Theorem 21.14.2.DCED1F266D4F Let $k\geq 0$ be an integer and $u\in \mathrm{PSH}(X_ n,k\alpha )$. Then $u=\pi ^*v$ for a unique $v\in \mathrm{PSH}(\mathbb P^{n-1},k\sigma )$. For every positive integer $m$, pullback identifies the weighted section spaces isometrically: for a section $s$ in the space on the base,
\[ \begin{split} H^0\bigl (\mathbb P^{n-1},\mathcal O(km)\otimes \mathcal I(mv)\bigr ) & \simeq H^0\bigl (X_ n,L_ k^ m\otimes \mathcal I(mu)\bigr ),\\ \| \pi ^*s\| _{X_ n,m,u}^2& =\| s\| _{\mathbb P^{n-1},m,v}^2. \end{split} \]
Here the norm on the base is defined with the Fubini–Study metric and the volume form $\sigma ^{n-1}$. The kernels $K$ and, off their pole sets, $B$ on $X_ n$ are the pullbacks of the corresponding kernels on the base. Pullback also identifies the bounded weighted section algebras $\mathcal W_\bullet (v)$ and $\mathcal W_\bullet (u)$.
Proof.
The descent of the potential follows from proposition 5.43.5. By the proof of proposition 19.8.13, the unweighted section spaces on both manifolds consist of the same homogeneous polynomials of degree $km$, and the pointwise norms of corresponding sections agree because the metrics correspond. For every nonnegative Borel function $F$ on the base, fiber integration gives
\[ \int _{X_ n}(F\circ \pi )\, \mathrm{d}V =\frac12\int _{\mathbb P^{n-1}}F\sigma ^{n-1} \int _{\pi ^{-1}(y)}\zeta =\int _{\mathbb P^{n-1}}F\sigma ^{n-1}. \]
Applied to $F=|s|^2\mathrm{e}^{-mv}$, with arbitrary values on the pole set, this formula gives equality of the two possibly infinite integrals. Since finiteness of the weighted norm characterizes the multiplier-ideal section spaces, this proves the asserted isometry. An orthonormal basis therefore pulls back to an orthonormal basis, which proves the kernel identities. Finally, a nonnegative function on the base is bounded if and only if its pullback is bounded, so pullback also identifies the bounded weighted sections, compatibly with multiplication.
$\square$
Pome
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