PomeProposition 19.4.26.97A03B9F1F87 The partial Okounkov body $\Delta _{\nu }(L,h)$ depends only on $\mathrm{dd}^{\mathrm{c}}h$, not on the explicit choices of $L,h_0,h$.
19.4.25 Basic properties of partial Okounkov bodies1FAA4BD1CBCB
\[ \Delta _{\nu }(T)\coloneqq \Delta _{\nu }(\theta ,\varphi ) \]
if $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ for some $\varphi \in \mathrm{PSH}(X,\theta )$.
Proof. There are two different claims to prove, as detailed in the two steps below.
Step 1. Let $h_0'$ be another Hermitian metric on $L$. Set $\theta '=c_1(L,h_0')$. Write $\mathrm{dd}^{\mathrm{c}}f=\theta -\theta '$. Let $\varphi '=\varphi +f\in \mathrm{PSH}(X,\theta ')$. Then
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\begin{equation} \Delta _{\nu }(\theta ,\varphi )=\Delta _{\nu }(\theta ',\varphi '). \end{equation}
This is obvious since $\Gamma (\theta ,\varphi )=\Gamma (\theta ',\varphi ')$.
Step 2. Let $L'$ be another big line bundle on $X$. By Step 1, we may assume that the reference Hermitian metric $h'_0$ on $L'$ is such that $c_1(L',h'_0)=\theta $.
Let $h'$ be a plurisubharmonic metric on $L'$ with $c_1(L,h)=c_1(L',h')$. Then
\[ \Delta _{\nu }(L,h)=\Delta _{\nu }(L',h'). \]
By the approximation step in the construction of $\Delta _\nu (L,h)$ and the continuity of the extension in theorem 19.2.10, it is enough to prove the assertion when $c_1(L,h)$ has analytic singularities. After taking a birational resolution, it suffices to deal with the case where $c_1(L,h)$ has analytic singularities along an effective $\mathbb {Q}$-divisor $D$. By rescaling, we may also assume that $D$ is a divisor. By 19.4.23, we further reduce to the case where $c_1(L,h)$ is not singular.
In this case, the assertion is proved in proposition 19.4.9. $\square$
Proposition 19.4.28.1934C9D19460 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that $\varphi \preceq _{\mathcal{I}} \psi $. Then
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\begin{equation} \Delta _{\nu }(\theta ,\varphi ) \subseteq \Delta _{\nu }(\theta ,\psi ). \end{equation}
\[ \Delta _{\nu }(\theta ,\varphi )\subseteq \Delta _{\nu }(L). \]
Proof. This follows from corollary 19.2.13. $\square$
Theorem 19.4.30.30EEF3A985B7 The Okounkov body map
\[ \Delta _{\nu }(\theta ,\bullet ):\left( \mathrm{PSH}(X,\theta )_{>0},d_ S\right)\rightarrow \left(\mathcal{K}_ n,d_{\mathrm{Haus}}\right) \]
is continuous.
Proof. Let $\varphi _ j\to \varphi $ be a $d_ S$-convergent sequence in $\mathrm{PSH}(X,\theta )_{>0}$. We want to show that
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19.4.31
\begin{equation} \Delta _{\nu }(\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{Haus}}}\Delta _{\nu }(\theta ,\varphi ). \end{equation}
Replacing the potentials by their $P$-envelopes does not change their partial Okounkov bodies, since $P$-equivalent potentials are $\mathcal{I}$-equivalent, nor does it change their $d_ S$-classes. Thus we may assume that all $\varphi _ j$'s and $\varphi $ are model potentials.
