21.13 Toric pluripotential theory on big line bundlesF1FA15E12648

Proof. Recall that

See [Proposition 4.3.3, CLS11] for example. So we only need to understand the common zeros of $D+\operatorname{div}\chi ^ m$ for all $m\in M\cap P_ D$. But we know that

Our assertion follows. $\square$

Proof. It follows from proposition 21.13.8 that the stable base locus of $D$ is given by the union of the $V(\tau )$'s, where $\tau $ runs over elements of $\Sigma $ satisfying (21.13.9) for all $m\in M_{\mathbb {Q}}\cap P_ D$. That is,

Since the latter part is a rational polytope, this statement is equivalent to

Our first assertion follows. $\square$

Proof. Step 1. We first prove the existence of $F_{\theta }$ for a specific choice of $h$.

Write $D$ as the difference of two toric-invariant ample divisors, say $D_1-D_2$. Let $h_1$ and $h_2$ be the associated Guillemin's metrics on $D_1$ and $D_2$ respectively. We define $h=h_1\otimes h_2^{-1}$.

In this case, our assertion follows since it holds in the case of Guillemin's metrics.

Step 2. In general, fix $h_0$ as in Step 1. Then the general $h$ can be written as $h_0\exp (-g)$ for some smooth function $g$ on $X$. Since both metrics are $T_ c$-invariant, so is $g$, and on the open torus we may write $g=\operatorname{Trop}^* r$ for some smooth function $r$ on $N_{\mathbb {R}}$. Hence it suffices to modify $F_{\theta }$ in Step 1 by $r$ to conclude. $\square$

Proof. Let $F_0$ be a smooth convex function on $N_{\mathbb {R}}$ such that $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^* F_0$ can be extended to a Kähler form on $X$. For example, Guillemin's construction (21.12.9) with respect to a suitable Delzant polytope gives such an example.

Write $\omega $ for a Kähler form on $X$ extending $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^*F_0$. Then for any large enough $C>0$, $\theta +C\omega $ is a Kähler form. So we conclude from proposition 21.12.24 that

By the multi-linearity of the non-pluripolar product proposition 13.11.4 6, the left-hand side is a polynomial in $C$. By the expansion formula for the mixed real Monge–Ampère measure theorem 21.11.58 1, the right-hand side is also a polynomial in $C$. We conclude that the same holds for $C=0$. Therefore, (21.13.29) follows.

(21.13.30) is a direct consequence of (21.13.29). $\square$

Proof. We first reduce to the case where $D|_{U_{\sigma }}=0$. In fact, replacing $D$ by $D+\operatorname{div}\chi ^{m_{\sigma }}$ would result in changing $P_ D$ to $P_ D-m_{\sigma }$. So in view of (21.13.6), we may assume that $D|_{U_{\sigma }}=0$ and hence $m_{\sigma }=0$.

Fix $k\in \mathbb {Z}_{>0}$. Let $s\in \mathrm{H}^0(X,L^ k)$ be a non-zero toric eigen-section, say $\chi ^ m$ for some $m\in kP_ D\cap M$. The zero-divisor of $s$ on $U_{\sigma }$ is given by

see [Proposition 4.1.2, CLS11]. Therefore,

Conversely, every section of $L^ k$ is a finite linear combination of such characters. In the coordinates on $U_{\sigma }$ determined by $u_{\rho _1},\ldots ,u_{\rho _ n}$, the Okounkov valuation of a non-zero linear combination is the lexicographically first exponent among the characters with non-zero coefficient. Hence arbitrary sections give no valuations beyond the right-hand side of (21.13.34), and (21.13.34) follows. $\square$

Proof. Apply Demailly's Bergman-kernel approximation theorem [Theorem 13.21, Dem12] to the singular Hermitian metric $h\exp (-\varphi )$ on $L$. Since $\theta _{\varphi }$ is a Kähler current, the theorem applies to the Hilbert spaces

for $q\gg 1$, endowed with the $L^2$-norm induced by $h^ q\exp (-q\varphi )$ and a fixed Kähler volume form. The usual decreasing extraction in Demailly's construction gives a quasi-equisingular approximation; taking $q$ sufficiently large at each step ensures that the approximants still have Kähler Chern currents.

