PomeTheorem 19.4.16.A653D079A694 We have
\[ \Gamma (\theta ,\varphi )\in \overline{\mathcal{S}'(\Delta _{\nu }(L))}_{>0}. \]
19.4.15 Construction of partial Okounkov bodies2A53BABD6994
Definition 19.4.17.FAA070EFD0CC The partial Okounkov body of $(L,h)$ is defined as
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\begin{equation} \Delta _{\nu }(L,h)=\Delta _{\nu }(\theta ,\varphi )\coloneqq \Delta \left(\Gamma (\theta ,\varphi )\right). \end{equation}
Note that when $h$ has minimal singularities, we have
\[ \Delta _{\nu }(L,h)=\Delta _{\nu }(L). \]
So partial Okounkov bodies generalize Okounkov bodies.
Corollary 19.4.19.56E59F7A3150 We have
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\begin{equation} \operatorname{vol}\Delta _{\nu }(\theta ,\varphi )=\frac{1}{n!}\operatorname{vol}\theta _{\varphi }. \end{equation}
We will prove theorem 19.4.16 and corollary 19.4.19 at the same time. The proof relies on the pseudometric $d_{\mathrm{sg}}$ introduced in (19.2.1).
Proof. Step 1. We first assume that $\varphi $ has analytic singularities and $\theta _{\varphi }$ is a Kähler current.
We claim that
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\begin{equation} d_{\mathrm{sg}}(\Gamma ^{\infty }(\theta ,\varphi ),\Gamma (\theta ,\varphi ))=0. \end{equation}
Observe that for each $\epsilon \in \mathbb {Q}_{>0}$, we have
\[ \mathrm{H}^0\Bigl (X,L^ k\otimes \mathcal{I}_{\infty }(k\varphi )\Bigr )\subseteq \mathrm{H}^0\Bigl (X,L^ k\otimes \mathcal{I}(k\varphi )\Bigr )\subseteq \mathrm{H}^0\left(X,L^ k\otimes \mathcal{I}_{\infty }(k(1-\epsilon )\varphi )\right) \]
for all large enough $k$. This is a consequence of 5.18.15. Therefore, it suffices to show that
\[ \lim _{\mathbb {Q}\ni \epsilon \to 0+}\operatorname{vol}\Gamma ^{\infty }(\theta ,(1-\epsilon )\varphi )=\operatorname{vol}\Gamma ^{\infty }(\theta ,\varphi ). \]
This follows from the explicit formula in proposition 19.4.11.
Step 2. We next handle the case where $\theta _{\varphi }$ is a Kähler current.
Let $(\varphi _ j)_ j$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. Then $\varphi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}$ by corollary 18.1.14.
In this case, it suffices to prove that
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\begin{equation} \Gamma (\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{sg}}} \Gamma (\theta ,\varphi ). \end{equation}
In fact, by theorem 18.1.46, we have
\[ \begin{aligned} & d_{\mathrm{sg}}\left(\Gamma (\theta ,\varphi _ j),\Gamma (\theta ,\varphi )\right) \\ =& \varlimsup _{k\to \infty } k^{-n} \left( h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _ j)\right)-h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)\right)\\ = & \lim _{k\to \infty } k^{-n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _ j)\right)-\lim _{k\to \infty } k^{-n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right) \\ =& \frac{1}{n!}\operatorname{vol}\theta _{\varphi _ j}-\frac{1}{n!}\operatorname{vol}\theta _{\varphi }. \end{aligned} \]
Letting $j\to \infty $, we conclude (19.4.22) by theorem 16.9.57.
Step 3. Now we only assume that $\operatorname{vol}\theta _{\varphi }>0$. We may replace $\varphi $ with $P_{\theta }[\varphi ]_{\mathcal{I}}$ and then assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$.
Take a potential $\psi \in \mathrm{PSH}(X,\theta )$ such that $\psi \leq \varphi $ and $\theta _{\psi }$ is a Kähler current. The existence of $\psi $ is proved in 13.13.47. For each $\epsilon \in (0,1)$, let $\varphi _{\epsilon }=(1-\epsilon )\varphi +\epsilon \psi $. It suffices to show that
\[ \Gamma (\theta ,\varphi _{\epsilon })\xrightarrow {d_{\mathrm{sg}}}\Gamma (\theta ,\varphi ) \]
as $\epsilon \to 0+$. We compute using theorem 18.1.46:
\[ \begin{aligned} & d_{\mathrm{sg}}\left(\Gamma (\theta ,\varphi _{\epsilon }),\Gamma (\theta ,\varphi )\right)\\ =& \varlimsup _{k\to \infty } k^{-n} \left( h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)- h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _{\epsilon })\right) \right)\\ =& \lim _{k\to \infty }k^{-n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right)-\lim _{k\to \infty }k^{-n}h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _{\epsilon })\right)\\ =& \frac{1}{n!}\operatorname{vol}\theta _{\varphi }- \frac{1}{n!}\operatorname{vol}\theta _{\varphi _{\epsilon }}\\ \to & 0 \end{aligned} \]
by theorem 16.9.57, as $\epsilon \to 0+$. Combining the three steps, $\Gamma (\theta ,\varphi )$ lies in $\overline{\mathcal S'(\Delta _\nu (L))}_{>0}$. This proves theorem 19.4.16. Moreover, by the continuity of the volume on $\overline{\mathcal S}(\Delta _\nu (L))$ and by theorem 18.1.46,
\[ \operatorname{vol}\Gamma (\theta ,\varphi )=\lim _{k\to \infty }k^{-n}h^0(X,L^ k\otimes \mathcal I(k\varphi ))=\frac{1}{n!}\operatorname{vol}\theta _\varphi . \]
Since $\operatorname{vol}\Delta (\Gamma (\theta ,\varphi ))=\operatorname{vol}\Gamma (\theta ,\varphi )$ by theorem 19.2.10, (19.4.20) follows. $\square$
Remark 19.4.23.555C64B2D134 It follows from the proof that if $\varphi $ has analytic singularities and $\theta _{\varphi }$ is a Kähler current, then (19.4.21) holds.
If we take a modification $\pi \colon Y\rightarrow X$ such that $\pi ^*\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$ on $Y$, then
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\begin{equation} \Delta _{\nu }(\theta ,\varphi )=\Delta _{\nu }(\pi ^*L-D)+\nu (D). \end{equation}
This is a very special case of theorem 19.5.26.
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