By proposition 16.9.17, after passing to a subsequence we can sandwich $\varphi _ j$ between a decreasing sequence and an increasing sequence, both converging to $\varphi $ in $d_ S$. Using proposition 19.4.28 and theorem 19.7.4, it is enough to prove (19.4.31) when $(\varphi _ j)_ j$ is monotone. By theorem 16.9.49, replacing $\varphi _ j$ and $\varphi $ by their $\mathcal{I}$-envelopes preserves this reduction, so we may further assume that the $\varphi _ j$'s and $\varphi $ are $\mathcal{I}$-model. In both cases, we claim that
\[ \Gamma (\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{sg}}} \Gamma (\theta ,\varphi ) \]
as $j\to \infty $. In fact, using theorem 18.1.46, we can compute
\[ \begin{aligned} d_{\mathrm{sg}}\left(\Gamma (\theta ,\varphi _ j), \Gamma (\theta ,\varphi )\right) = & \varlimsup _{k\to \infty } k^{-n}\left| h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _ j)\right)-h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right) \right|\\ =& \frac{1}{n!} \left| \operatorname{vol}\theta _{\varphi _ j}-\operatorname{vol}\theta _{\varphi }\right|, \end{aligned} \]
which converges to $0$ by theorem 16.9.57. $\square$
Proposition 19.4.32.CFE94018A80A Let $\pi \colon Y\rightarrow X$ be a modification. Then
\[ \Delta _{\nu }(\pi ^*L,\pi ^*h)=\Delta _{\nu }(L,h). \]
Proof. Thanks to proposition 16.6.10, we may assume that $\varphi $ is $\mathcal{I}$-model. By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ with analytic singularities in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ j\xrightarrow {d_ S}\varphi $. It is clear that $\pi ^*\varphi _ j\xrightarrow {d_ S}\pi ^*\varphi $. By theorem 19.4.30, we may then reduce to the case where $\varphi $ has analytic singularities. In this case, it suffices to apply 19.4.23. $\square$
Proposition 19.4.33.86E922DD5140 Let $(L', h')$ be another Hermitian big line bundle on $X$. Then
\[ \Delta _{\nu }(L,h)+\Delta _{\nu }(L', h')\subseteq \Delta _{\nu }(L\otimes L',h\otimes h'). \]
Proof. Take a Hermitian metric $h_0'$ on $L'$ and let $\theta '=c_1(L',h_0')$. We identify $h'$ with $\varphi '\in \mathrm{PSH}(X,\theta ')$. Then we need to show
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\begin{equation} \Delta _{\nu }(\theta ,\varphi )+\Delta _{\nu }(\theta ',\varphi ')\subseteq \Delta _{\nu }(\theta +\theta ',\varphi +\varphi '). \end{equation}
We observe that
\[ P_{\theta }[\varphi ]_{\mathcal{I}}+P_{\theta '}[\varphi ']_{\mathcal{I}}\sim _{\mathcal{I}} \varphi +\varphi '. \]
Thus, after replacing $\varphi $ and $\varphi '$ by their $\mathcal{I}$-envelopes, in view of proposition 19.4.28, we may assume that $\varphi $ and $\varphi '$ are $\mathcal{I}$-good.
By theorem 18.1.2, we can find sequences $(\varphi _ j)_ j$ and $(\varphi '_ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ and $\mathrm{PSH}(X,\theta ')_{>0}$ respectively such that
9C52F43A7B75$\varphi _ j$ and $\varphi '_ j$ both have analytic singularities for all $j\geq 1$, and
3F2CA81FC88F$\varphi _ j\xrightarrow {d_ S}\varphi $, $\varphi '_ j\xrightarrow {d_ S}\varphi '$.
