ m}^{(q)}(v)| \leq \frac{q!d_ m}{m^ q}\| v\| _\infty . \]
These statements include the case of zero-dimensional spaces.
Proof.
Extend an orthonormal basis of $W_ m$ to an orthonormal basis of $V_ m$ for the norm at the weight $v$. The difference of the kernels is the sum of the squared weighted evaluations of the $d_ m$ added basis vectors. It is nonnegative, and its integral against $\nu $ is $d_ m$, which gives the total variation formula. For the determinant functionals, apply the first-variation formula of proposition 18.34.1 to each of the two fixed spaces, and integrate the difference along $t\mapsto tv$, $0\leq t\leq 1$. By what we have just proved, at each $t$ the difference of the kernel measures is positive with total mass $q!d_ m/m^ q$, so its integral against $v$ is at most $q!d_ m\| v\| _\infty /m^ q$ in absolute value. Since both functionals vanish at zero, this proves the last assertion.
$\square$
Return to a submersion $\pi :X\to B$ and a pulled-back line bundle $L=\pi ^*A$ as in section 18.36, and write $f=n-q$ for the fiber dimension. Fix distinct points $p_1,\ldots ,p_ N$ of $X$ and positive rational numbers $a_1,\ldots ,a_ N$. Suppose that $\varphi \in \mathrm{QPSH}(X)$ is locally bounded away from these points and that, in local coordinates $x$ centered at $p_ j$,
DA25AD427AC7
18.37.2
\begin{equation} \varphi (x)=a_ j\log |x|^2+O(1). \end{equation}
Write
\[ Z=\{ \pi (p_1),\ldots ,\pi (p_ N)\} ,\qquad a_ b=\max _{\pi (p_ j)=b}a_ j\quad (b\in Z), \]
and, for an integer $m\geq 1$,
\[ r_{m,b}=\max \{ \lfloor ma_ b\rfloor -n+1,0\} ,\qquad s_{m,b}=\max \{ \lfloor ma_ b\rfloor -q+1,0\} . \]
Let $\mathfrak m_ b$ denote the ideal sheaf of a point $b\in B$; its zeroth power is $\mathcal O_ B$.
Proposition 18.37.3.BEF6F7AD320C Pullback identifies the selected space $H^0(X,L^ m\otimes \mathcal I(m\varphi ))$ with
\[ V_ m=H^0\left(B,A^ m\otimes \bigcap _{b\in Z}\mathfrak m_ b^{r_{m,b}}\right). \]
The space
\[ W_ m=H^0\left(B,A^ m\otimes \bigcap _{b\in Z}\mathfrak m_ b^{s_{m,b}}\right) \]
satisfies $W_ m\subseteq V_ m$ and
\[ 0\leq \dim V_ m-\dim W_ m \leq \sum _{b\in Z}\sum _{j=r_{m,b}}^{s_{m,b}-1} \binom {j+q-1}{q-1} =O(m^{q-1}), \]
where an empty inner sum is zero. The implied constant depends only on the points and their coefficients, not on $m$.
Proof.
Lemma 5.37.1, applied with a point as center, gives
\[ \mathcal I(m\varphi )_{p_ j}=\mathfrak m_{p_ j}^{\max \{ \lfloor ma_ j\rfloor -n+1,0\} }; \]
away from the points $p_ j$ the ideal is the full structure sheaf.
Every section of $L^ m$ descends to $B$, by the argument in proposition 18.36.1. Near $p_ j$ choose submersion coordinates $(z,\tau )$ with $\pi (z,\tau )=z$. A pulled-back section has a local coefficient $g(z)$ that does not depend on $\tau $, and its vanishing order at $(0,0)$ equals the vanishing order of $g$ at $z=0$. The conditions imposed by the points of one fiber are therefore vanishing conditions at the same base point, and the strongest of them is vanishing to order $r_{m,b}$. This proves the identification.
For the dimension bound, map each section in $V_ m$ to its jets modulo $\mathfrak m_ b^{s_{m,b}}$ at the finitely many points $b\in Z$. The kernel of this map is $W_ m$, so local trivializations of $A^ m$ give an injection
\[ V_ m/W_ m\hookrightarrow \bigoplus _{b\in Z}\mathfrak m_ b^{r_{m,b}}/ \mathfrak m_ b^{s_{m,b}}. \]
Surjectivity of the jet map is not needed. In $q$ variables there are $\binom {j+q-1}{q-1}$ monomials of degree $j$, which gives the dimensions of these quotients. Finally $0\leq s_{m,b}-r_{m,b}\leq f$ and $s_{m,b}=O(m)$, which proves the bound, also when $q=1$ or $f=0$.