It remains to note that this construction is compatible with the torus action. Indeed, both $h$ and $\varphi $ are $T_ c$-invariant, hence the above Hilbert spaces and their $L^2$-inner products are $T_ c$-invariant. Since $T_ c$ is compact and abelian, each of these finite-dimensional Hilbert spaces admits an orthonormal basis consisting of toric eigen-sections. With such a basis, the corresponding Bergman potential has the displayed form and is $T_ c$-invariant. This proves the lemma. $\square$

Proof. We first reduce to the case where $D|_{U_{\sigma }}=0$. In fact, changing $D$ to $D+\operatorname{div}\chi ^{m_{\sigma }}$ would result in changing $F_{\theta }$ to $F_{\theta }-m_{\sigma }$. Hence, $F_{\varphi }$ changes to $F_{\varphi }-m_{\sigma }$. Therefore, $\Delta (\theta ,\varphi )$ becomes $\Delta (\theta ,\varphi )-m_{\sigma }$. Taking (21.13.6) into consideration, we may assume that $m_{\sigma }=0$.

Step 1. We first reduce to the case where $\theta _{\varphi }$ is a Kähler current.

By 13.13.47, we can find $\psi \in \mathrm{PSH}(X,\theta )$ such that $\psi \leq \varphi $ and $\theta _{\psi }$ is a Kähler current. Taking the average along $T_ c$, we may assume that $\psi $ is $T_ c$-invariant.

For each $t\in (0,1)$, we let

Suppose that Kähler current case is known. Then we get

for any $t\in (0,1)$. It follows from theorem 21.11.56 that

for any $t\in (0,1)$. Thanks to theorem 19.4.30, we have

Comparing the volumes of both sides using proposition 21.13.28, (19.4.20), and the $\mathcal{I}$-goodness of toric singularities example 18.1.56, we find that

In particular, we conclude (21.13.40).

Step 2. We handle the case where $\theta _{\varphi }$ is a Kähler current.

Let $(\varphi _ j)_ j$ be the quasi-equisingular approximation given by 21.13.38.

Now assume that the result is known for each $\varphi _ j$. Then

In particular, thanks to proposition 19.4.28,

for each $j\geq 1$. It follows from theorem 19.4.30 that

Comparing the volumes of both sides using proposition 21.13.28, (19.4.20), and example 18.1.56, we conclude (21.13.40).

Step 3. It remains to prove the assertion for the analytic approximants appearing in Step 2.

By 21.13.38, these approximants may be chosen in the form

where $q\in \mathbb Z_{>0}$ and $s_1,\ldots ,s_ a\in \mathrm{H}^0(X,L^ q)$ are toric eigen-sections.

Let $m_1,\ldots ,m_ a\in qP_ D\cap M$ be the lattice points corresponding to $s_1,\ldots ,s_ a$.

Observe that

where we have applied (21.13.23) on the second line and 21.11.62 on the third line. Since $m_1,\ldots ,m_ a\in qP_ D$, the preceding formula also shows that

Thus, if $m\in k\Delta (\theta ,\varphi )\cap M$, then $m\in kP_ D\cap M$, and $\chi ^ m$ indeed determines a global section of $L^ k$.

