CAE331B64333
Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a big cohomology class. Fix a model potential $\phi \in \mathrm{PSH}(X,\theta )_{>0}$.
Definition 17.14.1.6CC72AA0EA90 A test curve $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$ is $\mathcal{I}$-model if for any $\tau <\Gamma _{\max }$, the potential $\Gamma _{\tau }$ is $\mathcal{I}$-model.
The subset of $\mathcal{I}$-model test curves in $\mathrm{TC}(X,\theta ;\phi )$ is denoted by $\mathcal{E}^{\mathrm{NA}}(X,\theta ;\phi )$. When $\phi =V_{\theta }$, we omit $\phi $ and write $\mathcal{E}^{\mathrm{NA}}(X,\theta )$ instead.
The union of the sets of $\mathcal{I}$-model test curves in $\mathrm{PSH}(X,\theta )$ for all model potentials $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ is denoted by $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.
Note that $\Gamma _{\Gamma _{\max }}$ is automatically $\mathcal{I}$-model, since
\[ \Gamma _{\Gamma _{\max }}\leq P_{\theta }\left[\Gamma _{\Gamma _{\max }}\right]_{\mathcal{I}}\leq \inf _{\tau <\Gamma _{\max }}P_{\theta }\left[\Gamma _{\tau }\right]_{\mathcal{I}}=\inf _{\tau <\Gamma _{\max }}\Gamma _{\tau }=\Gamma _{\Gamma _{\max }}. \]
Here we write $\mathrm{NA}$ with non-Archimedean in mind. The precise relation with non-Archimedean pluripotential theory will be clear in section 22.1. This section and the next can be skipped on a first reading; the necessary results can be consulted when reading section 22.1.
Proposition 17.14.2.683BF9BB4D63 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$. Then $\Gamma _{-\infty }$ is an $\mathcal{I}$-model potential.
Proof.
This follows from proposition 16.6.30.
$\square$
Proposition 17.14.3.A9AB5CE3BCAB Let $\theta '$ be another smooth closed real $(1,1)$-form on $X$ representing the same cohomology class as $\theta $. Then there is a canonical bijection
\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\xrightarrow {\sim } \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}. \]
This bijection satisfies the obvious cocycle condition.
Proof.
Let $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$. By proposition 17.12.14, the canonical map sends $\Gamma $ to the test curve with levels
\[ \Gamma '_\tau =P_{\theta '}[\Gamma _\tau -g]. \]
If $\Gamma _\tau $ is $\mathcal I$-model, then it is $\mathcal I$-good by example 18.1.11. Note that $\Gamma _\tau -g$ is $\mathcal I$-good since $\Gamma _{\tau }$ is. Hence
\[ P_{\theta '}[\Gamma _\tau -g]=P_{\theta '}[\Gamma _\tau -g]_{\mathcal I}, \]
so $\Gamma '_\tau $ is $\mathcal I$-model. The same argument applied to $-g$ gives the inverse map, and the cocycle condition follows from the one in proposition 17.12.14.
$\square$
Proposition 17.14.4.92E487FB0DA8 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Then the pointwise pull-back induces a bijection
\[ \pi ^*\colon \mathcal{E}^{\mathrm{NA}}(X,\theta ;\phi )\xrightarrow {\sim } \mathcal{E}^{\mathrm{NA}}(Y,\pi ^*\theta ;\pi ^*\phi ). \]
Proof.
This is an immediate consequence of proposition 17.12.15 and proposition 16.6.10.
$\square$
Definition 17.14.5.03BED679B3AD Given $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$, we define its $\mathcal{I}$-envelope $P_{\theta }[\Gamma ]_{\mathcal{I}}$ as the map
\[ (-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta ),\quad \tau \mapsto P_{\theta }\left[\Gamma _{\tau }\right]_{\mathcal{I}}. \]
More generally, for any closed real smooth positive $(1,1)$-form $\omega $ on $X$, we define $P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}$ as the map
\[ (-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta +\omega ),\quad \tau \mapsto P_{\theta +\omega }\left[\Gamma _{\tau }\right]_{\mathcal{I}}. \]
Proposition 17.14.6.BF40FB71C62C Let $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$. Then
\[ P_{\theta }[\Gamma ]_{\mathcal{I}}\in \mathcal{E}^{\mathrm{NA}}(X,\theta ;P_{\theta }[\phi ]_{\mathcal{I}}). \]
More generally, for any closed real smooth positive $(1,1)$-form $\omega $ on $X$, we have
\[ P_{\theta +\omega }[\Gamma ]_{\mathcal{I}}\in \mathcal{E}^{\mathrm{NA}}(X,\theta +\omega ;P_{\theta +\omega }[\phi ]_{\mathcal{I}}). \]
Proof.
