16.9 The $d_ S$-pseudometric45BED73AEFE3

Proof. This follows immediately from theorem 17.10.81. $\square$

Proof. We claim that

Recall that $\lor $ is defined in definition 17.10.85. Note that this assertion implies our desired inequality by 17.10.90.

In proving this assertion, we may assume that $\varphi ,\psi \leq 0$ since

for any $C\in \mathbb {R}$.

In fact, it is clear that

so the $\leq $ direction in (16.9.5) holds.

Conversely, if $\ell '\in \mathcal{R}^1(X,\theta )$ and $\ell '\geq \ell ^{\varphi }\lor \ell ^{\psi }$, then for each $t\geq 0$,

See (17.10.96) for the first inequality. Therefore, thanks to the definition of the Perron envelopes (cf. (17.10.21)), we have

for any $0\leq s\leq t$. It follows from (17.10.96) that $\ell '_ s\geq \ell ^{\varphi \lor \psi }_ s$ for any $s\geq 0$. $\square$

Proof. By 16.8.13, we have $\varphi \sim _ P \psi $ if and only if $\varphi \sim _ P \varphi \lor \psi $ and $\psi \sim _ P \varphi \lor \psi $. By 16.9.4, $d_ S(\varphi ,\psi )=0$ if and only if $d_ S(\varphi ,\varphi \lor \psi )=0$ and $d_ S(\psi ,\varphi \lor \psi )=0$. So it suffices to prove the assertion when $\varphi \leq \psi $. Assuming this, by proposition 17.10.83, 2 holds if and only if

where $\mathbf{E}$ is introduced in definition 17.10.82. But by (17.10.94), this holds if and only if

Thanks to the monotonicity theorem theorem 13.13.11, this holds if and only if for all $j=0,\ldots ,n$,

which is equivalent to 1 by proposition 16.8.7. $\square$

Proof. Set $\chi \coloneqq \varphi \lor \psi $. By 16.8.13, we have $\chi \sim _ P\psi $. Hence proposition 16.9.6 gives $d_ S(\chi ,\psi )=0$, and the pseudometric property gives

Since $\varphi \leq \chi $, (17.10.94) and proposition 17.10.83 give

Finally, proposition 16.8.7 applied to $\psi \leq \chi $ gives

for all $j=0,\ldots ,n$. $\square$

Proof. This is an immediate consequence of 16.9.7 and the monotonicity theorem proposition 16.8.22. $\square$

Proof. The estimates (16.9.10) follow from the combination of 16.9.7 and 16.9.4.

Suppose that $\varphi _ i\xrightarrow {d_ S}\varphi $. Then $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ by 16.9.4. Therefore, theorem 13.13.11 and 16.9.7 imply that

for any $j=0,\ldots ,n$. The last assertion now follows from (16.9.10) and theorem 13.13.11. $\square$

Proof. It suffices to observe that the middle term in (16.9.10) is invariant under $\pi ^*$. $\square$

Proof. Write $\varphi =\operatorname *{sup}_{j\in I}{}\! {}^{*}\varphi _ j$. Recall that by proposition 5.14.2, $\varphi \in \mathrm{PSH}(X,\theta )$. By 16.9.7, it suffices to show that for each $k=0,\ldots ,n$, we have

The latter follows from corollary 13.13.12. $\square$

Proof. We compute

where the first line is just the triangle inequality, the second line follows from corollary 16.9.9 and the third line follows from 16.9.4. $\square$

Proof. 1 $\iff $ 2. This follows immediately from 16.9.7.

Assume that $\int _ X \theta _{\varphi }^ n>0$.

2 $\implies $ 1. This is the case $k=n$ in (16.9.15).

1 $\implies $ 2. Assume (3). If for some $i\in I$, $\int _ X \theta _{\varphi _ i}^ n= \int _ X \theta _{\varphi }^ n$, then for all $j\geq i$ we have

Hence $\int _ X\theta _{\varphi _ j}^ n=\int _ X\theta _{\varphi }^ n$ by proposition 16.8.22. It follows from proposition 16.8.7 that $\varphi _ j\sim _ P\varphi $, and (2) follows from proposition 16.3.10. So in the sequel, we shall assume that for all $i\in I$, we have $\int _ X \theta _{\varphi _ i}^ n>\int _ X \theta _{\varphi }^ n$. In particular, each $\varphi _ i$ has positive mass.

