Corollary 18.1.1536E768E33AFB

Proof. Let $(\epsilon _ j)_ j$ and $\omega $ be as in definition 6.17.5, so that $\epsilon _ j\searrow 0$ and $\varphi _ j\in \mathrm{PSH}(X,\theta +\epsilon _ j\omega )$ for all $j$. Choose a Kähler form $\tilde\omega $ on $Y$ and a constant $C>0$ such that $\pi ^*\omega \leq C\tilde\omega $. Then

Moreover, the sequence $(\pi ^*\varphi _ j)_ j$ is decreasing, has analytic singularities, and decreases to $\pi ^*\varphi $.

It remains to verify the quasi-equisingular condition definition 6.17.5 4. Fix a constant $A>0$. Then $(\varphi _ j)_ j$ is also a quasi-equisingular approximation of $\varphi $ with respect to $\theta +A\omega $. Since $\theta _{\varphi }+A\omega $ is a Kähler current, corollary 18.1.14 gives

Viewing the convergence in a fixed enlarged background class, pulling back and using corollary 16.9.11, proposition 16.6.10, we get

The domination $\pi ^*\omega \leq C\tilde\omega $ also gives

so the preceding convergence is in the setting of corollary 18.1.14 on $Y$. The potential $\pi ^*\varphi $ has positive mass with respect to $\pi ^*(\theta +A\omega )$ by the bimeromorphic invariance of non-pluripolar products proposition 13.13.4. Applying corollary 18.1.14 again on $Y$ shows that $(\pi ^*\varphi _ j)_ j$ satisfies the quasi-equisingular condition definition 6.17.5 4 for $\pi ^*\varphi $. Combining this with the first paragraph proves the assertion with respect to $\pi ^*\theta $. $\square$