In particular, by the same separation argument as in 21.11.60, for any $k\in \mathbb {Z}_{>0}$ and any $m\in k\Delta (\theta ,\varphi )\cap M$, we have

is bounded from above on $T(\mathbb {C})$. In other words, the section $s$ of $L^ k$ corresponding to $m$ satisfies

Therefore,

where $\Delta _ k$ is defined in section 19.4. Hence,

Letting $k\to \infty $ and applying theorem 19.4.41, we find that

Comparing the volumes of both sides using proposition 21.13.28 and (19.4.20), we conclude that the equality holds and (21.13.40) follows. $\square$

Proof. This follows immediately from theorem 21.13.39 and theorem 19.4.51. In fact, since $X$ is projective and smooth, there is always a $T$-invariant smooth flag $Y_{\bullet }$ with $Y_1=E$. $\square$

Proof. 2 $\iff $ 3. This follows from the definition of $Q_{\sigma }$ in (21.13.7).

1 $\iff $ 2. Let $\rho _1,\ldots ,\rho _ r$ be the rays of $\sigma $. Up to replacing $D$ by a translation, we may assume that $m_{\sigma }=0$. Hence, we may take $m_{\rho _ i}=0$ for all $i$.

Let $\pi \colon Z\rightarrow X$ be the blow-up of $X$ along $Y$. See [Page 132, CLS11] for the basic properties of the toric blow-up. Take the divisor $\pi ^*D$ on $Z$. We choose the pull-back metric $\pi ^*h$ on $\pi ^*L$. Then $F_{\pi ^*\theta }$ can be taken as $\pi ^*F_{\theta }$ by (21.13.23). It follows that $\Delta (\theta ,\varphi )=\Delta (\pi ^*\theta ,\pi ^*\varphi )$. On the other hand, the ray corresponding to the exceptional divisor $E$ is generated by $u_{\rho _1}+\cdots +u_{\rho _ r}$. Since $X$ is smooth, this vector is primitive.

Recall that the support function of $\pi ^*D$ is the same as the support function of $D$, see [Proposition 6.2.7, CLS11]. In particular, we can take the Cartier datum $m_{\rho }=m_{\sigma } \mod M(\rho )$, where $\rho $ is the ray corresponding to $E$.

It follows from corollary 21.13.42 and 5.16.17 that

Our assertion follows in view of (21.13.5). $\square$

Proof. Taking $\varphi =V_{\theta }$ in theorem 21.13.39, we find

where we applied proposition 21.13.33 in the last equality. Therefore, (21.13.49) follows. In particular, (21.13.47) follows from proposition 21.13.28.

Next we prove (21.13.48). Take a smooth function $\chi \colon N_{\mathbb {R}}\rightarrow [0,1]$ so that

1. 37909700D0C1

   $\operatorname{Supp}\chi $ is compact;

2. C1605365568C

   $\chi \equiv 1$ on a neighbourhood of $0$.

Fix a $T_ c$-invariant volume form $\mu $ on $T(\mathbb {C})$. Then $(\operatorname{Trop}^*\chi )\mu $ can be regarded as a $T_ c$-invariant smooth non-negative measure on $X$. After choosing a suitable normalizing constant $C>0$, there is a unique solution

This is proved in [Theorem B, BEGZ10]. The uniqueness of $\varphi $ further guarantees its $T_ c$-invariance. Since $\varphi $ has minimal singularities, $F_{\varphi }\sim F_{V_{\theta }}$. Next, using proposition 21.13.28, the Monge–Ampère equation (21.13.50) implies

It follows from the regularity theorem [Theorem 2.19, BB13] and (21.13.49) that $F_{\varphi }\sim \operatorname{Supp}_{P_ D}$. Hence, (21.13.48) follows. $\square$

Proof. The proof is similar to that of theorem 21.12.13, but due to its importance, we give the details. Again, the correspondence between 2 and 3 follows easily from proposition 21.11.38.

Given $\varphi $, we can construct $F_{\varphi }$ in 2 as explained earlier in (21.13.26). Conversely, suppose that $F\in \mathcal{P}(N_{\mathbb {R}},P_ D)$, then $F\preceq F_{V_{\theta }}$ by theorem 21.13.45. Then

by 21.12.7. Since $F\preceq F_{V_{\theta }}$, we see that $\operatorname{Trop}^*(F-F_{\theta })$ is bounded from above. It follows that Grauert–Remmert's extension theorem theorem 5.14.10 is applicable, and this function extends to a unique $\theta $-psh function $\varphi $. The uniqueness of the extension guarantees that $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$.