Let $(b_ j)_{j\in I}$ be a net converging to $\infty $ such that

This can be achieved for example by taking $b_ j$ as the midpoint of this interval. By theorem 16.8.23, for each $j\in I$, we can find $\eta _ j\in \mathrm{PSH}(X,\theta )$ such that

It follows from the monotonicity theorem theorem 13.13.11 that for any $k=0,\ldots ,n$,

Taking the limit, we conclude the $\leq $ direction in (16.9.15). The $\geq $ direction follows from proposition 16.8.22.

Finally, we argue (16.9.16). Let $\psi _ j=P_{\theta }[\varphi _ j]$. It follows from corollary 16.3.20 that $\psi _ j$ is a model potential. Since the net $(\varphi _ j)_{j\in I}$ is decreasing with respect to the $P$-preorder, $(\psi _ j)_{j\in I}$ is decreasing pointwise. Let

By proposition 16.4.3, $\psi $ is a model potential. Since $P_{\theta }[\varphi ]\leq \psi \leq \psi _ j$ for all $j\in I$, the monotonicity theorem theorem 13.13.11 and proposition 16.3.10 give

Hence $\psi $ has the same mass as $P_{\theta }[\varphi ]$. As $\psi $ is model and $\psi \geq P_{\theta }[\varphi ]$, theorem 16.3.15 gives $\psi =P_{\theta }[\varphi ]$. $\square$

Proof. First note that $\int _ X \theta _{\varphi }^ n\geq \delta $ as well, due to the $d_ S$-continuity of non-pluripolar masses as proved in corollary 16.9.13. Replacing $\varphi $ by $P_{\theta }[\varphi ]$, we may then assume that $\varphi $ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$.

We are free to replace $(\varphi _ j)_ j$ by a subsequence. So we may assume that

where $C_ n$ is the constant in corollary 16.9.9, $D_ n$ is the constant in corollary 16.9.13.

In particular, by corollary 16.9.13,

Step 1. We handle the $\psi _ j$'s. For each $j\geq 1$ and $k\geq 1$, by 16.9.4 we have

By iteration, we find

Using corollary 16.9.12, we have

as $k\to \infty $. Hence

We conclude that $\psi _ j\xrightarrow {d_ S}\varphi $.

Moreover, we observe that

Indeed, set $\tilde{\psi }_ j=P_{\theta }[\psi _ j]$. Then $(\tilde{\psi }_ j)_ j$ is a decreasing sequence of model potentials, and $d_ S(\tilde{\psi }_ j,\psi _ j)=0$ by proposition 16.9.6. Hence $\tilde{\psi }_ j\xrightarrow {d_ S}\varphi $. If $\chi =\inf _ j\tilde{\psi }_ j$, then proposition 16.4.4 and corollary 16.9.14 give $\tilde{\psi }_ j\xrightarrow {d_ S}\chi $. Thus $d_ S(\chi ,\varphi )=0$. Since both $\chi $ and $\varphi $ are model potentials, proposition 16.9.6 gives $P_{\theta }[\chi ]=P_{\theta }[\varphi ]$, hence $\chi =\varphi $, proving (16.9.21).

Step 2. We consider the $\eta _ j$'s.

For each $j\geq 1$ and $k\geq 0$, we let

Using (16.9.18), (16.9.20) and corollary 16.9.13, we have

for all $j\geq 1$. Taking (16.9.19) into account, we have

for $j>1$.

Step 2.1. We claim that for a fixed $j\geq 1$, for any $k\in \mathbb {N}$, we have $\eta _ j^ k\in \mathrm{PSH}(X,\theta )$ and

We argue by induction on $k\geq 0$. The case $k=0$ is trivial. When $k>0$, assume that the case $k-1$ is known. Then

where the first inequality follows from the inductive hypothesis and (16.9.22).

Observe that

it follows from proposition 16.3.25 that $\eta _ j^ k\in \mathrm{PSH}(X,\theta )$. By theorem 16.3.35, we deduce that

where the second inequality follows from the inductive hypothesis and (16.9.22). Therefore, (16.9.23) follows.

