Proposition 18.35.3.FDE01F2B4B01 Let the line bundle $L$ with metric $h$, the selecting potential $\varphi \in \mathrm{QPSH}(X)$ and the continuous weight $v\in C(X)$ be as in section 18.30, and use the volume $\nu _ X$ on $X$. On $Y$, use the line bundles $\pi ^*L^ m\otimes R_\pi $ with the metrics $\pi ^*h^ m\otimes h_ R$, the selecting potential $\varphi \circ \pi $, the continuous weight $v\circ \pi $ and the volume $\nu _ Y$, and define the kernels and kernel measures as in section 18.30, with the fixed factor $(R_\pi ,h_ R)$ added. Then the identification $s\mapsto J_\pi \pi ^*s$,
\[ H^0(X,L^ m\otimes \mathcal I(m\varphi )) \simeq H^0(Y,\pi ^*L^ m\otimes R_\pi \otimes \mathcal I(m\varphi \circ \pi )), \]
is an isometry for the continuous weighted norms. The kernels and kernel measures on $Y$ and $X$ satisfy
\[ B_{Y,m}=|J_\pi |_{h_ R}^2(B_{X,m}\circ \pi ), \qquad \pi _*\beta _{Y,m}^{(r)}=\beta _{X,m}^{(r)} \]
for the same normalization integer $r$, also when the section spaces are zero.
Proof.
The identification is lemma 5.43.22 with $M=L^ m$ and $a=m\varphi $. By (18.35.2) and the change of variables formula,
\[ \int _ Y|J_\pi \pi ^*s|_{\pi ^*h^ m\otimes h_ R}^2\mathrm{e}^{-mv\circ \pi }\, \mathrm{d}\nu _ Y =\int _ X|s|_{h^ m}^2\mathrm{e}^{-mv}\, \mathrm{d}\nu _ X. \]
The selecting potential has already determined the section spaces and does not enter the norms. Hence the identification maps an orthonormal basis to an orthonormal basis, and summing the squared pointwise norms of the basis vectors gives the kernel formula, also at the zeros of $J_\pi $. Multiplying by $r!/m^ r$ and integrating a continuous function pulled back from $X$ proves the measure formula.
$\square$
Pome
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