21.12.6 Toric plurisubharmonic functionsE5B138E4CBB0

Proof. We may choose an identification $N\cong \mathbb {Z}^ n$ so that we have an identification $T(\mathbb {C})\cong \mathbb {C}^{*n}$. Then $\operatorname{Trop}$ is identified with the map

1 $\implies $ 2. By the standard convolution regularization of finite convex functions, let $F_ k\in C^{\infty }(\mathbb {R}^ n)\cap \mathrm{Conv}(\mathbb {R}^ n)$ be a decreasing sequence with limit $F$. It follows from a straightforward computation that

So $\operatorname{Trop}^* F_ k$ is plurisubharmonic. It follows from proposition 5.14.2 that $\operatorname{Trop}^* F$ is plurisubharmonic.

2 $\implies $ 1. It follows from 5.14.14 that $F$ is finite. Moreover, by radial regularization, we may find a decreasing sequence $\varphi _ k$ of $(S^1)^ n$-invariant smooth psh functions on $\mathbb {C}^{*n}$ with limit $\operatorname{Trop}^*F$. Write $\varphi _ k=\operatorname{Trop}^* F_ k$ for some function $F_ k\colon \mathbb {R}^ n\rightarrow \mathbb {R}$; it follows from (21.12.8) that $F_ k$ is convex for all $k$. Therefore, $F$ is convex by 21.11.13. $\square$

Proof. The bijection between 2 and 3 is the classical Legendre duality. Given $F$ as in 2, we construct $G=F^*$ and vice versa, see proposition 21.11.38.

The map from 1 to 2 is given as follows: Given $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$, since $\varphi $ is $T_ c$-invariant, we can find $f\colon N_{\mathbb {R}}\rightarrow [-\infty ,\infty )$ such that

We then define $F=f+F_0$. Then $\operatorname{Trop}^* F\in \mathrm{PSH}(T(\mathbb {C}))$. By 21.12.7, $F(n)$ is finite for any $n\in N_{\mathbb {R}}$ and $F$ is convex. Moreover, $F\preceq \operatorname{Supp}_ P$ since $\varphi $ is bounded from above on the compact space $X$ and $F_0\preceq \operatorname{Supp}_ P$.

Conversely, given a map $F\in \mathcal{P}(N_{\mathbb {R}},P)$, then

Since $F\preceq \operatorname{Supp}_ P$ and $F_0\sim \operatorname{Supp}_ P$, the difference $F-F_0$ is bounded from above. Hence the corresponding local psh potentials are locally bounded from above near $X\setminus T(\mathbb {C})$. It follows from theorem 5.14.10 that this function can be extended uniquely to an $\omega $-psh function on $X$. The uniqueness of the extension guarantees its $T_ c$-invariance.

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

Proof. The equivalence between 1 and 2 follows from the definition (21.12.15). The equivalence between 2 and 3 follows from the definition of the Legendre transform. $\square$

Proof. The first equality follows from (21.12.15). The second is obtained by taking the Legendre transform. $\square$

Proof. By proposition 16.3.2, we have $\varphi \land \psi \in \mathrm{PSH}(X,\omega )$. Since $\varphi $ and $\psi $ are $T_ c$-invariant, the set of candidates in the definition of $\varphi \land \psi $ is preserved by the action of $T_ c$. Hence $\varphi \land \psi $ is $T_ c$-invariant as well. So $\varphi \land \psi $ is the biggest element in $\mathrm{PSH}_{\mathrm{tor}}(X,\omega )$ which is dominated by both $\varphi $ and $\psi $. In view of theorem 21.12.13 and proposition 21.12.19, $G_{\varphi \land \psi }$ is the smallest closed proper convex function $G$ on $M_{\mathbb {R}}$ dominating both $G_{\varphi }$ and $G_{\psi }$, which is just $G_{\varphi }\lor G_{\psi }$.