Step 2.2. It follows by induction from proposition 16.3.29 that for any $j\geq 1$, $k\geq 0$,

By proposition 16.4.4, we have

for any $j\geq 1$. Letting $k\to \infty $ in (16.9.23), we find that

for $j\geq 1$. Observe that we also have

for $j\geq 1$ by theorem 13.13.11. It follows from these inequalities and (16.9.19) that

For $k\geq 1$, we have

Letting $k\to \infty $ gives $\eta _ j\leq \eta _{j+1}$. Hence $(\eta _ j)_ j$ is increasing. Hence corollary 13.13.12 gives

where $\eta =\operatorname *{sup}_{j\geq 1}{}\! {}^{*} \eta _ j$. For fixed $m$, if $j\geq m$, then $\eta _ j\leq \varphi _ j\leq \psi _ m$, whereas if $j<m$, then $\eta _ j\leq \eta _ m\leq \psi _ m$. Thus $\eta _ j$ is a candidate in the definition of $P_{\theta }[\psi _ m]$ for all $j,m$, and (16.9.21) gives $\eta \leq \varphi $. It follows from proposition 16.8.7 that $\eta \sim _ P \varphi $. By corollary 16.9.12 and proposition 16.9.6, we have $\eta _ j\xrightarrow {d_ S}\varphi $. $\square$

Proof. In a pseudometric space, sequential completeness implies completeness for Cauchy nets. So for the first assertion, we may assume that $(\varphi _ j)_ j$ is a Cauchy sequence.

We first treat the case of a decreasing sequence $(\gamma _ j)_ j$ with $\int _ X\theta _{\gamma _ j}^ n\geq \delta $ for all $j$. Put $\tilde\gamma _ j=P_\theta [\gamma _ j]$. Then $(\tilde\gamma _ j)_ j$ is a decreasing sequence of model potentials, $d_ S(\gamma _ j,\tilde\gamma _ j)=0$, and

by proposition 16.3.10. By proposition 16.4.4, the decreasing limit $\tilde\gamma =\inf _ j\tilde\gamma _ j$ is model and

Hence $\tilde\gamma _ j\to \tilde\gamma $ in $d_ S$ by corollary 16.9.14, and therefore $\gamma _ j$ also converges in $d_ S$.

Next let $(\varphi _ j)_{j>0}$ be a $d_ S$-Cauchy sequence with masses bounded from below by $\delta $. Put

By corollary 16.3.20, example 16.8.6, proposition 16.9.6, the $\tilde{\varphi }_ j$'s are model potentials, $d_ S(\varphi _ j,\tilde{\varphi }_ j)=0$, and

by proposition 16.3.10. Replacing $\varphi _ j$ by $\tilde{\varphi }_ j$, we may therefore assume that the $\varphi _ j$'s are model potentials. Passing to a subsequence, we may also assume that

where $C_ n$ is the constant in corollary 16.9.9. Define

for each $j>0$. As in the proof of proposition 16.9.17 Step 1, especially (16.9.20), we know that

The sequence $(\psi _ j)_ j$ is decreasing and satisfies $\int _ X\theta _{\psi _ j}^ n\geq \delta $ by theorem 13.13.11. By the first paragraph, $(\psi _ j)_ j$ converges in $d_ S$. Hence the chosen subsequence of $(\varphi _ j)_ j$ converges, and the original Cauchy sequence converges as well.

Finally, suppose that $(\varphi _ j)_{j\in I}$ is a decreasing net with masses bounded from below by $\delta $. For each $k=0,\ldots ,n$, the net

is decreasing and bounded from below, hence Cauchy as a real-valued net. If $j'\geq j$, then $\varphi _{j'}\leq \varphi _ j$, and 16.9.7 expresses $d_ S(\varphi _{j'},\varphi _ j)$ as a finite sum of the corresponding differences of these mixed masses. For two sufficiently large indices, we choose a common upper bound in $I$ and use the triangle inequality. It follows that $(\varphi _ j)_{j\in I}$ is $d_ S$-Cauchy. The first assertion gives its convergence. $\square$

Proof. After adding a constant, we may assume that $\varphi \leq 0$. Since $\theta _{\varphi }$ is a Kähler current, $(1-\epsilon )\varphi \in \mathrm{PSH}(X,\theta )$ for all small enough $\epsilon >0$. Fix such an $\epsilon $. Take any Kähler form $\omega $ on $X$ so that $\omega -\theta $ is still a Kähler form. Note that the choice of $\omega $ depends only on $X$ and $\theta $.