The two maps are inverse to each other by construction. $\square$

Proof. By (16.3.7),

Hence $P_{\theta }[\varphi ]\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ by proposition 21.13.55, proposition 21.13.56, proposition 21.13.57. The same results give

Put

If $x\in \operatorname{Dom}G_{\varphi }$, then $G_{\varphi }(x)-C\leq G_{V_{\theta }}(x)$ for all large enough $C$, so $K(x)=G_{V_{\theta }}(x)$. If $x\notin \operatorname{Dom}G_{\varphi }$, then $K(x)=\infty $. By proposition 21.11.34, we have $\Delta (\theta ,\varphi )=\overline{\operatorname{Dom}G_{\varphi }}$. Moreover, theorem 21.13.45 gives $F_{V_{\theta }}\sim \operatorname{Supp}_{P_ D}$, hence $G_{V_{\theta }}\sim \chi _{P_ D}$ by Legendre duality. In particular, $G_{V_{\theta }}$ is finite and continuous on $P_ D$. Thus the lower semicontinuous regularization of $K$ is equal to $G_{V_{\theta }}$ on $\Delta (\theta ,\varphi )$ and equal to $\infty $ outside $\Delta (\theta ,\varphi )$. $\square$

Proof. 1 $\iff $ 2. This follows from the $\mathcal{I}$-goodness of $\varphi $ and $\psi $, see example 18.1.56.

1 $\iff $ 3. When $\varphi ,\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )_{>0}$, this follows from proposition 21.13.58.

In general, fix an ample toric-invariant divisor $H$ on $X$ and a toric-invariant Hermitian metric $h_ H$ on $\mathcal{O}_ X(H)$ such that $\omega _ H\coloneqq \mathrm{dd}^{\mathrm{c}}h_ H$ is a Kähler form. We perturb $L$ to $L+m^{-1}H$ for $m\in \mathbb {Z}_{>0}$, then thanks to 21.11.47, we find that

Then since $\varphi ,\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta +m^{-1}\omega _ H)_{>0}$, we know that $\varphi \preceq _ P \psi $ if and only if

for any $m\in \mathbb {Z}_{>0}$. The latter condition is equivalent to $\Delta (\theta ,\varphi )\subseteq \Delta (\theta ,\psi )$. $\square$

Proof. Thanks to corollary 21.13.60, we have the equality

and the map

is well-defined and injective.

In order to see the surjectivity, take $K\in \mathcal{K}_ n$, $K\subseteq P_ D$. Let $\chi _ K$ be the characteristic function of $K$ as defined in example 21.11.5. Since $\chi _ K$ satisfies Condition 3 of theorem 21.13.53, we can find $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ so that $\chi _ K=G_{\varphi }$. Then $\Delta (\theta ,\varphi )=K$ due to proposition 21.11.34. $\square$

Proof. We begin with a subgeodesic $(\varphi _ t)_{t\in (0,1)}$ from $\varphi _0$ to $\varphi _1$. Then we define $F\colon N_{\mathbb {R}}\times (0,1)\rightarrow \mathbb {R}$ as follows:

Define $F_ t$ as in (21.13.63), we have

Let $t_ j\to 0+$. Since $\varphi _{t_ j}\to \varphi _0$ in $L^1$, after passing to a subsequence we have almost-everywhere convergence on $X$. Restricted to the open torus and using the tropical coordinates, this gives $F_{t_ j}\to F_{\varphi _0}$ almost everywhere on $N_{\mathbb R}$. Since the $F_{t_ j}$'s are convex and locally uniformly bounded above, theorem 21.11.20 upgrades this to pointwise convergence everywhere. As the limit is independent of the subsequence, $F_ t\to F_{\varphi _0}$ everywhere as $t\to 0+$. Similarly, $F_ t\to F_{\varphi _1}$ everywhere as $t\to 1-$.

