Let $\pi :X\to Y$ be a surjective holomorphic submersion from a compact complex manifold $X$, with connected fibers of dimension $f\geq 1$, and put $N=q+f=\dim X$. Fix a Hermitian form $\omega _ X$ on $X$. Use $E=\pi ^*L$ with the metric $h_ E=\pi ^*h$, and a smooth positive volume form $\nu _ X$ satisfying
A73831DBE999
18.44.10
\begin{equation} \pi _*\nu _ X=\nu _ Y. \end{equation}
Choose $p\in \pi ^{-1}(x)$ and submersion coordinates $(z,w)$ centered at $p$, in which $\pi (z,w)=z$. Let $\chi _ p$ be a function equal to $\log (|z|^2+|w|^2)$ near $p$, smooth on $X\setminus \{ p\} $, and zero outside a larger coordinate ball. Define the selecting potentials, which depend on $m$,
\[ \varphi _ m=\frac{k_ m+N-1}{m}\chi _ p,\qquad \mathcal I_ m=\mathcal I(m\varphi _ m). \]
As in definition 6.8.5, $\mathcal I_ m$ consists of the germs $F$ for which $|F|^2\mathrm e^{-m\varphi _ m}$ is locally integrable. The selected sections are measured with the smooth norm defined by $h_ E^ m$ and $\nu _ X$.
Theorem 18.44.11.154EDD25787A For all large $m$, $\varphi _ m\in \mathrm{PSH}(X,\omega _ X)$ and $\mathcal I_ m=\mathfrak m_ p^{k_ m}$. Neither $\varphi _ m$ nor $\mathcal I_ m$ is pulled back from $Y$, but pullback gives an isometry of Hilbert spaces and an identity of densities
\[ H^0(X,E^ m\otimes \mathcal I_ m)=\pi ^*V_ m,\qquad B_{X,m}=B_ m\circ \pi . \]
Thus the local laws above hold for every sequence of points of $X$ whose projections satisfy their hypotheses. Moreover $\log N_ m/\log m\to q<\dim X$, and
1D6E0EF28851
18.44.12
\begin{equation} m^{-q}B_{X,m}\nu _ X\longrightarrow \left(\frac{\theta ^ q/q!}{\nu _ Y}\circ \pi \right)\nu _ X \quad \hbox{in total variation}. \end{equation}
Normalizing by $N_ m$ instead gives the limit divided by $D$, a probability measure.
Proof.
The function $\chi _ p$ satisfies $\mathrm{dd}^{\mathrm{c}}\chi _ p\geq -C\omega _ X$, because it is plurisubharmonic near its pole and smooth elsewhere. Since its coefficient $(k_ m+N-1)/m$ tends to zero, $\varphi _ m$ is $\omega _ X$-plurisubharmonic for large $m$. A germ of order $\ell $ at $p$ has a nonzero initial homogeneous term, so the integral of its squared modulus over the sphere of radius $s$ is $cs^{2\ell }(1+o(1))$ with $c>0$. Local integrability is therefore equivalent to
\[ \int _0^{s_0}s^{2\ell -2(k_ m+N-1)+2N-1}\, \mathrm ds =\int _0^{s_0}s^{2\ell -2k_ m+1}\, \mathrm ds<\infty , \]
that is, to $\ell \geq k_ m$. Away from $p$ the ideal is the unit ideal. This proves $\mathcal I_ m=\mathfrak m_ p^{k_ m}$. The ideal is proper at $p$ but trivial at the other points of the same fiber, which is impossible for an ideal pulled back from $Y$; likewise, the pole of $\varphi _ m$ occurs only at $p$.
By the maximum modulus principle, every section of $\pi ^*L^ m$, written in a trivialization pulled back from the base, is constant on each compact connected fiber. Local holomorphic sections of the submersion show that the resulting coefficient on the base is holomorphic, and constancy along the fibers makes these coefficients glue. In the submersion coordinates, a pulled-back coefficient depends only on $z$, so its order at $p$ equals its order at $x$. This proves the identity of the selected spaces. Fiber integration and (18.44.10) give
\[ \int _ X|\pi ^*s|_{h_ E^ m}^2\nu _ X=\int _ Y|s|_{h^ m}^2\nu _ Y. \]
Pulling back an orthonormal basis proves the identity of densities. If $b_ m$ and $b$ are the densities, with respect to $\nu _ Y$, of the measures in the total-variation limit of theorem 18.44.2, then the corresponding densities on $X$ are $b_ m\circ \pi $ and $b\circ \pi $, and their $L^1(\nu _ X)$ distance equals $\int _ Y|b_ m-b|\nu _ Y$. This proves (18.44.12). The dimension formula (18.44.4) gives $\log N_ m/\log m\to q$ and the final normalization.
$\square$
Volume forms satisfying (18.44.10) exist on every such submersion. For any smooth positive volume form $\mu _ X$, write $\pi _*\mu _ X=b\nu _ Y$. Submersion charts and a finite partition of unity express $b$ as a sum of integrals of smooth compactly supported densities in the fiber variables, and differentiation under the integral sign shows that $b$ is smooth. Since the fibers are nonempty, $b>0$. Thus $\nu _ X=\mu _ X/(b\circ \pi )$ satisfies (18.44.10). For a product $X=Y\times F$, the fiber $F$ may be any compact connected complex manifold, Kähler or not. On a product, both the volumes $\nu _ Y\otimes \nu _ F$ with $\nu _ F(F)=1$ and the nonproduct volumes $W(y,u)\nu _ Y\nu _ F$, with $W>0$ and $\int _ FW(y,u)\nu _ F=1$, satisfy (18.44.10).
For an arbitrary smooth positive volume $\nu '_ X$, the pulled-back norm becomes the norm on the base for the volume $\pi _*\nu '_ X$, whereas the local laws above were proved for the specific volume $\nu _ Y$. Since the selecting potentials vary with $m$, theorem 18.44.11 does not settle 24.10.2 for fixed singularities. Weights with mixed or angular remainder terms, observation points with some $\alpha _ j=0$, and other ideals on the base are not treated here. The lifted measure keeps the chosen conditional fiber volume; it is not a Monge–Ampère measure in the dimension of $X$.
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