Since $\varphi \preceq (1-\epsilon )\varphi $, by 16.9.7 and the multi-linearity of the non-pluripolar product proposition 13.11.4 3, we can compute

where, for $1\leq m\leq n$,

Note that

where we used the monotonicity theorem theorem 13.13.11. Hence the whole expression is of the order of $\mathcal{O}(\epsilon )$, and our assertion follows. $\square$

Proof. Fix $t\in (0,1]$. We write

for any $i\in I$.

By corollary 16.9.9, it suffices to show that for each $j=0,\ldots ,n$,

Observe that

So after binomial expansion, (16.9.28) follows from corollary 16.9.9. $\square$

Proof. Since $\varphi \preceq \varphi _ t$, by 16.9.7, we need to show that for each $j=1,\dots ,n$, we have

For this purpose, we compute

As $t\to 0+$, the right-hand side clearly tends to $0$. $\square$

Proof. Since $d_ S$ is a pseudometric, in order to establish the continuity of mixed masses, it suffices to consider sequences instead of nets. So we may assume that $I=\mathbb {Z}_{>0}$ as ordered sets.

Step 1. We reduce to the case where $\varphi _ j^ k,\varphi _ j$ all have positive masses and there is a constant $\delta >0$ such that for all $j=1,\ldots ,n$ and $k\in I$,

Assuming in this case, the theorem has been proved, let us handle the general case. Take $t\in (0,1)$. By 16.9.27, we have

as $k\to \infty $ for each $j=1,\ldots ,n$. Note that for all $k\in I$,

Assume that we have proved the special case of the theorem, we have

More explicitly, for fixed $k$ the left-hand side is the value at $t$ of a polynomial $Q_ k(t)$ of degree at most $n$, and the right-hand side is the value of a polynomial $Q(t)$ of degree at most $n$. Choose $n+1$ distinct numbers $t_0,\ldots ,t_ n\in (0,1)$. Lagrange interpolation expresses $Q_ k(0)-Q(0)$ as a fixed linear combination of the numbers $Q_ k(t_ a)-Q(t_ a)$, which tend to $0$. Hence the same formula holds at $t=0$. From this, (16.9.31) follows.

Step 2. We now assume that the assumptions mentioned at the beginning of Step 1 are met. Note that we may assume that $\varphi _ j^ k$, $\varphi _ j$ are model potentials for all $j=1,\ldots ,n$, $k>0$ by proposition 16.9.6 and corollary 16.3.20.

We first handle the following monotone situation. Assume that, for each fixed $i$, the sequence $(\varphi _ i^ k)_ k$ is either increasing or decreasing. Let $\tilde{\varphi }_ i$ denote its pointwise limit, namely

in the decreasing case and

in the increasing case. We assume moreover that $\tilde{\varphi }_ i\sim _ P\varphi _ i$ for every $i$. Since mixed non-pluripolar masses are invariant under replacing each entry by a $P$-equivalent potential, it suffices in this monotone situation to prove convergence to the mixed mass of the $\tilde{\varphi }_ i$'s.

After reordering the indices, we may assume that there is $i_0\in \{ 0,\ldots ,n\} $ such that $(\varphi _ i^ k)_ k$ is decreasing for $i\leq i_0$ and increasing for $i>i_0$. Therefore, for each $k>0$, using theorem 13.13.11, we have

Using corollary 13.13.12, we therefore conclude that

It remains to prove

By theorem 13.13.11, for each $k>0$, we have

When proving (16.9.32), we may replace $\varphi _ j^ k$ by $\tilde{\varphi }_ j$ whenever $j>i_0$, $k>0$. Thus, we are reduced to the case where for all $i$, $(\varphi _ i^ k)_ k$ is decreasing.

