Proposition 17.12.14.B015D2EA4B4F Let $\theta '$ be another smooth closed real $(1,1)$-form on $X$ representing the same cohomology class as $\theta $. Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. Let $\phi '\in \mathrm{PSH}(X,\theta ')_{>0}$ be the unique model potential satisfying $\phi \sim \phi '$.
Then there is a canonical bijection
\[ \mathrm{TC}(X,\theta ;\phi )\xrightarrow {\sim } \mathrm{TC}(X,\theta ';\phi '). \]
This bijection induces the following bijections:
\[ \mathrm{TC}^1(X,\theta ;\phi )\xrightarrow {\sim }\mathrm{TC}^1(X,\theta ';\phi '),\quad \mathrm{TC}^{\infty }(X,\theta ;\phi )\xrightarrow {\sim }\mathrm{TC}^{\infty }(X,\theta ';\phi '). \]
These bijections satisfy the obvious cocycle conditions.
Proof.
Choose $g\in C^{\infty }(X)$ such that $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$. Given any $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$, we observe that $\Gamma '\colon (-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta ')$ defined as
\[ \tau \mapsto P_{\theta '}[\Gamma _{\tau }-g] \]
lies in $\mathrm{TC}(X,\theta ';\phi ')$. In fact, the concavity and monotonicity of $\Gamma '$ follow from proposition 16.4.2. Moreover, applying proposition 16.4.9 to the increasing net obtained as $\tau \to -\infty $ gives
\[ \Gamma '_{-\infty }=P_{\theta '}[\Gamma _{-\infty }-g]=P_{\theta '}[\phi -g]=\phi '. \]
Thus $\Gamma '\in \mathrm{TC}(X,\theta ';\phi ')$.
Moreover, the choice of $g$ is irrelevant since for any other choice of $g$, say $g'$, we have
\[ \Gamma _{\tau }-g\sim \Gamma _{\tau }-g' \]
for all $\tau <\Gamma _{\max }$. The inverse construction is obtained by replacing $g$ with $-g$. The boundedness condition is clearly preserved. Finally, by proposition 16.3.10, for every $\tau <\Gamma _{\max }$,
\[ \int _ X(\theta '+\mathrm{dd}^{\mathrm{c}}\Gamma '_\tau )^ n=\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _\tau )^ n, \]
and the same holds for $\phi $ and $\phi '$. Hence the finite energy condition is preserved. The cocycle condition is immediate from the uniqueness of the model representative in a fixed singularity type.
$\square$
Pome
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