Pome

Lemma 13.13.473AA4A74713F0

Lemma 13.13.47.3AA4A74713F0 For any $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, there is $\psi \in \mathrm{PSH}(X,\theta )$ such that

  1. B1CBA0B9F2D3

    $\theta _{\psi }$ is a Kähler current, and

  2. 9583FAA82351

    $\psi \leq \varphi $.

In particular, there is an increasing sequence $(\varphi _ i)_{i>0}$ in $\mathrm{PSH}(X,\theta )$ converging almost everywhere to $\varphi $ such that $\theta _{\varphi _ i}$ is a Kähler current for all $i\geq 1$.

Proof. Using 13.13.30, we can find $\epsilon \in (0,1)$ and $\gamma \in \mathrm{PSH}(X,\theta )$ such that

\[ \epsilon V_{\theta }+(1-\epsilon )\gamma \leq \varphi . \]

We observe that the cohomology class $\{ \theta \} $ is big as a consequence of proposition 13.13.3. Therefore, we can take $\eta \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\eta }$ is a Kähler current and $\eta \leq 0$. Since $\eta \leq V_{\theta }$, we may take

\[ \psi \coloneqq \epsilon \eta +(1-\epsilon )\gamma . \]

Then $\psi $ clearly satisfies 1 and 2.

For the latter claim, it suffices to take

\[ \varphi _ i=\left(1-(i+1)^{-1}\right)\varphi +(i+1)^{-1}\psi \]

for each $i>0$. $\square$

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