19.4 Algebraic partial Okounkov bodiesC9B9B0985C6B

Proof. Step 1. We first show that there is $\Delta \in \mathcal{K}_ n$ such that $\Delta _ k(L)\subseteq \Delta $. For this purpose, using 19.3.3, we may assume that $\nu $ is induced by an admissible flag $Y_{\bullet }$ on $X$. We may further assume that each $Y_ i$ is smooth.

Fix $s\in \mathrm{H}^0(X,L^ k)^{\times }$ for some $k\in \mathbb {Z}_{>0}$. Assume that $s\neq 0$. We need to show that for each $i=1,\ldots ,n$, $\nu (s)_ i\leq Ck$ for some constant $C>0$, independent of the choices of $k$ and $s$.

Fix an ample divisor $H$ on $X$. Take a large enough integer $b_1>0$ such that

Then $\nu (s)_1\leq b_1 k$. Next take a large enough integer $b_2$ such that, for every $a\in [0,b_1]$,

After removing the first vanishing order of $s$ along $Y_1$, the resulting section on $Y_1$ is a section of $k((L-aY_1)|_{Y_1})$ for some $a\in [0,b_1]$. Hence $\nu (s)_2\leq b_2k$. Continuing in this manner and choosing the constants uniformly over the compact ranges of the preceding normalized vanishing orders, we conclude that $\nu (s)_ i/k$ is bounded for each $i$.

Here the expressions involving $L-aY_1$ are understood numerically. At the $i$-th step the previous normalized vanishing orders lie in a compact polytope, and the intersection number is continuous in these parameters; hence the constants $b_ i$ can be chosen uniformly.

Step 2. Observe that $\Gamma (L)$ is clearly a semigroup. It remains to show that $\Gamma (L)$ generates $\mathbb {Z}^{n+1}$ as an Abelian group.

For this purpose, take two very ample divisors $A$ and $B$ so that $L=\mathcal{O}_ X(A-B)$. After choosing $A$ and $B$ ample enough, we may guarantee that there exist sections $s_0\in \mathrm{H}^0(X,A)$, $t_ i\in \mathrm{H}^0(X,B)$ for $i=0,\dots ,n$ such that

and $\nu (t_ i)$ is the $i$-th unit vector $e_ i\in \mathbb {R}^ n$ for $i=1,\ldots ,n$.

Since $L$ is big, we can find $m_0>0$ such that for any $m\geq m_0$, we can find an effective divisor $F_ m$ on $X$ linearly equivalent to $mL-B$. Let $f_ m=\nu ([F_ m])$. Then

Since $(m+1)L$ is linearly equivalent to $A+F_ m$, we have

It follows that $\Gamma (L)$ generates $\mathbb {Z}^{n+1}$. $\square$

Proof. This follows immediately from proposition 19.4.7 and theorem 19.2.5. $\square$

Proof. By theorem 5.18.8, we can choose a modification $\pi \colon Y\to X$ such that $\pi ^*\varphi $ has log singularities along an effective simple normal crossing $\mathbb Q$-divisor $D$. We regard $\nu $ as a valuation of $\mathbb C(Y)=\mathbb C(X)$. By the definition of $\mathcal I_\infty $, the section spaces defining $\Gamma ^\infty (\theta ,\varphi )$ are naturally identified with

Thus we may compute $\Gamma ^\infty (\theta ,\varphi )$ on $Y$.

In this case,

Since $\pi ^*L-D$ is nef and big by 5.18.5, our assertion follows from the same argument as proposition 19.4.7. $\square$

Proof. As $L$ is big, we can take $k_0\in \mathbb {Z}_{>0}$ so that

1. 6BF24D61A02C

   $T^{-1}\otimes L^{k_0}$ admits a non-zero global holomorphic section $s_0$, and

2. 0D44384F7F7B

   $T\otimes L^{k_0}$ admits a non-zero global holomorphic section $s_1$.

