Example 17.14.1020A8C7BFE0BB

Example 17.14.10.20A8C7BFE0BB Let $L$ be a big line bundle on $X$ and $\mathcal{F}$ be a filtration on $R(X,L)$. Fix a smooth Hermitian metric $h$ on $L$ and write $\theta =c_1(L,h)$.

We introduce a few auxiliary functions. For each $k\in \mathbb {Z}_{>0}$, we introduce

When $k\tau \leq \tau _ k(\mathcal{F})$, we know that $\mathcal{F}_ k^{k\tau }\neq 0$. Moreover, proposition 5.20.4 and proposition 5.14.2 imply that

Observe that for $k,k'\in \mathbb {Z}_{>0}$, we have

In particular, by Fekete's lemma,

exists for any $\tau <\tau (\mathcal{F})$.

We define $(\Gamma ^{\mathcal{F}}_{\tau })_{\tau <\tau (\mathcal{F})}$ as follows:

We claim that $\Gamma ^{\mathcal{F}}\in \mathcal{E}^{\mathrm{NA}}(X,\theta )$ and is bounded.

It is clear that $(-\infty ,\tau (\mathcal{F}))\ni \tau \mapsto \Gamma ^{\mathcal{F}}_{\tau }$ is decreasing. We prove its concavity. By proposition 16.4.2, it suffices to show that

is concave. In other words, we need to prove the following: Given $\tau _0<\tau _1<\tau (\mathcal{F})$ and $t\in (0,1)$, we have

But thanks to proposition 5.14.6 and proposition 5.14.5, it suffices to show that

for all $t\in (0,1)$. Take $s_ i\in \mathcal{F}^{k_ i\tau _ i}_{k_ i}$ for $i=0,1$ with $|s_ i|_{h^{k_ i}}^2\leq 1$, where $k_0,k_1\in \mathbb {Z}_{>0}$. We need to prove that

Approximating $t$ by rational numbers $t_ m\searrow t$, and using the decreasingness of the filtration in the parameter $\lambda $, we reduce to the case $t\in \mathbb Q$. Write $t=p/q$ with $p,q\in \mathbb {Z}_{>0}$. Then

and

So (17.14.12) follows.

Note that for each $k\in \mathbb {Z}_{>0}$, $\tau \leq k^{-1}\tau _ k(\mathcal{F})$, we know that $k^{-1}\Gamma _{\tau }^{\mathcal{F},k}$ is $\mathcal{I}$-good by proposition 18.1.19. The family is bounded above by $0$, so it follows from the same proposition that the obstacle in the definition of $\Gamma ^{\mathcal{F}}_{\tau }$ is $\mathcal{I}$-good. Hence the $P$- and $\mathcal{I}$-envelopes of this obstacle agree, so $\Gamma ^{\mathcal{F}}_{\tau }$ is $\mathcal{I}$-model for each $\tau <\tau (\mathcal{F})$.

It remains to show that the test curve $\Gamma ^{\mathcal{F}}$ is bounded and lies in $\mathcal{E}^{\mathrm{NA}}(X,\theta )$. Fix $\tau \leq -C$, where $C$ is as in (17.14.9). We will show that

This follows from the Bergman kernel technique. Based on the theory developed so far, we can also give an elementary argument.

Fix $k\in \mathbb Z_{>0}$. Since $\tau \leq -C$, we have $\mathcal F^{k\tau }_ k=\mathrm H^0(X,L^ k)$. Let $s\in \mathrm H^0(X,L^ k)$. After multiplying $s$ by a constant, we may assume that $|s|_{h^ k}^2\leq 1$. By the definition of $\Gamma ^{\mathcal F,k}_\tau $,

Hence

Since

we get

Thus

for every $k\in \mathbb Z_{>0}$. By theorem 18.1.46,

As $\Gamma ^{\mathcal F}_\tau $ is $\mathcal I$-model, this volume is equal to $\int _ X\left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\mathcal F}_\tau \right)^ n$. Hence $\Gamma ^{\mathcal F}_\tau $ has full Monge–Ampère mass. Since it is also model, it must be equal to $V_\theta $. This proves (17.14.13).