Next we show that $F$ is convex. Let $p_1\colon X\times S\rightarrow X$ be the projection, where

Since $(\varphi _ t)_{t\in (0,1)}$ is a subgeodesic, its complexification $\Phi $ is $p_1^*\theta $-psh. Recall that $\Phi $ is defined as

In particular, $\Psi \colon T(\mathbb {C})\times S\rightarrow \mathbb {R}$ defined by

is plurisubharmonic and $T_ c\times S^1$-invariant. Fix a small enough $\epsilon >0$. We can find a decreasing sequence of $T_ c\times S^1$-invariant plurisubharmonic functions $\Psi _ i$ on $T(\mathbb {C})\times S_{\epsilon }$ converging to $\Psi $ everywhere, where

Let us write

for some $F_ i\colon N_{\mathbb {R}}\times (\epsilon ,1-\epsilon )\rightarrow \mathbb {R}$.

The same computation as in 21.12.7 shows that $F_ i$ is convex. It follows that $F$, as the decreasing limit of $F_ i$, is also convex on $N_{\mathbb {R}}\times (\epsilon ,1-\epsilon )$. Since $\epsilon >0$ is arbitrary, we conclude that $F$ is convex on $N_{\mathbb {R}}\times (0,1)$.

Conversely, suppose that we are given $F$ in 2. We define $\Phi \colon T(\mathbb {C})\times S\rightarrow \mathbb {R}$ using (21.13.64). The arguments in the previous part can be reversed to show that $\Phi $ is $p_1^*\theta |_{T(\mathbb {C})\times S}$-psh.

By our assumption, for each $t\in (0,1)$, we have

for some constant $C\in \mathbb {R}$ independent of the choice of $t$. Therefore, $\Phi $ is bounded from above and hence, by theorem 5.14.10, admits a unique $p_1^*\theta $-psh extension to $X\times S$, which we still denote by $\Phi $. We let

for all $t\in (0,1)$ and $x\in X$. We claim that $(\varphi _ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$.

For this purpose, we only need to show that $(\varphi _ t)_{t\in (0,1)}$ has the correct boundary value. But from our assumption in 2, we know that as $t\to 0+$ (resp. $t\to 1-$), $\varphi _ t\to \varphi _0$ (resp. $\varphi _ t\to \varphi _1$) almost everywhere. In particular, $\sup _ X \varphi _ t\geq -C'$ for some large constant $C'>0$ independent of $t\in (0,1)$. Therefore, together with (21.13.65), we deduce from proposition 5.17.3 that $\{ \varphi _ t\} _{t\in (0,1)}$ is a relatively compact family with respect to the $L^1$-topology. We need to show that each cluster point $\psi $ as $t\to 0+$ is equal to $\varphi _0$. But we already know that $\psi =\varphi _0$ almost everywhere. Hence we deduce $\psi =\varphi _0$ from proposition 5.14.6. As $t\to 0+$, we have $\varphi _ t\xrightarrow {L^1}\varphi _0$. Similarly, as $t\to 1-$, we have $\varphi _ t\xrightarrow {L^1}\varphi _1$.

The two constructions are inverse to each other by construction. $\square$

Proof. Let us begin with a subgeodesic $(\psi _ t)_{t\in (0,1)}$ as in 1. Let $F$ be the convex function as in proposition 21.13.62. We extend $F$ to a function $F\colon N_{\mathbb {R}}\times \mathbb {R}\rightarrow \mathbb {R}$ as follows: For any $n\in N_{\mathbb {R}}$, we define

Then $F$ is a proper closed convex function on $N_{\mathbb {R}}\times \mathbb {R}$. Let $\Psi $ be the Legendre transform of $F$. Then $\Psi $ is a proper closed convex function on $M_{\mathbb {R}}\times \mathbb {R}$ by theorem 21.11.24. By (21.11.23), for any $m\in M_{\mathbb {R}}$ and $s\in \mathbb {R}$, we have

