Pome14F47AA5A61BThe comparison of theorem 18.39.3 does not need the exact local form (18.39.2). It suffices to bound the selecting potential from below by a divisorial pole on all of $X$, and from above by an anisotropic model in a single chart over the divisor. The lower bound gives the inclusion of the reference spaces into the selected spaces. The upper bound forces vanishing along the divisor to an order that falls short of the reference order by a bounded amount: separate powers of the normal variables replace the shift by the fiber dimension with a weighted shift. The selecting potential may then be singular along a positive-dimensional subset of a fiber, and it need not descend to the base.
Let $\pi :X\to B$ be a surjective holomorphic submersion of connected compact complex manifolds with connected fibers of dimension $f\geq 1$, and let $B$ be Kähler of dimension $q\geq 1$. Fix a Hermitian form $\omega $ on $X$, a holomorphic line bundle $A$ on $B$ with a smooth Hermitian metric $h$, and $\varphi \in \mathrm{PSH}(X,\omega )$. Write $\theta =c_1(A,h)$, and use holomorphic descent to regard the selected spaces as subspaces of $H^0(B,A^ m)$:
Let $D\subset B$ be a nonempty smooth connected reduced divisor. Fix a smooth metric $H_ D$ on $\mathcal O_ B(D)$, and write $s_ D$ for its canonical section and $\eta =c_1(\mathcal O_ B(D),H_ D)$ for its curvature.
Lemma 18.39.7.BBA6C67F74CF Let $r\geq 1$ and $\ell \geq 0$ be integers, let $p,b_1,\ldots ,b_ r$ be positive integers, and let $P>0$. The function
of $(z,w)\in \mathbb C\times \mathbb C^ r$ is locally integrable near the origin if and only if
In the case of equality, the integral diverges logarithmically.
Proof. After integrating over the angles, substitute $x_0=|z|^{2p}$ and $x_ j=|w_ j|^{2b_ j}$. Up to a positive constant, the integral becomes
over a box near the origin. This box contains a small simplex $\{ \sum _ jx_ j<\delta \} $ and is contained in a larger one. On either simplex use the coordinates $\rho =\sum _ jx_ j$ and $x_ j=\rho y_ j$. The integral over $y$ is finite and positive, because all its exponents exceed $-1$, and the integral over $\rho $ is
This gives the threshold and the logarithmic divergence at equality. Bounded factors in the weight, holomorphic units and additional tangential variables do not affect this local test. $\square$
Theorem 18.39.8.A79139E2A3C6 Suppose that $c>0$ and $u\in C^\infty (B,\mathbb R)$ satisfy
for a constant $C$. Suppose further that there is one submersion polydisk over an open subset of $D$, with coordinates
where $1\leq r\leq f$, on which
for positive integers $p,b_1,\ldots ,b_ r$ and a constant $C'$. Put
Then $\psi \in \mathrm{PSH}(B,\theta )$, its non-pluripolar mass is $\int _ B\beta ^ q\in (0,\infty )$, and for every $m\geq 1$,
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. $\square$
Corollary 18.39.9.094794C14195 Keep the positivity condition $\beta >0$ of theorem 18.39.8. Suppose that a nonempty compact smooth submanifold $Z\subset \pi ^{-1}(D)$ projects submersively onto $D$, and that near every point of $Z$ there are coordinates as in that theorem in which
with fixed $c,p,b_ j$. Suppose also that $\varphi $ is locally bounded outside $Z$. Then $\varphi $ satisfies the hypotheses of theorem 18.39.8, so its conclusions hold, and moreover
Proof. The local model is bounded below by $c\log |z|^2$ up to a constant. Near points of $\pi ^{-1}(D)$ outside $Z$, the function $\varphi $ is locally bounded, so the same lower bound holds there; away from $\pi ^{-1}(D)$ both functions are locally bounded. By compactness, finitely many charts give the global lower bound. The upper bound holds in any chart of the model. Conversely, if a section is divisible by $z^{K_ m}$, its residual coefficient is bounded and holomorphic in each chart of the model, and the strict inequality $K_ m+1>mc-a$ together with lemma 18.39.7 gives integrability there. Outside $Z$ the function $\varphi $ is locally bounded, which proves the reverse inclusion. $\square$
