21.13.31 Partial Okounkov bodies962BB5F87752

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. $\square$