Descent also identifies the spaces of holomorphic sections selected by multiplier ideals, together with their weighted $L^2$ norms. In particular, sections of pulled-back line bundles grow quadratically, as on the two-dimensional base. For integers $a,b$, write $\mathcal O_ B(a,b)$ for the tensor product of the pullbacks of $\mathcal O_{\mathbb P^1}(a)$ and $\mathcal O_{\mathbb P^1}(b)$ from the two factors of $B$.
Proposition 10.19.6.CDBA8166D33B Let $a,b$ be positive integers, let $L_ B=\mathcal O_ B(a,b)$ and $L=\pi ^*L_ B$. For every quasi-plurisubharmonic $v$ on $B$ and positive integer $m$, pullback gives an isomorphism
\[ H^0\bigl (B,L_ B^ m\otimes \mathcal I(mv)\bigr ) \longrightarrow H^0\bigl (X,L^ m\otimes \mathcal I(mv\circ \pi )\bigr ). \]
For $v=0$ these spaces have dimension $(am+1)(bm+1)$.
Equip $L_ B$ with a smooth Hermitian metric and $L$ with its pullback, choose a continuous real function $\varphi $ on $B$, let $\mathrm{d}V_ X$ be a smooth positive volume form on $X$, and put $\nu =\pi _*\mathrm{d}V_ X$. Use the norms
\[ \| s\| _ B^2=\int _ B|s|^2\mathrm{e}^{-m\varphi }\, \mathrm{d}\nu , \qquad \| \pi ^*s\| _ X^2=\int _ X|\pi ^*s|^2 \mathrm{e}^{-m\varphi \circ \pi }\, \mathrm{d}V_ X. \]
Then the isomorphism is isometric. Let $K_{m,B}$ and $K_{m,X}$ be the corresponding partial Bergman kernels, that is, the sums of the weighted pointwise squared norms $|s|^2\mathrm{e}^{-m\varphi }$ and $|\pi ^*s|^2\mathrm{e}^{-m\varphi \circ \pi }$ over orthonormal bases of the two spaces. Then
\[ K_{m,X}=K_{m,B}\circ \pi , \qquad \pi _*(K_{m,X}\, \mathrm{d}V_ X)=K_{m,B}\, \mathrm{d}\nu . \]
Proof.
The identification of the section spaces is proposition 18.36.1, applied to $L_ B$ and $v$ on the base. For $v=0$, the usual basis of bihomogeneous monomials on $\mathbb P^1\times \mathbb P^1$ has $(am+1)(bm+1)$ elements.
Integration along the fibers of the proper submersion shows that $\nu $ is a smooth positive volume form, and the defining identity $\int _ Bg\, \mathrm d\nu =\int _ X(g\circ \pi )\, \mathrm dV_ X$ proves equality of the two norms. Pulling back an orthonormal basis and comparing pointwise norms gives the first kernel identity, and the same integration identity gives the second. The assertions include the case of a zero selected space, for which both kernels vanish.
$\square$
Even without a singular selecting weight, the growth is therefore quadratic:
\[ \lim _{m\to \infty }m^{-2}h^0(X,L^ m)=ab, \qquad \lim _{m\to \infty }m^{-3}h^0(X,L^ m)=0. \]
The next calculation shows that the variational volume of a small Hermitian perturbation detects this missing dimension of growth.
Pome
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