We prove the first assertion; the second is identical, replacing $\theta $ by $\theta +\omega $.
Set
\[ \widetilde\Gamma _\tau \coloneqq P_\theta [\Gamma _\tau ]_{\mathcal I}. \]
The family is decreasing by 16.6.28. If $\tau =(1-s)\tau _0+s\tau _1$, then the concavity of $\Gamma $ gives
\[ \Gamma _\tau \geq (1-s)\Gamma _{\tau _0}+s\Gamma _{\tau _1}. \]
Hence, by 16.6.28 and proposition 16.6.27,
\[ \widetilde\Gamma _\tau \geq (1-s)\widetilde\Gamma _{\tau _0}+s\widetilde\Gamma _{\tau _1}. \]
Thus the family is concave. Finally,
\[ \operatorname *{sup}_{\tau <\Gamma _{\max }}{}\! {}^{*}\widetilde\Gamma _\tau = P_\theta [\phi ]_{\mathcal I} \]
follows from proposition 16.6.30. Since $P_\theta [\phi ]_{\mathcal I}\geq \phi $, its mass is positive by the monotonicity theorem theorem 13.13.11. This proves the assertion.
$\square$
Definition 17.14.7.6C763D5B798A Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. A geodesic ray $\ell \in \mathcal{R}(X,\theta ;\phi )$ is maximal if $\ell ^*$ is $\mathcal{I}$-model, namely, if $\ell ^*\in \mathcal E^{\mathrm{NA}}(X,\theta ;\phi )$.
An important class of $\mathcal{I}$-model test curves is given by filtrations. We briefly recall the corresponding terminology.
Definition 17.14.8.D6CC0BBA6D35 Let $L$ be a big line bundle. We write
\[ R(X,L)=\bigoplus _{k=0}^{\infty } \mathrm{H}^0(X,L^ k) \]
for the section ring of $L$.
A filtration on $R(X,L)$ is a decreasing family of graded linear subspaces $(\mathcal{F}^{\lambda })_{\lambda \in \mathbb {R}}$ of $R(X,L)$ with graded pieces
\[ \mathcal{F}^{\lambda }=\bigoplus _{k=0}^{\infty } \mathcal{F}^{\lambda }_ k, \]
such that the following conditions are satisfied:
The filtration is left-continuous: For any $\lambda \in \mathbb {R}$, we have
\[ \mathcal{F}^{\lambda }=\bigcap _{\lambda '<\lambda }\mathcal{F}^{\lambda '}; \]
the filtration is multiplicative: For any $\lambda ,\lambda '\in \mathbb {R}$ and any $k,k'\in \mathbb {N}$, we have
\[ \mathcal{F}^{\lambda }_ k\cdot \mathcal{F}^{\lambda '}_{k'}\subseteq \mathcal{F}^{\lambda +\lambda '}_{k+k'}; \]
there is an integer $C>0$ such that
F76F341D25A8
17.14.9
\begin{equation} \mathcal{F}^{Cm}_{m}=0,\quad \mathcal{F}^{-Cm}_ m=\mathrm{H}^0(X,L^ m) \end{equation}
for all $m\in \mathbb {Z}_{>0}$.
Given a filtration $\mathcal{F}$ on $R(X,L)$, we define
\[ \tau _ k(\mathcal{F})=\sup \left\{ \lambda \in \mathbb {R}: \mathcal{F}^{\lambda }_ k\neq 0\right\} . \]
By Fekete's lemma, we can introduce
\[ \tau (\mathcal{F})=\lim _{k\to \infty }\frac{1}{k}\tau _ k(\mathcal{F})=\sup _{k\in \mathbb {Z}_{>0}}\frac{1}{k}\tau _ k(\mathcal{F}). \]
Note that $\tau (\mathcal{F})$ is bounded from above by the constant $C$ in (
17.14.9), hence finite.
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
\[ \Gamma ^{\mathcal{F},k}_{\tau }\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\left\{ \log |s|_{h^ k}^2: s\in \mathcal{F}^{k\tau }_ k, |s|_{h^ k}^2\leq 1 \right\} . \]
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
\[ \Gamma ^{\mathcal{F},k}_{\tau }\in \mathrm{PSH}(X,k\theta ),\quad \tau \leq k^{-1}\tau _ k(\mathcal{F}). \]
Observe that for $k,k'\in \mathbb {Z}_{>0}$, we have
\[ \Gamma ^{\mathcal{F},k+k'}_{\tau }\geq \Gamma ^{\mathcal{F},k}_{\tau }+\Gamma ^{\mathcal{F},k'}_{\tau }. \]
In particular, by Fekete's lemma,
79C9CAB675E4
17.14.11
\begin{equation} \lim _{k\in \mathbb {Z}_{>0}}\frac{1}{k}\Gamma ^{\mathcal{F},k}_{\tau }=\sup _{k\in \mathbb {Z}_{>0}}\frac{1}{k}\Gamma ^{\mathcal{F},k}_{\tau } \end{equation}
exists for any $\tau <\tau (\mathcal{F})$.