By proposition 16.4.4, the masses $\int _ X\theta _{i,\varphi _ i^ k}^ n$ decrease to $\int _ X\theta _{i,\tilde{\varphi }_ i}^ n$. Hence, thanks to 13.13.30, for each $i=1,\ldots ,n$, we may take an increasing sequence $(b^ k_ i)_ k$ tending to $\infty $ satisfying

where the upper endpoint is understood as $\infty $ if the denominator is $0$. For each such $b_ i^ k$, 13.13.30 gives $\rho _ i^ k\in \mathrm{PSH}(X,\theta _ i)$ with

Then by theorem 13.13.11 again,

Letting $k\to \infty $, we conclude (16.9.32). This proves the monotone situation.

We now return to the general case. It suffices to prove that any subsequence of

has a further subsequence converging to the desired limit. Starting with an arbitrary subsequence, we pass to a further subsequence, still indexed by $k$, so that the construction in proposition 16.9.17 applies simultaneously to the finitely many sequences $(\varphi _ i^ k)_ k$, $i=1,\ldots ,n$.

Thus, after discarding finitely many terms and relabeling, for each $i$ there are model potentials $\eta _ i^ k,\psi _ i^ k\in \mathrm{PSH}(X,\theta _ i)$ such that $(\eta _ i^ k)_ k$ is increasing, $(\psi _ i^ k)_ k$ is decreasing, and

For the upper sequence, we take $\psi _ i^ k=P_{\theta _ i}[\operatorname *{sup}_{\ell \geq k}{}\! {}^{*}\varphi _ i^\ell ]$; as in (16.9.21), this gives $\inf _ k\psi _ i^ k=\varphi _ i$. For the lower sequence, the construction in proposition 16.9.17 together with proposition 16.4.3 shows that the $\eta _ i^ k$'s are model after discarding finitely many terms, and the proof of proposition 16.9.17 shows that their increasing limit is $P$-equivalent to $\varphi _ i$.

By theorem 13.13.11, for every $k$,

The monotone situation just proved applies to both the lower and upper sequences, and both limiting mixed masses are equal to

The squeeze theorem therefore gives the desired convergence along this further subsequence. This concludes the proof. $\square$

Proof. The equivalence between 2 and 3 follows directly from 16.9.7.

1 $\implies $ 2. That $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ follows from corollary 16.9.9. While (16.9.34) follows from theorem 16.9.30.

2 $\implies $ 1. By (16.9.10), we need to show that for each $j=0,\ldots ,n$, we have

This follows from theorem 16.9.30 and (16.9.34). $\square$

Proof. 1 $\implies $ 2. It suffices to show that for each $j=0,\ldots ,n$, we have

Note that this quantity is a linear combination of terms of the following form:

where $r=0,\ldots ,j$. By theorem 16.9.30, it suffices to show that $\varphi \lor \varphi _ i \xrightarrow {d_ S}\varphi $. But this follows from corollary 16.9.33.

2 $\implies $ 1. From the direction we already proved, for each $C\geq 1$, we have

By theorem 16.9.30, it follows that

for all $j=0,\ldots ,n$. It follows that

By corollary 16.9.33, it remains to show that $\varphi _ i\lor \varphi \xrightarrow {d_{S,\theta }}\varphi $. By corollary 16.9.33 again, we know that $\varphi _ i\lor \varphi \xrightarrow {d_{S,\theta +\omega }}\varphi $. So it suffices to apply (16.9.37) to $\varphi _ i\lor \varphi $ instead of $\varphi _ i$, and we conclude by 16.9.7. $\square$

Proof. First note that for each $j=0,\ldots ,n$,

So thanks to corollary 16.9.33, it remains to argue that for all $j=0,\ldots ,n$,

Observe that for $t\in (0,1)$, we have

so the desired limit follows by the same binomial expansion. $\square$

Proof. By corollary 16.9.36, we may replace $\theta $ by $\theta +\omega $, where $\omega $ is a Kähler form on $X$. In particular, we may assume that $\varphi _ i,\psi _ i,\eta _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. By proposition 16.9.6, we may assume that $\varphi _ i,\psi _ i,\eta _ i$ are model potentials for all $i\in I$ and hence $\psi _ i \leq \varphi _ i\leq \eta _ i$ for all $i\in I$.

