PomeLemma 16.28.9.5C02C050C6B5 The map $u\mapsto u-\delta \chi /2$, followed by the canonical extension across the hyperplane at infinity, is a bijection from $\mathcal L_\delta $ onto $\mathrm{PSH}(\mathbb P^ n,\theta _\delta )$.
Lemma 16.28.95C02C050C6B5
Proof. Use homogeneous coordinates $[Z_0:\cdots :Z_ n]$ and the affine chart $Z_0\ne 0$. On a chart $Z_ i\ne 0$ meeting $H=\{ Z_0=0\} $, the local plurisubharmonic representative of $p=u-\delta \chi /2$ is
\[ p+\frac\delta 2\log \frac{\sum _{j=0}^ n|Z_ j|^2}{|Z_ i|^2} =u+\delta \log |Z_0/Z_ i|. \]
This function is locally bounded above near $H$, by the growth bound on $u$, because the norm of the affine coordinates is at most a constant, depending on the chart, divided by $|Z_0/Z_ i|$. By lemma 5.24.7, it therefore extends canonically across $H$. Since the local potentials of $\theta _\delta $ differ by smooth functions on overlaps, the extensions agree there.
Conversely, every $p\in \mathrm{PSH}(\mathbb P^ n,\theta _\delta )$ is bounded above, since $\mathbb P^ n$ is compact. The function $u=p+\delta \chi /2$ is plurisubharmonic on the affine chart, and
\[ 0\leq \tfrac 12\chi -\log ^+\lVert z\rVert \leq \tfrac 12\log 2 \]
gives the required growth bound. Neither function is identically $-\infty $ on the affine chart. Uniqueness of the canonical extension shows that the two maps are inverse to each other. $\square$
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