Then for $k\in \mathbb {Z}_{>k_0}$, we have injective linear maps

It follows that

Using theorem 19.2.5, we conclude. $\square$

Proof. Observe that for each $k\in \mathbb {N}$, we have

So our assertion follows immediately from theorem 19.2.5. $\square$

Proof. This is an immediate consequence of theorem 19.2.5. $\square$

Proof. Step 1. We first assume that $\varphi $ has analytic singularities and $\theta _{\varphi }$ is a Kähler current.

We claim that

Observe that for each $\epsilon \in \mathbb {Q}_{>0}$, we have

for all large enough $k$. This is a consequence of 5.18.15. Therefore, it suffices to show that

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

In fact, by theorem 18.1.46, we have

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

as $\epsilon \to 0+$. We compute using theorem 18.1.46:

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,

Since $\operatorname{vol}\Delta (\Gamma (\theta ,\varphi ))=\operatorname{vol}\Gamma (\theta ,\varphi )$ by theorem 19.2.10, (19.4.20) follows. $\square$

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

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

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. $\square$

Proof. This follows from corollary 19.2.13. $\square$

Proof. Let $\varphi _ j\to \varphi $ be a $d_ S$-convergent sequence in $\mathrm{PSH}(X,\theta )_{>0}$. We want to show that

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

as $j\to \infty $. In fact, using theorem 18.1.46, we can compute

which converges to $0$ by theorem 16.9.57. $\square$

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. $\square$

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

We observe that

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

1. 9C52F43A7B75

   $\varphi _ j$ and $\varphi '_ j$ both have analytic singularities for all $j\geq 1$, and

2. 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. $\square$

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

By theorem 19.2.5 and 19.4.23, (19.4.36) follows. $\square$

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. $\square$

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,

It follows that

On the other hand,

As $\delta \to 0+$, the right-hand side converges to

thanks to proposition 18.1.27. Our assertion therefore follows. $\square$

Proof. Up to replacing $X$ by a modification and twisting $T$ accordingly, we may assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$, see proposition 19.4.32 and theorem 5.18.8.

Take $\epsilon \in \mathbb {Q}\cap (0,1)$. In this case, for large enough $k\in \mathbb {Z}_{>0}$ we have

Take an integer $N\in \mathbb {Z}_{>0}$ so that $ND$ is a divisor and $N\epsilon $ is an integer.

Let $\Delta '$ be the limit of a subsequence of $(\Delta _{k,T}(\theta ,\varphi ))_ k$, say the sequence defined by the indices $k_1,k_2,\ldots $. Thanks to theorem 19.7.4, it suffices to show that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$.

There exists $t\in \{ 0,1,\ldots ,N-1\} $ such that $k_ i\equiv t$ modulo $N$ for infinitely many $i$, up to replacing $(k_ i)_ i$ by a subsequence, we may assume that $k_ i\equiv t$ modulo $N$ for all $i$. Write $k_ i=N g_ i+t$. Then for large enough $i$, we have

So

Letting $i\to \infty $, by proposition 19.4.12,

Letting $\epsilon \to 0+$, using the continuity of classical Okounkov bodies in the big cone recalled in theorem 19.5.4, we find that

where we applied 19.4.23 as well. Our assertion follows. $\square$

Proof. By proposition 19.4.33 and proposition 19.4.28, we have

for any small enough $\beta \in \mathbb {Q}_{>0}$. In particular, if $\Delta '$ is the Hausdorff limit of a subnet of $(\Delta ((1-\beta )\theta ,\varphi ))_{\beta }$, then $\Delta '\subseteq \Delta _{\nu }(\theta ,\varphi )$. But

Since we have not developed the theory of nef b-divisors yet, we give a direct proof as well. Take a Kähler form $\omega $ so that $\theta _{\varphi }\geq \omega $. Let $\psi =P_{\theta -\omega }[\varphi ]_{\mathcal{I}}$. Then $\varphi \sim _{\mathcal{I}}\psi $. In order to establish the last equality, we may replace $\varphi $ by $\psi $ and hence assume that $\varphi $ is $\mathcal{I}$-good. For small enough $\beta >0$, the form $(1-\beta )\theta -(\theta -\omega )=\omega -\beta \theta $ is Kähler, so this remains true in the class $(1-\beta )\{ \theta \} $ by 18.1.8. Thus the equality of masses in an $\mathcal{I}$-equivalence class reduces the desired equality to

which is obvious.