It follows from theorem 13.13.11 that for each $k=0,\ldots ,n$, we have

for all $i\in I$. By theorem 16.9.30, the limits, with respect to $i\in I$, of the two outer terms are $\int _ X \theta _{\varphi }^ k\wedge \theta _{V_{\theta }}^{n-k}$. It follows that

By corollary 16.9.33, it remains to prove that $\varphi _ i \lor \varphi \xrightarrow {d_ S}\varphi $. By corollary 16.9.33 and proposition 16.8.28, up to replacing $\psi _ i$ (resp. $\varphi _ i$, $\eta _ i$) by $\psi _ i\lor \varphi $ (resp. $\varphi _ i\lor \varphi $, $\eta _ i\lor \varphi $), we may assume from the beginning that $\psi _ i,\varphi _ i,\eta _ i\geq \varphi $. Now $\varphi _ i\xrightarrow {d_ S}\varphi $ by (16.9.40) and 16.9.7. $\square$

Proof. We have $V_{\lambda \theta }=\lambda V_{\theta }$. The middle expression in (16.9.10) for the pair $(\lambda \varphi _ i,\lambda \varphi )$ in the class $\lambda \theta $ is therefore $\lambda ^ n$ times the corresponding expression for $(\varphi _ i,\varphi )$ in the class $\theta $. The equivalence follows from corollary 16.9.9. $\square$

Proof. Since $d_ S$ is a pseudometric, we may assume that both nets are actually sequences and $I=\mathbb {Z}_{>0}$. By corollary 16.9.36, we may assume that the masses $\int _ X \theta _{\varphi }^ n>0$, $\int _ X \theta _{\psi }^ n>0$.

By proposition 16.9.6 and proposition 16.8.28, replacing all potentials by their $P$-envelopes changes neither the hypotheses nor the conclusion. Thus we may assume that the $\varphi _ j$'s, the $\psi _ j$'s, $\varphi $ and $\psi $ are all model. After discarding finitely many terms, the masses of the two sequences are bounded below by a positive constant. Applying proposition 16.9.17 to both sequences and using the sandwich criterion corollary 16.9.39, it is enough to treat the case where each of $(\varphi _ j)_ j$ and $(\psi _ j)_ j$ is monotone. In this reduced situation, $(\varphi _ j)_ j$ (resp. $(\psi _ j)_ j$) converges to $\varphi $ (resp. $\psi $) almost everywhere due to proposition 16.9.47, corollary 16.9.12 and proposition 16.4.9.

We handle three cases separately.

Step 1. Assume that both sequences are increasing.

In this case, we have $\varphi _ j\lor \psi _ j\nearrow \varphi \lor \psi $ almost everywhere. Therefore, $\varphi _ j\lor \psi _ j\xrightarrow {d_ S} \varphi \lor \psi $ by corollary 16.9.12.

Step 2. Assume that one sequence, say $(\varphi _ j)_ j$ is increasing while the other is decreasing. Then we have

Thanks to corollary 16.9.39, it suffices to show that both sides converge to $\varphi \lor \psi $ with respect to $d_ S$. So we reduce to the case where both sequences are decreasing.

Step 3. Assume that both sequences are decreasing.

In this case, due to corollary 16.9.14, it suffices to show that

The $\geq $ direction follows from theorem 13.13.11; it remains to argue the $\leq $ direction.

Thanks to 13.13.30, we may find a sequence $(\epsilon _ j)_ j$ in $(0,1)$ with limit $0$ and a sequence $(\eta _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ such that

It follows that for each $j\geq 1$, we have

Therefore by theorem 13.13.11,

Letting $j\to \infty $, we find that

Therefore, in order to prove (16.9.43), we may assume that one of the sequences is constant; let us say $\psi _ j=\psi $ for all $j$. Repeating the same argument as before and constructing $(\epsilon _ j)_ j$, $(\eta _ j)_ j$ as above, we get

Letting $j\to \infty $, we conclude (16.9.43). $\square$

Proof. It follows from proposition 16.9.42 that

By 16.8.13, we have $\varphi _ i\lor \psi _ i\sim _ P \psi _ i$ for all $i\in I$. In particular, by proposition 16.9.6,

By proposition 16.9.6 again, $\varphi \lor \psi \sim _ P\psi $ and hence $\varphi \preceq _ P \psi $ by 16.8.13. $\square$

Proof. By corollary 16.9.36, we may assume that $\theta _1,\theta _2$ are both Kähler forms. We denote them by $\omega _1,\omega _2$ instead. Let $\omega =\omega _1+\omega _2$.

