Pome

Proposition 17.14.3A9AB5CE3BCAB

Proposition 17.14.3.A9AB5CE3BCAB Let $\theta '$ be another smooth closed real $(1,1)$-form on $X$ representing the same cohomology class as $\theta $. Then there is a canonical bijection

\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\xrightarrow {\sim } \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}. \]

This bijection satisfies the obvious cocycle condition.

Proof. Let $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$. By proposition 17.12.14, the canonical map sends $\Gamma $ to the test curve with levels

\[ \Gamma '_\tau =P_{\theta '}[\Gamma _\tau -g]. \]

If $\Gamma _\tau $ is $\mathcal I$-model, then it is $\mathcal I$-good by example 18.1.11. Note that $\Gamma _\tau -g$ is $\mathcal I$-good since $\Gamma _{\tau }$ is. Hence

\[ P_{\theta '}[\Gamma _\tau -g]=P_{\theta '}[\Gamma _\tau -g]_{\mathcal I}, \]

so $\Gamma '_\tau $ is $\mathcal I$-model. The same argument applied to $-g$ gives the inverse map, and the cocycle condition follows from the one in proposition 17.12.14. $\square$

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