Therefore, the latter half of (21.13.67) follows. Next recall that

as follows from proposition 17.10.7. Therefore,

Conversely, let us begin with a function $\Psi $ as in 2. Let $F$ be the Legendre transform of $\Psi $. We first observe that $F(n,t)=\infty $ for all $n\in N_{\mathbb {R}}$ and $t\not\in [0,1]$. Indeed, the upper bound $\Psi \leq G+(s\lor 0)$ gives

and the last supremum is infinite exactly when $t\notin [0,1]$.

For $t\in [0,1]$, the lower bound $\Psi \geq G_{\varphi _0}\lor (G_{\varphi _1}+s)$ implies

Consequently,

for all $t\in [0,1]$ and $n\in N_{\mathbb {R}}$. On the other hand, the upper bound $\Psi \leq G+(s\lor 0)$ gives

The convexity of $F$ gives one-sided boundary limits

The estimates above give $F_0\leq F_{\varphi _0}$ and $F_1\leq F_{\varphi _1}$, and also show that $F_0,F_1\in \mathcal{P}(N_{\mathbb {R}},P_ D)$. Let $\psi _0,\psi _1\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ be the corresponding potentials given by theorem 21.13.53. Then $\psi _0\leq \varphi _0$ and $\psi _1\leq \varphi _1$. Applying proposition 21.13.62 to the restriction of $F$ to $N_{\mathbb {R}}\times (0,1)$, we get a subgeodesic $(\psi _ t)_{t\in (0,1)}$ from $\psi _0$ to $\psi _1$. Thus $(\psi _ t)_{t\in (0,1)}$ satisfies 1. The two operations are inverse to each other by construction. $\square$

Proof. The equivalence between 1 and 2 follows from a general fact theorem 16.8.32. In the toric case, 2 $\implies $ 1 can also be argued more directly.

Assume that these conditions are satisfied. Let $(\varphi _ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0$ to $\varphi _1$. Since the geodesic is the upper envelope of toric subgeodesics, $\varphi _ t\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ for all $t\in (0,1)$. Let $\Psi '$ be the proper convex function on $M_{\mathbb {R}}\times \mathbb {R}$ defined by corollary 21.13.66. Then $\Psi '$ is the minimum of all $\Psi $ satisfying (21.13.67). We claim that

It suffices to show that the right-hand side is proper, namely, $G_{\varphi _0}\lor G_{\varphi _1}$ is not identically $\infty $. But recall that by proposition 17.10.7, we have $\varphi _0\land \varphi _1\in \mathrm{PSH}(X,\theta )$. Therefore, by proposition 21.13.56,

In particular, (21.13.70) follows.

Now by construction,

for all $t\in (0,1)$. So (21.13.69) follows. $\square$

Proof. First observe that given $g$ as in 2, (21.13.72) indeed induces a geodesic ray in $\mathcal{R}_{\mathrm{tor}}(X,\theta ;\phi )$ thanks to corollary 21.13.68. Therefore, we have a map from 2 to 1.

Conversely, given $\ell \in \mathcal{R}_{\mathrm{tor}}(X,\theta ;\phi )$, it follows from corollary 21.13.68 that $G_{\ell _ t}$ is linear in $t\geq 0$ after restriction to $\operatorname{Int}\{ G_{\phi }<\infty \} $. Let

Note that for $m\in \operatorname{Int}\{ G_{\phi }<\infty \} $, we actually have

Hence, $g'$ is a proper convex function on $M_{\mathbb {R}}$. Define $g=\operatorname{cl}g'$. Then $g$ is a proper closed convex function on $M_{\mathbb {R}}$, and $\overline{\operatorname{Dom}g}=\Delta (\theta ,\phi )$. We have therefore a map from 1 to 2.