1+2 $\implies $ 3. It suffices to show that for each $r=0,\ldots ,n$,

Observe that for each $i\in I$,

The expression with the actual rooftop is non-negative and is bounded above by the same expression with this larger potential in place of $(\varphi _ i+\psi _ i)\lor (\varphi +\psi )$. Thus, it suffices to show that

The left-hand side is a linear combination of

with $a=0,\ldots ,r$. Observe that $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ and $\psi _ i\lor \psi \xrightarrow {d_ S}\psi $ by corollary 16.9.9, each term tends to $0$ by theorem 16.9.30.

1+3 $\implies $ 2. For each $C\geq 1$, from the direction we already proved together with 16.9.41,

By theorem 16.9.30, for each $j=0,\ldots ,n$,

Both sides are polynomials in $C$. Comparing the constant terms gives

Therefore, 2 follows if $\psi _ i\geq \psi $ for each $i$ by 16.9.7.

Next we prove the general case. By the direction that we already proved, we know that $\varphi _ i+\psi \xrightarrow {d_{S}} \varphi +\psi $. By proposition 16.9.42, we have

It follows from the special case above that $\psi _ i\lor \psi \xrightarrow {d_{S}}\psi $. It follows from (16.9.46) and corollary 16.9.33 that 2 holds.

2+3 $\implies $ 1. This is similar. $\square$

Proof. First observe that by our assumption, $(P_{\theta }[\varphi _ i])_{i\in I}$ is a decreasing net. By the continuity of non-pluripolar masses corollary 16.9.13, we have

Replacing $I$ by a tail, we may therefore assume that $\varphi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for each $i\in I$. Since $\varphi _ i\sim _ P P_\theta [\varphi _ i]$, replacing $\varphi _ i$ by $P_\theta [\varphi _ i]$ does not change its $d_ S$-class by proposition 16.9.6. We may assume that each $\varphi _ i$ is model and $(\varphi _ i)_{i\in I}$ is decreasing. In this case, by proposition 16.4.4, the potential

is model in $\mathrm{PSH}(X,\theta )_{>0}$ with

Therefore, by corollary 16.9.14, we find $\varphi _ i\xrightarrow {d_ S}\psi $. In particular, $\varphi \sim _ P \psi $ by proposition 16.9.6. Therefore,

Therefore, (16.9.48) follows. $\square$

Proof. Let $(\varphi _ i)_{i\in \mathbb {Z}_{>0}}$ be a sequence in $\mathrm{PSH}(X,\theta )_{>0}$ such that $\varphi _ i\xrightarrow {d_ S}\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. We want to show that

We may assume that the $\varphi _ i$'s and $\varphi $ are all model potentials by proposition 16.9.6.

By proposition 16.9.17 and corollary 16.9.39, we may assume that $(\varphi _ i)_ i$ is either increasing or decreasing. In the increasing case, we apply proposition 16.6.30 and corollary 16.9.12, while in the decreasing case, we apply proposition 16.6.29, proposition 16.4.4 and corollary 16.9.14. $\square$

Proof. There are two assertions to prove, as detailed in the two steps.

Step 1. We first handle the case where $\epsilon \to 0+$.

In this case (16.9.51) means

as $\epsilon \to 0+$. So thanks to corollary 16.9.14 and proposition 16.4.4, it suffices to prove the following:

First observe that

In fact, the $\leq $ direction is trivial. As for the reverse inequality, it suffices to observe that the right-hand side lies in $\mathrm{PSH}(X,\theta )$. Therefore, due to proposition 16.4.4,

Therefore, for all $\epsilon >0$ small enough, we can find $\eta _{\epsilon }\in \mathrm{PSH}(X,\theta +\epsilon \omega )$ and $a_{\epsilon }>0$ decreasing to $0$ so that

Therefore, thanks to proposition 16.4.2,

Since the envelopes $P_{\theta +\omega }[\eta _\epsilon ]$ have supremum $0$, they are uniformly bounded in $L^1$ by proposition 5.17.3; hence $a_\epsilon P_{\theta +\omega }[\eta _\epsilon ]\to 0$ in $L^1$. Letting $\epsilon \to 0$ and using proposition 5.14.6, we conclude (16.9.52).