For the statement about $F$, we use proposition 21.11.36:

Since $\varphi \land \psi \not\equiv -\infty $, $F_{\varphi \land \psi }=(G_\varphi \lor G_\psi )^*$ is finite on all of $N_{\mathbb R}$. It follows that $F_\varphi \land F_\psi >-\infty $, and hence is finite on all of $N_{\mathbb R}$ as well. Being finite and convex on $N_{\mathbb R}$, it is continuous, hence closed. Therefore

Proof. Thanks to 21.11.13 and proposition 21.11.8, in both cases, the statement for $F$ is clear. The corresponding statement for $G$ is obtained via proposition 21.11.36. When $I$ is finite, the convex function $\bigvee _ i F_{\varphi _ i}$ is finite on all $N_{\mathbb R}$, hence closed. Therefore no additional closure is needed. $\square$

Proof. We only need to prove (21.12.25). By proposition 21.11.45, we can find a decreasing sequence

with limit $F_\varphi $. By theorem 21.12.13, each $F_ j$ corresponds to a bounded toric potential $\varphi _ j\in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$.

By theorem 13.10.4 and theorem 21.11.55, it suffices to establish (21.12.25) for the $\varphi _ j$'s. We may therefore reduce to the case where $F_{\varphi }$ is smooth. We write $F=F_{\varphi }$ to simplify the notations. The notations $a_ i=\log |z_ i|^2$ will be used, where $i=1,\ldots ,n$.

Next we fix an identification $N=\mathbb {Z}^ n$. Fix a test function $f\in C^{0}_ c(N_{\mathbb {R}})$. We need to show that

Using proposition 21.11.52 and (21.12.8), this reduces to

Expanding the bracket, we get

where $\mathrm{d}\log z_ i=z_ i^{-1}\mathrm{d}z_ i$ and $\mathrm{d}\log \overline{z_ i}=\overline{z_ i}^{-1}\mathrm{d}\overline{z_ i}$ are understood.

Using the apparent symmetry, the expression on the right-hand side becomes

where $\mathfrak {S}_ n$ is the permutation group on $\{ 1,\ldots ,n\} $ and $\operatorname {sgn}(\tau )$ is the sign of $\tau $.

Next, switch to polar coordinates for each $z_ i$: Let $z_ i=r_ i\exp (\mathrm{i}\theta _ i)$ and recall that $r_ i=\exp (a_ i/2)$. Then the left-hand side of (21.12.26) becomes

which is exactly what we have expected. $\square$

Proof. By (16.3.7), we have

It follows from proposition 21.12.20, proposition 21.12.21 and proposition 21.12.23 that $P_{\omega }[\varphi ]\in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$. Moreover, by the same propositions, we have

We spell out the last closure. Set

If $x\in \operatorname{Dom}G_{\varphi }$, then $G_{\varphi }(x)-C\leq G_0(x)$ for $C$ large enough, hence $K(x)=G_0(x)$. If $x\notin \operatorname{Dom}G_{\varphi }$, then $K(x)=\infty $. Since $\Delta (\omega ,\varphi )=\overline{\operatorname{Dom}G_{\varphi }}$ and $G_0$ is continuous on $P$, the lower semicontinuous regularization $\operatorname{cl}K$ is equal to $G_0$ on $\Delta (\omega ,\varphi )$ and equal to $\infty $ outside $\Delta (\omega ,\varphi )$. This is exactly (21.12.30). $\square$

Proof. Put $\Delta =\Delta (\omega ,\varphi )$. By proposition 16.6.18, proposition 21.12.29, we may replace $\varphi $ by $P_{\omega }[\varphi ]$ without changing its multiplier ideal or Newton body. We may therefore assume that $F_{\varphi }-\operatorname{Supp}_{\Delta }$ is bounded on $N_{\mathbb {R}}$. Set

Since $\Delta $ is compact, $H$ is Lipschitz and positively homogeneous.

Observe also that $\mathrm{H}^0(X,L\otimes \mathcal{I}(\varphi ))$ is a $T_ c$-invariant subspace of $\mathrm{H}^0(X,L)$. Since the $T_ c$-weight spaces in $\mathrm{H}^0(X,L)$ are the lines $\mathbb {C}\chi ^ m$, it is the direct sum of suitable $\mathbb {C}\chi ^ m$'s.