We define $(\Gamma ^{\mathcal{F}}_{\tau })_{\tau <\tau (\mathcal{F})}$ as follows:
\[ \Gamma ^{\mathcal{F}}_{\tau }\coloneqq P_{\theta }\left[ \operatorname *{sup}_{k\in \mathbb {Z}_{>0}}{}\! {}^{*} \frac{1}{k} \Gamma ^{\mathcal{F},k}_{\tau }\right]{}^{\href{#sgpt-a0000078461}{2}}. \]
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
\[ \Bigl (-\infty ,\tau (\mathcal{F})\Bigr )\ni \tau \mapsto \operatorname *{sup}_{k\in \mathbb {Z}_{>0}}{}\! {}^{*} \frac{1}{k} \Gamma ^{\mathcal{F},k}_{\tau } \]
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
\[ \operatorname *{sup}_{k\in \mathbb {Z}_{>0}}{}\! {}^{*} \frac{1}{k} \Gamma ^{\mathcal{F},k}_{t\tau _1+(1-t)\tau _0}\geq t\operatorname *{sup}_{k\in \mathbb {Z}_{>0}}{}\! {}^{*} \frac{1}{k} \Gamma ^{\mathcal{F},k}_{\tau _1}+(1-t)\operatorname *{sup}_{k\in \mathbb {Z}_{>0}}{}\! {}^{*} \frac{1}{k} \Gamma ^{\mathcal{F},k}_{\tau _0}. \]
But thanks to proposition 5.14.6 and proposition 5.14.5, it suffices to show that
\[ \sup _{k\in \mathbb {Z}_{>0}} \frac{1}{k}\Gamma ^{\mathcal{F},k}_{t\tau _1+(1-t)\tau _0}\geq t\sup _{k\in \mathbb {Z}_{>0}} \frac{1}{k}\Gamma ^{\mathcal{F},k}_{\tau _1}+(1-t)\sup _{k\in \mathbb {Z}_{>0}} \frac{1}{k}\Gamma ^{\mathcal{F},k}_{\tau _0} \]
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
9BF5C28825D8
17.14.12
\begin{equation} \sup _{k\in \mathbb {Z}_{>0}} \frac{1}{k}\Gamma ^{\mathcal{F},k}_{t\tau _1+(1-t)\tau _0}\geq \frac{1-t}{k_0}\log |s_0|_{h^{k_0}}^2+\frac{t}{k_1}\log |s_1|_{h^{k_1}}^2. \end{equation}
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
\[ s\coloneqq s_0^{k_1(q-p)}\otimes s_1^{k_0p}\in \mathcal{F}_{k_0k_1 q}^{k_0k_1\tau _0(q-p)+k_0k_1\tau _1 p}, \]
and
\[ \begin{aligned} & \frac{1}{k_0k_1 q} \log |s|_{h^{k_0k_1 q}}^2\\ =& \frac{1}{k_0k_1 q}\left(k_1(q-p)\log |s_0|^2+ k_0p\log |s_1|^2\right)\\ = & \frac{1-t}{k_0}\log |s_0|_{h^{k_0}}^2+\frac{t}{k_1}\log |s_1|_{h^{k_1}}^2. \end{aligned} \]
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
EA5B88AFDFF1
17.14.13
\begin{equation} \Gamma ^{\mathcal{F}}_{\tau }=V_{\theta }. \end{equation}
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 $,
\[ \Gamma ^{\mathcal F,k}_\tau \geq \log |s|_{h^ k}^2. \]
Hence
\[ s\in \mathrm H^0\left(X,L^ k\otimes \mathcal I(\Gamma ^{\mathcal F,k}_\tau )\right). \]
Since
\[ k\Gamma ^{\mathcal F}_\tau \geq \Gamma ^{\mathcal F,k}_\tau , \]
we get
\[ s\in \mathrm H^0\left(X,L^ k\otimes \mathcal I(k\Gamma ^{\mathcal F}_\tau )\right). \]
Thus
\[ \mathrm H^0\left(X,L^ k\otimes \mathcal I(k\Gamma ^{\mathcal F}_\tau )\right)=\mathrm H^0(X,L^ k) \]
for every $k\in \mathbb Z_{>0}$. By theorem 18.1.46,
\[ \operatorname{vol}\left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\mathcal F}_\tau \right)=\operatorname{vol}L. \]
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).
Pome
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