Proposition 18.28.514C368CFA09EB

Proof. The equality part in (18.28.52) follows from corollary 18.28.14. It remains to prove the weak convergence.

Let $\tau $ be the weak limit of a subsequence of $(\beta ^ k_{v,\varphi ,\omega ^ n})_ k$. In view of (18.28.48), it suffices to show that

We can obtain an upper bound of $\tau $ via the observation:

After replacing $\varphi $ by $V_{\theta }$, the corresponding problem is handled by [Theorem 1.2, Ber11], giving us

where the last identity is due to proposition 16.3.41.

We obtain that

Now observe that it suffices to show that

Assuming this, then (18.28.54) further gives

where the equality follows from proposition 16.3.41, and (18.28.53) follows.

It now remains to argue (18.28.55). Fix a Kähler form $\omega $ on $X$. Observe that

for all $\epsilon \geq 0$ by 18.28.15 and proposition 16.6.22. The family $P_{\theta +\epsilon \omega ,X}[\varphi ](v)$ is decreasing as $\epsilon \to 0+$. Let

Then $\chi \in \mathrm{PSH}(X,\theta )$ since $\chi \geq P_{\theta ,X}[\varphi ](v)$. Moreover $\chi \leq v$ and $\chi \preceq \varphi $. Hence

Thus, equality holds, and

In other words,

So in order to show (18.28.55), we may fix $\epsilon >0$ and argue that

Let $k\geq 1$, and $s \in \mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal I(k\varphi ))$ be a section such that $N^ k_{v,\omega ^ n}(s) \leq 1$. Then by [Lemma 4.1, Ber11], there exists $C>0$ such that

This implies that

We define $\varphi _ k$ as in proposition 5.20.17. Take $\alpha _ k\nearrow 1$ as in proposition 5.20.17. After adding a constant to $\varphi $, we may assume that $\varphi \leq 0$ on $X$. After replacing $\alpha _ k$ by a subsequence, we also have $\alpha _ k\varphi \in \mathrm{PSH}(X,\theta +\epsilon \omega )$ for all $k>0$, and $\alpha _ k\varphi \searrow \varphi $. Then

We notice that $\frac{1}{k}\log h^ k \otimes h_ T(s,s) \in \mathrm{PSH}(X, \theta + \epsilon \omega )$ for all $k \geq k_0(\epsilon )$. Indeed,

by proposition 5.20.4.

In particular,

Now taking supremum over all candidates $s$, we obtain that

Now we prove (18.28.56). First observe that $\alpha _ k\varphi \xrightarrow {d_ S}\varphi $ by 16.9.25. Thus, by proposition 18.28.35,

and we get that

It suffices to show that $\tau $ does not put mass on the set $\{ P_{\theta +\epsilon \omega ,X}[\alpha _ m \varphi ](v) < v\} $ for each fixed $m$. Fix a positive rational $\delta >0$, it suffices to show that $\tau $ puts no mass on $F\coloneqq \{ P_{\theta +\epsilon \omega ,X}[\alpha _ m \varphi ](v) < v-\delta \} $. Note that $F$ is open, and (18.28.57) gives $\beta ^\ell _{v,\varphi ,\omega ^ n}(F)\to 0$. Therefore, $\tau $ puts no mass on $F$ as well, proving (18.28.56). $\square$