16.4 Properties of the $P$-envelope666167293D79

Proof. By proposition 5.17.8, every $\chi \in \mathrm{PSH}(Y,\pi ^*\theta )$ is of the form $\chi =\pi ^*\psi $ for a unique $\psi \in \mathrm{PSH}(X,\theta )$. Since $\pi $ is surjective, $\chi \leq 0$ if and only if $\psi \leq 0$. Moreover,

Thus the candidates defining $P_{\pi ^*\theta }[\pi ^*\varphi ]$ are exactly the pull-backs of the candidates defining $P_\theta [\varphi ]$, and hence

Finally, the non-pluripolar mass is preserved under bimeromorphic pull-back by proposition 13.13.4. Therefore $\varphi $ is model of positive mass if and only if $\pi ^*\varphi $ is. $\square$

Proof. 1 This follows by rescaling the candidates in the definition of the $P$-envelope.

2 Suppose that $\psi _1\in \mathrm{PSH}(X,\theta _1)$ and $\psi _2\in \mathrm{PSH}(X,\theta _2)$ satisfy

for $i=1,2$. Then

It follows from (16.3.6) that

Since $\psi _1$ and $\psi _2$ are arbitrary, we conclude.

3 This follows by inclusion of the defining candidate sets. $\square$

Proof. Since $P_{\theta }[\varphi _ j]=\varphi _ j$, we have $\sup _ X\varphi _ j=0$. By corollary 5.17.5, we have $\varphi \in \mathrm{PSH}(X,\theta )$, and $\varphi \leq 0$. Therefore, for each $j\in I$,

Therefore, $\varphi =P_{\theta }[\varphi ]$. $\square$

Proof. We first observe that $\varphi \in \mathrm{PSH}(X,\theta )$. In fact, for each $i_0\in I$, the tail $(\varphi _ i)_{i\geq i_0}$ lies in $\mathrm{PSH}(X,\theta +\epsilon _{i_0}\omega )$ and decreases to $\varphi $. Since $\sup _ X\varphi _ i=0$ by (16.4.5), the limit is not identically $-\infty $. Thus $\varphi \in \mathrm{PSH}(X,\theta +\epsilon _{i_0}\omega )$ for every $i_0\in I$. Since $\epsilon _{i_0}\to 0$, it follows that $\varphi \in \mathrm{PSH}(X,\theta )$. For every $i\in I$ we have

Therefore, $P_{\theta }[\varphi ]\leq \varphi $. The reverse inequality is trivial. Hence $P_{\theta }[\varphi ]=\varphi $.

By the monotonicity theorem theorem 13.13.11 and the multi-linearity of the non-pluripolar product proposition 13.11.4 6, we have

We now argue the reverse inequality.

Fix $i_0\in I$. Since passing to the tail $i\geq i_0$ does not change the limsup, we have

where in the first line we used (16.4.5) and theorem 16.3.12, and in the last line we used the fact that $\varphi _ i \searrow \varphi $ on the tail $i\geq i_0$, and on this tail, $\varphi _ i \to \varphi $ in $L^1$ as well by corollary 5.17.5, and therefore, corollary 13.13.16 is applicable. Taking the limit with respect to $i_0$, we arrive at the desired conclusion:

where the equality follows from the multi-linearity of the non-pluripolar product proposition 13.11.4 6 and the last inequality follows from theorem 16.3.12. This finishes the proof of (16.4.6).

It remains to argue (16.4.7). Fix $i_0\in I$. By proposition 5.17.7, there is a constant $B>0$ such that

By 13.13.30 and (16.4.6), for any $\delta \in (0,1)$ there exists $i_{\delta }\geq i_0$ such that for all $i\geq i_{\delta }$ there exists $\psi _ i \in \mathrm{PSH}(X,\theta +\epsilon _ i\omega )$ satisfying $(1-\delta ) \varphi _ i + \delta \psi _ i \leq \varphi $. This implies that for $i\geq i_{\delta }$ we have

Since $\psi _ i\in \mathrm{PSH}(X,\theta +\epsilon _{i_0}\omega )$ for $i\geq i_{\delta }$, we get

Letting $\delta \searrow 0$ gives $\varliminf _{i\in I}\nu (\varphi _ i,E)\geq \nu (\varphi ,E)$. The reverse inequality follows from $\varphi \leq \varphi _ i$, and hence we conclude (16.4.7). $\square$

Proof. Set

Since $\varphi $ is model, $\varphi =P_{\theta }[\varphi ]\leq P_{\theta +\epsilon \omega }[\varphi ]$ for every $\epsilon >0$, hence $\varphi \leq \eta \leq 0$.

Next we prove the reverse direction. Applying proposition 16.4.4 to the decreasing net $(P_{\theta +\epsilon \omega }[\varphi ])_{\epsilon >0}$ and using proposition 16.3.10, we get

Thus $\eta $ is a candidate in the characterization theorem 16.3.15 of $P_\theta [\varphi ]$. Hence $\eta \leq P_\theta [\varphi ]=\varphi $. $\square$

Proof. We may assume that $I$ is infinite since otherwise, there is nothing to prove.

We write

Then it is clear that $\eta \leq P_{\theta }[\varphi ]$.

By corollary 13.13.12, we have

We may assume that for each $i\in I$, $\int _ X\theta _{\varphi _ i}^ n<\int _ X\theta _{\varphi }^ n$, since otherwise $P_{\theta }[\varphi _ i]=P_{\theta }[\varphi ]$ by theorem 16.3.15, and the same holds for all $j\geq i$; in that case there is nothing to prove.

So by 13.13.30, we can find a decreasing net $\epsilon _ i\searrow 0$ ($i\in I$) with $\epsilon _ i\in (0,1)$ and $\psi _ i\in \mathrm{PSH}(X,\theta )$ ($i\in I$) such that for all $i\in I$,

We may, for example, take

By the concavity property of the envelope proposition 16.4.2, we have

Each $P_{\theta }[\psi _ i]$ has supremum $0$, hence the family is bounded in $L^1$ by proposition 5.17.3. Thus $\epsilon _ iP_{\theta }[\psi _ i]\to 0$ in $L^1$. The previous inequality then gives $P_{\theta }[\varphi ]\leq \eta $ almost everywhere, and hence everywhere by proposition 5.14.6. $\square$