It remains to argue that this map is the converse of the preceding map from 2 to 1. The non-trivial point is to verify (21.13.72) holds for the $g$ we just constructed. Fix $t>0$. We need to show that

This holds on $\operatorname{Int}\{ G_{\phi }<\infty \} $ by corollary 21.13.68 and the definition of $g$, but then it holds everywhere thanks to proposition 21.11.18. Our assertion follows. $\square$

Proof. Fix $\tau \in \mathbb {R}$. Then

is clearly toric-invariant. Therefore,

Fix $m\in M_{\mathbb {R}}$. We compute

Our assertions follow. $\square$

Proof. The idea of the proof is that since 19.5.34, 19.5.36 describe how partial Okounkov bodies behave under restriction, and theorem 21.13.39 compares partial Okounkov bodies with Newton bodies, we can deduce the behavior of Newton bodies under restriction as well.

First we note that by our assumption, $L|_ Y$ is a big line bundle. In particular, if we set $r=\dim \sigma $, then $\dim Y=n-r$.

We also note that $\operatorname{Tr}_ Y^\theta (\varphi )$ is $T_{Y,c}$-invariant. Indeed, for any $g\in T_ c$, let $g_ Y$ be the induced automorphism of $Y$. Then $g_ Y^*\operatorname{Tr}_ Y(\varphi )$ is again a trace of $g^*\varphi =\varphi $ along $Y$. Hence $g_ Y^*\operatorname{Tr}_ Y(\varphi )\sim _ P\operatorname{Tr}_ Y(\varphi )$ by proposition 20.2.8. Since $\theta |_ Y$ is $T_{Y,c}$-invariant and the $P$-envelope depends only on the $P$-equivalence class, we get $g_ Y^*\operatorname{Tr}_ Y^\theta (\varphi )=\operatorname{Tr}_ Y^\theta (\varphi )$. As the image of $T_ c$ in $\operatorname{Aut}(Y)$ is $T_{Y,c}$, the assertion follows.

For this purpose, let $\sigma ^0$ be an $n$-dimensional cone of $\Sigma $ containing $\sigma $. The image $\overline{\sigma ^0}$ in $N(\sigma )$ is then an $(n-r)$-dimensional cone of $\mathrm{Star}(\sigma )$. We shall use these cones as the reference cones while defining the partial Okounkov bodies.

We list the rays in $\sigma ^0(1)$ as follows:

where $\rho _1,\ldots ,\rho _ r\in \sigma (1)$ and hence $\rho _{r+1},\ldots ,\rho _ n\not\in \sigma (1)$. In particular, the images

of the latter give a list of $\overline{\sigma ^0}(1)$.

We construct the flag $Y_{\bullet }$ on $X$ using the rays (21.13.77) and the flag $Z_{\bullet }$ on $Y$ using the rays (21.13.78). Note that

where $i=1,\ldots ,n-r$.

Next we compute the Cartier data associated with $\overline{\sigma ^0}$ for the divisor $D_ Y$. By definition, $m_{\overline{\sigma ^0}}\in M(\sigma )$ is the unique element satisfying

for all $j=r+1,\ldots ,n$. Since $m_{\sigma ^0}-m_{\sigma }$ vanishes on $\sigma $, it lies in $M(\sigma )$, and the preceding formula gives

Let $\Phi \colon M\rightarrow \mathbb {Z}^ n$ and $\Psi \colon M(\sigma )\rightarrow \mathbb {Z}^{n-r}$ be defined as

Observe that for $i=r+1,\ldots ,n$, we have

so

for $m\in M(\sigma )$. Therefore, we have a commutative diagram

where $L\colon \mathbb {R}^ n\rightarrow \mathbb {R}^{n-r}$ is the map

By theorem 21.13.39, we have

By corollary 21.13.43, the condition $\nu (\varphi ,Y)=0$ is equivalent to

In particular, the first $r$ valuation coordinates of $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ along the flag $Y_\bullet $ vanish. By 19.5.34 and 19.5.36,