It follows that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$. We conclude by theorem 19.7.4. $\square$

Proof of theorem 19.4.41. Fix a very ample line bundle $H$ on $X$ and a Kähler form $\omega \in c_1(H)$ such that $\omega \geq \theta $.

Step 1. We first handle the case where $\theta _{\varphi }$ is a Kähler current, say $\theta _{\varphi }\geq 2\delta \omega $ for some $\delta \in (0,1)$. Take a quasi-equisingular approximation $(\varphi _ j)_ j$ of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We may assume that $\theta _{\varphi _ j}\geq \delta \omega $ for all $j\geq 1$. After adding constants, we also assume that $\varphi \leq 0$ and $\varphi _ j\leq 0$ for all $j$.

Let $\Delta '$ be a limit of a subsequence of $(\Delta _{k,T}(\theta ,\varphi ))_ k$. Let us say the indices of the subsequence are $k_1<k_2<\cdots $. By theorem 19.7.4, it suffices to show that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$.

Observe that for each $j\geq 1$, we have $\Delta '\subseteq \Delta _{\nu }(\theta ,\varphi _ j)$ by 19.4.43. Letting $j\to \infty $, we find $\Delta '\subseteq \Delta _{\nu }(\theta ,\varphi )$ as a consequence of theorem 19.4.30. Therefore, it suffices to prove that

Fix an integer $N>\delta ^{-1}$. Observe that for any $j\geq 1$, we have $\varphi _ j\in \mathrm{PSH}(X,(1-N^{-1})\theta )$. Similarly, $\varphi \in \mathrm{PSH}(X,(1-N^{-1})\theta )$. By 19.4.44, it suffices to argue that

Step 1.1. We first reduce to the case where $N|k_ i$ for all $i$.

We are free to replace $(k_ i)_ i$ by a subsequence, so we may assume that $k_ i\equiv a$ modulo $N$ for all $i\geq 1$, where $a\in \{ 0,1,\ldots ,N-1\} $. We write $k_ i=g_ i N+a$. Observe that for each $i\geq 1$,

Up to replacing $T$ by $T\otimes L^{-N+a}$, we may therefore assume that $a=0$, so that $k_ i=g_ i N$.

Step 1.2. Write $k_ i=g_ i N$ for all $i$. We prove (19.4.45).

Since

we can apply 13.13.30 to find $j'\in \mathbb {Z}_{>0}$ such that for all $j\geq j'$, there is $\psi _ j\in \mathrm{PSH}(X,\theta )_{>0}$ satisfying

Fix $j\geq j'$. It suffices to show that

for some $v'\in \mathbb {R}^ n$. In fact, if this is true, we have

Letting $j\to \infty $ and applying corollary 16.9.36 and theorem 19.4.30, we conclude (19.4.45).

It remains to prove (19.4.47).

By 13.13.48, there is $g'>0$ such that for any $g\geq g'$, we can find a non-zero section

By 5.18.15, after possibly increasing $g'$, for all $g\geq g'$,

Moreover, the inequality (19.4.46) gives that for $g\geq g'$,

where the second inclusion follows directly from the definition of $\mathcal I_\infty $. Thus multiplication by $s_ g$ gives an injective linear map

In particular, when $g=g_ i$ for some $i$ large enough, we then find

We observe that the $(g_ i)^{-1}\nu (s_{g_ i})$'s are bounded, since both convex bodies appearing in this equation remain bounded as $i$ varies. After passing to a subsequence, we may assume that $(g_ i)^{-1}\nu (s_{g_ i})$ converges. Then 19.4.43, followed by division by $N$, gives a vector $v'\in \mathbb {R}^ n$ such that (19.4.47) holds.

Step 2. Next we handle the general case.

Let $\Delta '$ be the Hausdorff limit of a subsequence of $(\Delta _{k,T}(\theta ,\varphi ))_ k$, say the subsequence with indices $k_1<k_2<\cdots $. By theorem 19.7.4, it suffices to prove that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$.