$\square$
Equip $A$ with a smooth metric $h_ A$ and put $\theta =c_1(A,h_ A)$. Choose a smooth positive probability volume $\nu _ B$ on $B$, smooth fiber probabilities as in section 18.36, and the smooth positive volume $\nu _ X=\mathcal L\nu _ B$ on $X$. Use the pullback metric on $L$, and for $v\in C(B)$ use the weight $v\circ \pi $ in the norm on $X$. We write $\beta _{X,m,v\circ \pi ,\varphi }^{(q)}$ and $\mathcal F_{X,m}^{(q)}$ for the kernel measure and the determinant functional on $X$ defined by these data and the selecting potential $\varphi $.
Theorem 18.37.4.D9B0D46D42B2 Assume that $B$ is Kähler and that there is $\psi \in \mathrm{PSH}(B,\theta )$ with positive non-pluripolar mass, locally bounded off $Z$, such that
\[ \psi (z)=a_ b\log |z-b|^2+O(1)\quad \hbox{near each }b\in Z \]
in local coordinates $z$ on $B$. Let $E(v)=E_{\theta ,B}[\psi ]_{\mathcal I}(v)$. Then, for the selecting potential $\varphi $ of (18.37.2),
\[ \beta _{X,m,v\circ \pi ,\varphi }^{(q)} \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E(v))\bigr ). \]
More precisely, the total variation distance between $\beta _{X,m,v\circ \pi ,\varphi }^{(q)}$ and the lift of the kernel measure on $B$ selected by $\psi $ is $O(m^{-1})$. The determinant functionals satisfy
\[ \mathcal F_{X,m}^{(q)}(v\circ \pi )\longrightarrow \int _0^1\! \left(\int _ B v\, \mathrm{d}\mathrm{MA}_\theta (E(tv))\right)\mathrm{d}t. \]
The normalized dimensions $q!N_ m/m^ q$ converge to the total mass of the limit measure, which is finite and positive. If $n>q$, the kernel measures with the normalization $n!/m^ n$ tend to zero in total variation.
Proof.
By lemma 5.37.1 in dimension $q$, the space of sections of $A^ m$ selected by $\psi $ on $B$ is the space $W_ m$ of proposition 18.37.3. Corresponding sections in $V_ m$ and on $X$ have equal norms, since their pointwise norms and the continuous weight are pulled back from $B$ and every fiber probability has mass one. As in proposition 18.36.1, pulling back an orthonormal basis shows that the kernel measure on $X$ is the lift of the kernel measure of $V_ m$ on $B$.
By lemma 18.37.1, proposition 18.37.3, the $q$-normalized kernel measures of $V_ m$ and $W_ m$ on $B$ differ by a positive measure of total mass $O(m^{-1})$, and so do their lifts. Apply theorem 18.35.5 on $B$, with the identity as Kähler modification; its hypotheses are among ours. It gives the convergence of the kernel measures of $W_ m$, and since the difference tends to zero, the asserted limit on $X$ follows.
By the same lemma, the determinant functionals of $V_ m$ and $W_ m$ differ by at most $O(m^{-1})\| v\| _\infty $. The isometric identification of the norms and corollary 18.35.6 on $B$ give the limit of the functionals. Testing the constant function one proves the assertion about the dimensions. Since the $q$-normalized masses are bounded, multiplying them by $(n!/q!)m^{q-n}$ proves the last assertion.
$\square$
If $\theta >0$, lemma 5.37.2, applied on $B$ with the Hermitian form $\theta $, constructs the required $\psi $ whenever the coefficients $a_ b$ are sufficiently small. Off the poles, $\theta +\mathrm{dd}^{\mathrm{c}}\psi $ then dominates a positive multiple of $\theta $, so the non-pluripolar mass of $\psi $ is positive; it is finite because $\theta $ is closed and $B$ is Kähler. The same lemma, applied on $X$ with a Hermitian form, constructs $\varphi $. If the fibers have positive dimension, such a function $\varphi $ does not descend to $B$, since it is finite at the other points of a fiber containing a pole.
The section formula of proposition 18.37.3 also shows that the kernels cannot recover the singularities of the selecting potential. Moving an isolated pole within its fiber does not change any selected space, and neither does adding further poles in that fiber whose coefficients do not exceed the largest one. For the fixed pullback metric $h=\pi ^*h_ A$, continuous weight $v\circ \pi $ and lifted volume $\nu _ X$, all kernels and determinant functionals are therefore the same in every degree. The multiplier ideals on $X$ are nevertheless different: for large weights, the ideal of one potential is a proper ideal at a point where the ideal of the other is the full local ring. Under the hypotheses of theorem 18.37.4, this loss of information occurs together with a positive limit at the growth rate $m^ q$. In the closed theory, by contrast, the limit envelope has the multiplier-ideal type of the selecting potential (theorem 18.32.2); this property is lost when the selecting potential is plurisubharmonic only with respect to a Hermitian form.
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