If $m\notin P$, then $\chi ^ m$ is not a section of $L$. Suppose henceforth that $m\in M\cap P$. Membership in $\mathrm{H}^0(X,L\otimes \mathcal{I}(\varphi ))$ is equivalent to the finiteness of

Since $\operatorname{Trop}^*F_0+\varphi =\operatorname{Trop}^*F_{\varphi }$, the boundedness of $F_{\varphi }-\operatorname{Supp}_{\Delta }$ and proposition 21.12.24 show that this is equivalent to the finiteness of

Changing variables by $\nu =\nabla G_0(m')$, we obtain the equivalent integral

If $m\in \Delta $, its integrand is at most $1$, proving 1.

To prove 2, extend $\sigma $ to a maximal cone $\tau \in \Sigma $. Its primitive ray generators $u_1,\ldots ,u_ n$ form a basis of $N$, and we may arrange that $u_ i=u_{\rho _ i}$ for $1\leq i\leq p$. If $e_1,\ldots ,e_ n$ is the dual basis of $M$, then $z_ i=\chi ^{e_ i}$ are coordinates on the affine chart $U_{\tau }\cong \mathbb {C}^ n$.

Let $v\in P\cap M$ be the vertex corresponding to $\tau $. The character $\chi ^ v$ is a local frame of $L$ on $U_{\tau }$. Writing

the weight of the singular metric in this frame is $F_{\varphi }(\nu )-\langle v,\nu \rangle $. The local representative of $\chi ^ m$ is $\chi ^{m-v}$, so its squared norm is

On the unit polydisc, $\omega ^ n$ is comparable to Euclidean volume. Set $t_ i=-\log |z_ i|^2$. Integrating out the angular variables and using the boundedness of $F_{\varphi }-\operatorname{Supp}_{\Delta }$, the weighted integral over this polydisc is finite if and only if the following integral is finite:

Restrict this integral to the tube parametrized by $s>1$ and $0<y_ i<1$ for $2\leq i\leq n$, with

The Jacobian is $c_1>0$. Since $H$ is Lipschitz and positively homogeneous and $\sum _{i=1}^ p c_ i=1$, the exponent on this tube is

with a bounded error uniform in $s$ and the $y_ i$'s. The local integral is therefore bounded from below by a positive multiple of

which diverges under (21.12.32). This proves 2. $\square$

Proof. Take $k>0$. Let $F_{0,k}$ denote the Guillemin potential associated with the polytope $kP$. On $T(\mathbb C)$, set

Then $\widetilde\varphi _ k$ is the toric potential, with respect to the reference metric associated with $kP$, corresponding to the singular metric $h^ k\exp (-k\varphi )$ on $L^ k$. It differs from $k\varphi $ by the smooth toric function $\operatorname{Trop}^*(kF_0-F_{0,k})$. Hence

Moreover, the convex function associated with $\widetilde\varphi _ k$ is $kF_\varphi $, so its Legendre transform is

Thus the Newton body of $\widetilde\varphi _ k$ is $k\Delta (\omega ,\varphi )$.

Applying theorem 21.12.31 1 to the line bundle $L^ k$ and the potential $k\varphi $, whose Newton body is $k\Delta (\omega ,\varphi )$, we have

Here $\# $ denotes the number of elements. Therefore,

The left-hand side is just $\operatorname{vol}\Delta (\omega ,\varphi )$, as follows from theorem 19.7.20. So we conclude that

It remains to prove the reverse inequality.

We choose an identification $M\cong \mathbb {Z}^ n$. In particular, there is a natural distance $\mathrm{d}$ on $M_{\mathbb {R}}\cong \mathbb {R}^ n$. Given a convex body $K$ and $\epsilon >0$, we let

This is again a convex body, since it can be realized as the Minkowski sum of $K$ with a closed ball of radius $\epsilon $.