Take $\psi \in \mathrm{PSH}(X,\theta )$ such that $\theta _{\psi }$ is a Kähler current and $\psi \leq \varphi $. The existence of $\psi $ follows from 13.13.47.

Then for any $\epsilon \in \mathbb {Q}\cap (0,1)$,

for all $k\geq 1$. It follows from Step 1 that

As $\epsilon \to 0+$, we have

by 16.9.25. Hence theorem 19.4.30 gives

in the Hausdorff metric. It follows that $\Delta ' \supseteq \Delta _{\nu }(\theta ,\varphi )$.

It remains to establish that

Fix an integer $N>0$, it suffices to argue that

Assuming this, letting $N\to \infty $ and using 18.1.31 together with proposition 18.1.27 6, we conclude (19.4.48).

After adding a constant to $\varphi $, which does not change the multiplier ideals or the current $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $, we may assume that $\varphi \leq 0$.

Step 2.1. We first reduce to the case $N|k_ i$ for all $i$.

For this purpose, we are free to replace $k_1<k_2<\cdots $ by a subsequence. We may then assume that $k_ i\equiv a$ modulo $N$ for all $i\geq 1$ for some $a\in \{ 0,1,\ldots ,N-1\} $. We write $k_ i=g_ iN +a$. Observe that

Up to replacing $T$ by $T\otimes L^ a$, we may assume that $a=0$.

Step 2.2. We write $k_ i=g_ i N$ for all $i$.

Take a non-zero section $s\in \mathrm{H}^0(X,H)$.

We have an injective linear map

for each $g\geq 1$. In particular, for each $i\geq 1$,

Letting $i\to \infty $, by Step 1, we have

So

This completes the proof. $\square$

Proof. Replacing $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$, we may assume that $\varphi $ is $\mathcal{I}$-good.

Step 1. We first reduce to the case where $\varphi $ has analytic singularities.

By theorem 18.1.2, we can find a sequence $(\varphi _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ with analytic singularities such that $\varphi _ j\xrightarrow {d_ S}\varphi $. It follows from theorem 19.4.30 that

Therefore, by 19.7.10,

In view of theorem 16.9.55, it suffices to prove (19.4.52) with $\varphi _ j$ in place of $\varphi $.

Step 2. Assume that $\varphi $ has analytic singularities. In view of proposition 19.4.32 and theorem 5.18.8, after replacing $X$ by a modification, we may assume that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $F$.

Perturbing $L$ by an ample $\mathbb {Q}$-line bundle by proposition 19.4.39, we may assume that $\theta _{\varphi }$ is a Kähler current. Therefore, $L-F$ is ample by 5.18.5. Finally, by rescaling, we may assume that $F$ is a divisor and $L$ is a line bundle.

By theorem 19.4.41, we know that

By definition,

In view of proposition 5.16.15, it remains to show that

The $\geq $ direction is trivial, we prove the converse. Observe that

As $L-F$ is ample, for large enough $k$, the line bundle $k(L-F)$ is globally generated. Hence we can choose a section of $k(L-F)$ which does not vanish at the generic point of $E$. Viewed as a section of $L^ k\otimes \mathcal O_ X(-kF)$, it has vanishing order exactly $\operatorname{ord}_ E(kF)$ along $E$. Therefore,

Thus, (19.4.54) follows. $\square$

Proof. By theorem 19.4.51, the assumed inclusions imply that for every prime divisor $E$ on every modification $\pi $,

Hence $\varphi \preceq _{\mathcal{I}}\psi $ by proposition 16.8.9. $\square$

Proof. After replacing $X$ by a modification, we may assume that $E$ is a prime divisor on $X$. Note that since $X$ is projective, we can always find an admissible flag with $E$ being the first component.

Then (19.4.57) follows from theorem 19.4.51 and the fact that $\Delta _{Y_{\bullet }}(\theta ,V_{\theta })=\Delta _{Y_{\bullet }}(L)$ for any admissible flag $Y_{\bullet }$ on $X$. $\square$