Pome9C902F002FA9Keep the submersion $\pi :X\to B$, the line bundle $(A,h)$, the selecting potential $\varphi $ and the descended spaces $V_ m$ of subsection 18.39.6. Several divisorial singularities can be handled at once if the divisors cross normally. At their intersections the integrability test on the base separates in normal coordinates, and a bounded change of each divisorial order again costs only $O(m^{q-1})$ sections. For this we need the restriction estimate with all the divisorial twists at once.
Lemma 18.39.12.46F892FF98FE Let $M$ be a compact complex manifold of dimension $d\geq 0$, and let $E,F_1,\ldots ,F_ N$ be holomorphic line bundles on $M$. For each $C_0>0$ there is a constant $C_1$ such that
Proof. For $d=0$ the dimension is at most the number of points of $M$. For $d\geq 1$, use fixed smooth metrics, a smooth positive volume form and finitely many framed charts as in the proof of lemma 18.39.1. The local weight of the metric is $m\phi _ E+\sum _ i j_ i\phi _{F_ i}$, whose second derivatives are bounded by $Cm$ uniformly over the stated range. After its constant and real-linear terms at a point have been removed by a holomorphic affine change of frame, it is bounded by a fixed constant on a ball of radius $\varepsilon m^{-1/2}$ about that point. The submean inequality then gives $|s(x)|^2\leq Cm^ d\| s\| _{L^2}^2$, uniformly in $x$, $m$ and the integers $j_ i$. Since evaluation has a one-dimensional target, an orthonormal basis satisfies $\sum _\ell |s_\ell (x)|^2\leq Cm^ d$, and integration gives the estimate. The constants depend only on the fixed bundles, metrics, charts, volume and $C_0$. $\square$
Let $D_1,\ldots ,D_ N$ be distinct nonempty smooth connected reduced divisors on $B$ whose union has simple normal crossings. For each $i$, choose a smooth metric $H_ i$ on $\mathcal O_ B(D_ i)$, and write $s_ i$ for the canonical section and $\eta _ i$ for the curvature.
Theorem 18.39.13.A7CF6A5A6A5C Suppose that $c_1,\ldots ,c_ N>0$ and $u\in C^\infty (B,\mathbb R)$ satisfy
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
where $p_ i$ and $b_{ij}$ are positive integers and $C_ i$ is a constant. Put
Then $\psi \in \mathrm{PSH}(B,\theta )$ has non-pluripolar mass $\int _ B\beta ^ q\in (0,\infty )$, and for every $m\geq 1$,
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
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
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. $\square$
The upper bounds are required only away from the crossings; integrability at the crossings is controlled by the global lower bound. Exact models can nevertheless have fiber poles over the crossings, as in the following example, where different divisors use separate fiber coordinates.
Example 18.39.14.411D2D5FFC5D Let $B=(\mathbb P^1)^ q$, $X=B\times (\mathbb P^1)^ q$ and $A=\mathcal O_ B(1,\ldots ,1)$. For each $i$, let $D_ i\subset B$ be the inverse image of a marked point in the $i$th factor of $B$, and let $E_ i\subset X$ be the inverse image of a marked point in the $i$th factor of the fiber. Use normalized Fubini–Study metrics on the corresponding line bundles, and write $\omega _{B,i},\omega _{F,i}$ for the pullbacks to $X$ of the normalized Fubini–Study forms on the $i$th factors of $B$ and of the fiber. Put
Then $\varphi \in \mathrm{PSH}(X,\omega )$, and theorem 18.39.13 applies with $u=0$, $c_ i=1/2$, $p_ i=2$, $b_{i1}=3$ and $a_ i=2/3$. The reference potential $\psi $ has mass $q!/2^ q$. With $L_ m=\lfloor m/2\rfloor $ and $K_ m=\max \{ \lfloor m/2-2/3\rfloor ,0\} $, the selected and reference spaces are exactly
They agree for $m=1$, and for $m\geq 2$ the defect is
For $q\geq 2$ this is $q2^{1-q}m^{q-1}+O(m^{q-2})$, and for $q=1$ it is exactly one.
Proof. By the calculation in example 18.39.10, the complex Hessian of each summand of $\varphi $ is bounded below by $-\tfrac 12\omega _{B,i}-\tfrac 34\omega _{F,i}$, also across its pole set by the extension theorem after a quadratic correction. Summing proves $\varphi \in \mathrm{PSH}(X,\omega )$. The global lower bound follows from $R_ i^2+T_ i^3\geq R_ i^2$. For the upper bound over $D_ i$, away from the other divisors, place the $i$th fiber coordinate at its marked point; all the other summands are smooth there. The residual is $\beta =\tfrac 12\sum _ i\omega _{B,i}>0$, whose top integral is $q!/2^ q$.
The theorem gives the required generic orders along every $D_ i$. Conversely, near a point where several of the pairs $(R_ i,T_ i)$ vanish, the density $\mathrm{e}^{-m\varphi }$ is comparable to the product of the factors $(|z_ i|^4+|w_ i|^6)^{-m/4}$ in disjoint pairs of variables. A descended coefficient divisible by every $z_ i^{K_ m}$ has a bounded holomorphic residual. Each factor is integrable because $K_ m+1>m/2-2/3$, and Fubini's theorem gives integrability at the intersections; elsewhere the same argument uses fewer factors. This proves the identity for $V_ m$, and the theorem gives $W_ m$. A line bundle $\mathcal O_ B(d,\ldots ,d)$ with $d\geq 0$ has $(d+1)^ q$ independent sections, the products of monomials in the separate factors. Since $K_ m=L_ m-1$ for $m\geq 2$ and $m-L_ m=\lceil m/2\rceil $, subtraction gives the displayed defect and its leading term. $\square$
As before, the selecting potential takes finite values and the value $-\infty $ in the fibers over the divisors, so it does not descend. Theorem 18.39.13 extends the sufficient conditions for 24.10.2 to normal crossing divisors; it does not cover base ideals of other forms, and it gives no necessary conditions.
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