Applying theorem 21.12.31 2 to the polytope $kP$ and the potential $\widetilde\varphi _ k$, we can find $C>0$ uniform in $k$ such that for each $k\in \mathbb {Z}_{>0}$,

Indeed, if a point $m\in kP\cap M$ is at distance at least $C$ from $k\Delta (\omega ,\varphi )$, the separating hyperplane theorem gives a unit vector $v\in N_{\mathbb R}$ such that

Since the fan $\Sigma $ is finite and covers $N_{\mathbb R}$, there is a constant $\Lambda >0$ with the following property: every unit vector $v\in N_{\mathbb R}$ can be written in the form

where the rays $\rho _ i$ are contained in a single cone of $\Sigma $. Put

Since the support function is positively homogeneous, we have

Choosing $C>\Lambda $ makes (21.12.32) applicable, uniformly in $k$.

Fix $\epsilon >0$. Then for large enough $k$, we have

It follows that

Applying theorem 19.7.20 again, we conclude that

Letting $\epsilon \to 0+$ and applying theorem 19.7.5, we find

Our assertion follows. $\square$

Proof. This follows immediately from proposition 21.12.19. $\square$

Proof. 1 $\iff $ 3. By proposition 21.12.24,

where in the first equality, we use the fact that $\omega _{\varphi }^ n$ does not charge pluripolar sets proposition 13.11.4 4.

Therefore, 1 and 3 are equivalent.

2 $\iff $ 3. This follows from proposition 21.11.34. $\square$

Proof. For $\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$, write

Step 1. We assume that $\varphi $ is bounded.

Put $h=F_{\varphi }-F_0$, $\varphi _ t=t\varphi $ and

for $t\in [0,1]$. Then $F_ t=F_{\varphi _ t}$ and $G_ t\coloneqq F_ t^*=G_{\varphi _ t}$. Since $\varphi $ is bounded, $h$ is bounded and continuous; in particular, $F_ t\in \mathcal E^\infty (N_{\mathbb R},P)$ for every $t\in [0,1]$.

We claim that

Let $s>0$ and $t,t+s\in [0,1]$. At a point $m\in \operatorname{Int}P$ where both $G_ t$ and $G_{t+s}$ are differentiable, we have

Therefore,

Integrating the last inequality and using 21.11.57 gives

Letting $s\to 0+$ and using theorem 21.11.55 for $F_{t+s}\to F_ t$, we obtain the right derivative in (21.12.41). The left derivative follows in the same way. Hence (21.12.41) holds.

By proposition 21.12.24, the right-hand side of (21.12.41) is

On the other hand, 16.5.57 gives

Thus $E_{\omega }(\varphi _ t)-\mathcal C(\varphi _ t)$ is constant in $t$. It vanishes at $t=0$, hence (21.12.40) follows by taking $t=1$.

Step 2. We handle the general case.

Both sides of (21.12.40) change by the same additive constant when we replace $\varphi $ by $\varphi +A$, namely by

We may therefore assume that $\varphi \leq 0$. For $C>0$, put

Then $\varphi _ C\in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )\cap L^{\infty }(X)$ and $\varphi _ C\searrow \varphi $ as $C\to \infty $. By proposition 21.12.23, $G_{\varphi _ C}\nearrow G_{\varphi }$. Applying Step 1 to $\varphi _ C$, we get

Letting $C\to \infty $, the left-hand side converges to $E_{\omega }(\varphi )$ by proposition 16.5.20. On the right-hand side, since $\varphi _ C\leq 0$, we have $G_{\varphi _ C}\geq G_0$, and the monotone convergence theorem applied to $G_{\varphi _ C}-G_0\nearrow G_{\varphi }-G_0$ gives (21.12.40). $\square$

Proof. The equivalence between 1 and 3 follows from proposition 21.12.39. The equivalence between 2 and 3 is the definition of $\mathcal{E}^1(N_{\mathbb {R}},P)$, since $F_{\varphi }^*=G_{\varphi }$. $\square$

Proof. This follows from proposition 21.12.39, proposition 21.12.21 and definition 17.10.41. $\square$