In the third line, we used $\Delta (\theta ,\varphi )\subseteq P_ D$ from theorem 21.13.45, the identity (21.13.7), and the fact that $m_{\sigma ^0}-m_\sigma $ vanishes on $\sigma $. Since $\Psi _{\mathbb {R}}$ is an affine isomorphism, our assertion follows. $\square$

Proof. If $\nu (\varphi ,Y)>0$, then by corollary 21.13.43, $\Delta (\theta ,\varphi )\cap Q_\sigma =\varnothing $. By convention the right-hand side of (21.13.80) is $0$, and the trace is not defined, so the left-hand side is $0$.

Assume now that $\nu (\varphi ,Y)=0$. Take a toric-invariant ample line bundle $H$ on $X$ and a toric-invariant Kähler metric $\omega \in c_1(H)$. For $\epsilon \in \mathbb {Q}_{>0}$, write $Q_{\sigma ,\epsilon }$ for the face corresponding to $\sigma $ for the divisor $D+\epsilon H$, with the same convention on the Cartier datum along $\sigma $. Since $\theta _{\varphi }+\epsilon \omega $ is a toric Kähler current and $\nu (\varphi ,Y)=0$, its trace along $Y$ is defined and has positive volume. Hence theorem 21.13.75 gives

By 21.11.47,

We choose the Cartier datum of $D+\epsilon H$ along $\sigma $ to be the sum of the chosen datum of $D$ and $\epsilon $ times the chosen datum of $H$. With this convention, the face of $P_ D+\epsilon P_ H$ corresponding to $\sigma $ is $Q_\sigma +\epsilon Q_\sigma ^ H$. In particular,

where $Q^ H_{\sigma }$ denotes the corresponding face of the polytope of $H$. Indeed, the defining inequalities along the rays of $\sigma $ are non-negative on both summands, so equality for their sum forces equality on each summand. By example 20.2.17, the left-hand side converges to

with the convention that this limit is $0$ when the trace in the limiting class is not defined. The right-hand side converges to

Hence (21.13.80) follows. $\square$

Proof. Apply corollary 21.13.79 to $\varphi =V_{\theta }$. Together with Nakamaye's theorem [Theorem 5.7, ELMNP09] and theorem 20.2.34, this says that $V(\tau )$ lies in the augmented base locus exactly when the $\dim V(\tau )$-dimensional volume of $Q_{\tau }$ is zero, which is equivalent to $\dim Q_{\tau }<n-\dim \tau $. $\square$

Proof. It suffices to prove the following assertion:

In fact, assuming this, by induction, for each tuple of non-negative integers $C_1,\ldots ,C_ n$, we have

Comparing the volumes, we have

On the other hand, by proposition 21.13.28, the $\mathcal{I}$-goodness of toric singularities example 18.1.56, and proposition 18.1.27, we have

Comparing the coefficients of the above two expressions, we conclude (21.13.85).

It only remains to establish (21.13.86). For this purpose, we fix smooth $T_ c$-invariant Hermitian metrics $h_1$ and $h_2$ on $\mathcal{O}_ X(D_1)$ and $\mathcal{O}_ X(D_2)$ respectively. Correspondingly, take $h_1\otimes h_2$ as the reference Hermitian metric on $\mathcal{O}_ X(D_1+D_2)$. Write

We choose the rational section of $\mathcal O_ X(D_1+D_2)$ to be $s_{D_1}\otimes s_{D_2}$. With this choice and with the reference metric $h_1\otimes h_2$, the normalization (21.13.23) gives

Thus

Take $\varphi _ i\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta _ i)$ with $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for $i=1,2$. Then $F_{\varphi _1}+F_{\varphi _2}$ is a candidate for $F_{\varphi _1+\varphi _2}$; see (21.13.26). Therefore, (21.13.86) follows from the equality of closed gradient images proved in 21.11.47. $\square$