Step 2. We then handle the case where $\epsilon \to 0-$.

In this case, (16.9.51) simply means

as $\epsilon \to 0+$. But this follows from corollary 16.9.12 if we can prove

where we understand that $\epsilon $ is small enough so that $\theta -\epsilon \omega $ represents a big cohomology class. Since $\{ \theta \} $ is big, we can find $\varphi \in \mathrm{PSH}(X,\theta )$ so that

for some $\delta >0$. For a small $\epsilon >0$, we then have

Therefore,

Letting $\epsilon \to 0+$, we then find

The reverse inequality is trivial, hence (16.9.53) is established. $\square$

Proof. First observe that since $d_ S$ is a pseudometric, it suffices to prove (16.9.56) when $I=\mathbb {Z}_{>0}$ as partially ordered sets.

By corollary 16.9.36, after adding a Kähler form to $\theta $, we may assume that the masses of $\varphi _ j$ and of $\varphi $ are bounded from below by a positive constant.

By theorem 16.9.49, we may assume that $\varphi _ j$ and $\varphi $ are both $\mathcal{I}$-model and hence model. When proving (16.9.56), we are free to pass to subsequences.

By proposition 16.9.17 and a sandwich argument, we may assume that the sequence $(\varphi _ j)_ j$ is either increasing or decreasing. In the increasing case, the Lelong numbers $\nu (\varphi _ j,E)$ decrease to a limit at least $\nu (\varphi ,E)$, while $L^1$-convergence gives the opposite inequality by upper semicontinuity of Lelong numbers. In the decreasing case, (16.9.56) follows from proposition 16.4.4. $\square$

Proof. The latter part of (16.9.58) is just a special case of theorem 16.9.30. It remains to prove the former part.

We may therefore assume that $\int _ X \theta _{\varphi _ j}^ n>0$ for all $j\in I$ after passing to a tail. Then by theorem 16.9.49, we have

Therefore, the first part of (16.9.58) follows again from theorem 16.9.30. $\square$

Proof. Fix $\lambda '>\lambda >0$. We want to find $j_0>0$ so that for $j\geq j_0$, (16.9.60) holds.

Step 1. We first assume that $\varphi $ has analytic singularities.

Let $\pi \colon Y\rightarrow X$ be a log resolution of $\varphi $ and let $E_1,\ldots ,E_ N$ be all prime divisors in the polar locus of $\varphi $ on $Y$. Recall that by theorem 5.16.18, a non-zero local holomorphic function $f$ lies in the right-hand side of (16.9.60) if and only if for all $i=1,\ldots ,N$,

where $A_ X$ denotes the log discrepancy. Similarly, if $f$ lies in the left-hand side of (16.9.60), then for all $i=1,\ldots ,N$,

As Lelong numbers are continuous with respect to $d_ S$ by theorem 16.9.55, we can find $j_0>0$ so that when $j\geq j_0$, $\lambda '\nu (\varphi _ j,E_ i)\geq \lambda \nu (\varphi ,E_ i)$ for all $i$. In particular, (16.9.61) follows.

Step 2. We handle the general case.

By corollary 16.9.36, we are free to increase $\theta $ and assume that $\theta _{\varphi }$ is a Kähler current.

Take a quasi-equisingular approximation $(\psi _ k)_ k$ of $\varphi $ in $\mathrm{PSH}(X,\theta )$. The existence is guaranteed by theorem 5.18.11. Take $\lambda ''\in (\lambda ,\lambda ')$. Then by definition of quasi-equisingular approximations (cf. (5.18.9)), we can find $k>0$ so that

Observe that $\psi _ k\geq \varphi $ for all $k>0$. By proposition 16.9.42, we then find that for any $k>0$, we have $\varphi _ j\lor \psi _ k\xrightarrow {d_ S}\varphi \lor \psi _ k=\psi _ k$ as $j\to \infty $. By Step 1, we can find $j_0>0$ so that for $j\geq j_0$,

It follows that for $j\geq j_0$,

This completes the proof. $\square$