Then $\varphi _ j+\varphi '_ j\in \mathrm{PSH}(X,\theta +\theta ')_{>0}$ and $\varphi _ j+\varphi '_ j\xrightarrow {d_ S} \varphi +\varphi '$ by theorem 16.9.45. Thus, by theorem 19.4.30, we may assume that $\varphi $ and $\varphi '$ both have analytic singularities. Taking a birational resolution, we may further assume that they have log singularities. By 19.4.23, we reduce to the case without singularities, in which case the result is just proposition 19.4.13. $\square$
Theorem 19.4.35.10DA5B8BC320 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then for any $t\in (0,1)$,
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\begin{equation} \Delta _{\nu }\left(\theta ,t\varphi +(1-t)\psi \right)\supseteq t\Delta _{\nu }(\theta ,\varphi )+(1-t)\Delta _{\nu }(\theta ,\psi ). \end{equation}
Proof. We may assume that $t$ is rational as a consequence of theorem 19.4.30. Similarly, as in the proof of proposition 19.4.33, we may reduce to the case where both $\varphi $ and $\psi $ have analytic singularities. In this case, let $N>0$ be an integer such that $Nt$ is an integer. Then for any $s\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}_{\infty }(k\varphi ))$ and $r\in \mathrm{H}^0(X,L^ k\otimes \mathcal{I}_{\infty }(k\psi ))$, we have
\[ s^{tN}\otimes r^{N-tN}\in \mathrm{H}^0\left(X,L^{kN}\otimes \mathcal{I}_{\infty }\left(kN(t\varphi +(1-t)\psi )\right)\right). \]
Proposition 19.4.37.300A5A5F84BB For any $a\in \mathbb {Z}_{>0}$,
\[ \Delta _{\nu }(a\theta ,a\varphi )=a\Delta _{\nu }(\theta ,\varphi ). \]
Proof. As in the proof of proposition 19.4.33, we may assume that $\varphi $ has log singularities. Using 19.4.23, we reduce to the case without the singularities, which is proved in proposition 19.4.14. $\square$
In particular, if $S$ is a closed positive $(1,1)$-current on $X$ with $\int _ X S^ n>0$ and such that
\[ \{ S\} \in \mathrm{NS}^1(X)_{\mathbb {Q}}, \]
we can define
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\begin{equation} \Delta _{\nu }(S)\coloneqq a^{-1}\Delta _{\nu }(aS) \end{equation}
for a sufficiently divisible positive integer $a$. This definition is independent of the choice of $a$ thanks to proposition 19.4.37.
We also need the following perturbation. Let $A$ be an ample line bundle on $X$. Fix a Hermitian metric $h_ A$ on $A$ such that $\omega \coloneqq c_1(A,h_ A)$ is a Kähler form on $X$.
Proposition 19.4.39.63CE446DCB89 As $\mathbb {Q}_{>0}\ni \delta \searrow 0$, the convex bodies $\Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )$ are decreasing and
\[ \Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )\xrightarrow {d_{\mathrm{Haus}}} \Delta _{\nu }(\theta _{\varphi }). \]
Proof. Let $0\leq \delta <\delta '$ be two rational numbers. Take $C\in \mathbb {N}_{>0}$ divisible enough, so that $C\delta $ and $C\delta '$ are both integers. Then by proposition 19.4.33,
\[ \Delta _{\nu }(C\theta +C\delta \omega +C\mathrm{dd}^{\mathrm{c}}\varphi )\subseteq \Delta _{\nu }(C\theta +C\delta '\omega +C\mathrm{dd}^{\mathrm{c}}\varphi ). \]
It follows that
\[ \Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )\subseteq \Delta _{\nu }(\theta +\delta '\omega +\mathrm{dd}^{\mathrm{c}}\varphi ). \]
On the other hand,
\[ \operatorname{vol}\Delta _{\nu }(\theta +\delta \omega +\mathrm{dd}^{\mathrm{c}}\varphi )=\frac{1}{n!}\operatorname{vol}(\theta +\delta \omega )_{\varphi }. \]
As $\delta \to 0+$, the right-hand side converges to
\[ \operatorname{vol}\Delta _{\nu }(\theta ,\varphi )=\frac{1}{n!}\operatorname{vol}\theta _{\varphi }, \]
thanks to proposition 18.1.27. Our assertion therefore follows. $\square$
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