Corollary 20.2.558F47DBEBF9D3

Proof. Choose a closed smooth real $(1,1)$-form $\theta _0$ representing a big class such that $\varphi \in \mathrm{PSH}(X,\theta _0)$ and $\theta _0+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current. Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta _0)_{>0}$. Since $\varphi $ is $\mathcal I$-good, corollary 18.1.14, theorem 18.1.2, proposition 16.9.6 gives

Applying 20.2.54, we find a co-pluripolar set of $H\in \Lambda $ such that $H$ is connected and smooth, the restrictions $\varphi _ j|_ H$ and $\varphi |_ H$ are not identically $-\infty $, and

For such an $H$, the potentials $\varphi _ j|_ H$ have analytic singularities, hence are $\mathcal I$-good by example 18.1.10. It follows from corollary 18.1.9 that $\varphi |_ H$ is $\mathcal I$-good.

On the other hand, corollary 20.2.50 gives, after intersecting with another co-pluripolar set of parameters $H$, that $\nu (\varphi ,H)=0$ and

The trace $\operatorname{Tr}_ H(\varphi )$ is $\mathcal I$-good by proposition 20.2.8. Since both $\operatorname{Tr}_ H(\varphi )$ and $\varphi |_ H$ are $\mathcal I$-good, their $\mathcal I$-equivalence identifies their $\mathcal I$-envelopes, which agree with their $P$-envelopes. Hence

Finally corollary 20.2.52 shows, after intersecting with one more co-pluripolar set of $H\in \Lambda $, that $\operatorname{Tr}_ H(\varphi )$ has a representative in $\mathrm{PSH}(H,\theta |_ H)_{>0}$. $\square$