Such exact models exist with pole sets of positive dimension inside the fibers. Let $B$ be projective, let $A$ be ample, and let $X=B\times (\mathbb P^1)^ f$ with $\pi $ the first projection. Mark a point $o_ j$ in each of the first $r$ factors $\mathbb P^1$ of the fiber, and let $E_ j\subset X$ be the inverse image of $o_ j$. With smooth metrics on the corresponding line bundles, put
The common zero set of the terms of $Q$ is $D\times \{ o_1\} \times \cdots \times \{ o_ r\} \times (\mathbb P^1)^{f-r}$. Near it, the positive factors of the metrics give exactly the local model of corollary 18.39.9, and $\varphi $ is smooth elsewhere. The inequality $Q\geq \| \pi ^*s_ D\| ^{2p}$ gives the global lower bound. Locally, $\mathrm{dd}^{\mathrm{c}}\log Q$ is bounded below by a convex combination of the negatives of the smooth curvature forms of the terms; hence $\mathrm{dd}^{\mathrm{c}}\log Q\geq -C_0\omega $ for a constant $C_0$, first off the common zero set and then across it, after a quadratic correction, by the extension theorem for plurisubharmonic functions. Since $A$ is ample, there is $u$ with $\theta +\mathrm{dd}^{\mathrm{c}}u>0$. For sufficiently small $\lambda >0$ we then have $\lambda C_0\leq 1$, so that $\varphi \in \mathrm{PSH}(X,\omega )$, and $\theta +\mathrm{dd}^{\mathrm{c}}u-p\lambda \eta >0$. Thus the exact series of the corollary also occurs when $r<f$.
Example 18.39.10.EB221EFD409A Let $B=\mathbb P^ q$ with a hyperplane $D$, let $A=\mathcal O_ B(1)$, and let $X=B\times \mathbb P^1$. Let $\omega _ B,\omega _ F$ be the Fubini–Study forms normalized to have total mass one, and use the corresponding metrics on the divisor line bundles. For a point $o\in \mathbb P^1$ set
Then $p=2$, $b_1=3$, $c=1/2$, $a=2/3$, and the reference potential $\psi =\tfrac 12\log \| s_ D\| ^2$ has mass $2^{-q}$. The spaces $V_1$ and $W_1$ agree, and for every $m\geq 2$ the defect is
For $q=1$ the defect is exactly one for every $m\geq 2$.
Proof. Locally, the complex Hessian of the logarithm of a sum of two squared norms of holomorphic sections is bounded below by a convex combination of the negatives of their curvature forms. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$ off the pole set, and across it by the extension theorem after a quadratic correction. Thus $\varphi \in \mathrm{PSH}(X,\omega )$. The pole set is $D\times \{ o\} $, along which $\varphi $ has the exact local model of corollary 18.39.9. Here $u=0$, $\beta =\tfrac 12\omega _ B$, and $K_ m=L_ m-1$ for $m\geq 2$, while $K_1=L_1=0$. The spaces $V_ m$ and $W_ m$ therefore correspond to homogeneous polynomials of degrees $m-L_ m+1$ and $m-L_ m$, and since $m-L_ m=\lceil m/2\rceil $, the difference of their dimensions is the displayed binomial coefficient. $\square$
In this example $\varphi $ is finite on most of each fiber over $D$ and equals $-\infty $ on a proper subset of it, so $\varphi $ differs from every pullback of a function on $B$ by an unbounded amount. For $p=b_ j=1$ and $r=f$, the shift $a=f$ is the shift by the fiber dimension found in theorem 18.39.3. Theorem 18.39.8 thus gives sufficient conditions for the approximation asked for in 24.10.2; we do not know which other fiber singularities admit such an approximation.
The assumption $f\geq 1$ excludes only a trivial case. If the fibers have dimension zero, then $\pi $ is a covering, since $X$ is compact and $\pi $ is a submersion, and hence a biholomorphism, since the fibers are connected. If, in addition, the descended selecting potential is $\theta $-plurisubharmonic with finite positive mass, taking it as $\psi $ gives $V_ m=W_ m$.
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