5.5. The one-dimensional case [02US]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

5.5. The one-dimensional case

We now study in detail the one-dimensional case. Besides being a concrete example of the relationship between functions, models, metrics, and measures, it is also a crucial step in the proof that a toric metric is semipositive if and only if the corresponding function is concave. Of this equivalence, up to now we have proved only one implication and the reverse implication will be proved in the next section.

The only complete one-dimensional toric variety over a field is the projective line. Since, by Proposition 5.53 and Proposition 2.35 we know the effect of taking finite extensions of the field $K$ , we can use the following result to reduce any model of $\mathbb{P}^{1}$ to a simpler form.

Definition 5.54.

Let $K$ be a field complete with respect to an absolute value associated to a nontrivial discrete valuation. Let $K^{\circ}$ be the ring of integers. Let $X$ be a proper curve over $K$ . A semi-stable model of $X$ is a flat proper regular scheme ${\mathcal{X}}$ of finite type over $\operatorname{Spec}(K^{\circ})$ with an isomorphism $X\to{\mathcal{X}}_{\eta}$ , such that the special fibre ${\mathcal{X}}_{o}$ is a reduced normal crossing divisor.

Proposition 5.55.

Let $K$ be a field complete with respect to an absolute value associated to a nontrivial discrete valuation. Let $K^{\circ}$ be the ring of integers. Let ${\mathcal{X}}$ be a proper model over $K^{\circ}$ of $\mathbb{P}^{1}_{K}$ . Then there exists a finite extension $H$ of $K$ with ring of integers ${H}^{\circ}$ , a semi-stable model ${\mathcal{X}}^{\prime}$ of $\mathbb{P}^{1}_{H}$ , and a proper morphism of models $\mathcal{X}^{\prime}\to\mathcal{X}\times\operatorname{Spec}({H}^{\circ})$ .

Proof.

This follows, for instance, from [Liu06, Corollary 2.8]. ∎

Consider the toric variety $X_{\Sigma}\simeq\mathbb{P}^{1}$ . We can choose an isomorphism $N\simeq\mathbb{Z}$ and $N_{\mathbb{R}}\simeq\mathbb{R}$ . Then $\Sigma=\{\mathbb{R}_{-},\{0\},\mathbb{R}_{+}\}$ . Let $0$ denote the invariant point of $\mathbb{P}^{1}_{K}$ corresponding to the cone $\mathbb{R}_{+}$ and $\infty$ the invariant point corresponding to the cone $\mathbb{R}_{-}$ . Let $t$ denote the absolute coordinate of $\mathbb{P}^{1}$ given by the monomial $\chi^{1}$ .

Let $\mathcal{X}$ be a semi-stable model of $\mathbb{P}^{1}_{K}$ . By extending scalars if necessary, we may suppose that all the components of the special fibre are defined over $k=K^{\circ}/K^{\circ\circ}$ and contain a rational point. Since the special fibre $\mathcal{X}_{o}$ is connected and of genus zero, we deduce that the special fibre is a tree of rational curves, each isomorphic to $\mathbb{P}^{1}_{k}$ . Let $D_{0}$ and $D_{\infty}$ denote the horizontal divisors corresponding to the point $0$ and $\infty$ of $\mathbb{P}^{1}_{K}$ . Then, there is a chain of rational curves that links the divisor $D_{0}$ with $D_{\infty}$ that is contained in the special fibre. We will denote the irreducible components of the special fibre that form this chain by $E_{0},\dots,E_{k}$ , in such a way that the component $E_{0}$ meets $D_{0}$ , the component $E_{k}$ meets $D_{\infty}$ and, for $0<i<k$ , the component $E_{i}$ meets only $E_{i-1}$ and $E_{i+1}$ . The other components of $\mathcal{X}_{o}$ will be grouped in branches, each branch has its root in one of the components $E_{i}$ . We will denote by $F_{i,j}$ , $j\in\Theta_{i}$ the components that belong to a branch with root in $E_{i}$ . We are not giving any particular order to the sets $\Theta_{i}$ .

We denote by $E\cdot F$ the intersection product of two $1$ -cycles of ${\mathcal{X}}$ . Since the special fibre is reduced, we have

\operatorname{div}(\varpi)=\sum_{i=0}^{k}\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

Again by the assumption of semi-stability, the intersection product of two different components of $\mathcal{X}_{o}$ is either $1$ , if they meet, or zero, if they do not meet. Since the intersection product of $\operatorname{div}(\varpi)$ with any component of $\mathcal{X}_{o}$ is zero, we deduce that, if $E$ is any component of $\mathcal{X}_{o}$ , the self-intersection product $E\cdot E$ is equal to minus the number of components that meet $E$ . In particular, all components $F_{i,j}$ that are terminal, are $(-1)$ -curves. By Castelnuovo Criterion, we can successively blow-down all the components $F_{i,j}$ to obtain a new semi-stable model of $\mathbb{P}^{1}_{K}$ whose special fibre consist of a chain of rational curves. For reasons that will become apparent later we denote this model as ${\mathcal{X}}_{\mathbb{S}}$ .

Lemma 5.56.

If we view $t$ as a rational function on $\mathcal{X}$ , then there is an integer $a$ such that

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}(a-i)\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

Proof.

It is clear that

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}a_{i}E_{i}+\sum_{j\in\Theta_{i}}a_{i,j}F_{i,j}

for certain coefficients $a_{i}$ and $a_{i,j}$ that we want to determine as much as possible.

If a component $E$ of $\mathcal{X}_{0}$ , with coefficient $a$ , does not meet $D_{0}$ nor $D_{\infty}$ , but meets $r\geq 1$ other components, and the coefficients of $r-1$ of these components are equal to $a$ , while the coefficient of the remaining component is $b$ , we obtain that

0=\operatorname{div}(t)\cdot E=aE\cdot E+a(r-1)+b=-ra+a(r-1)+b=b-a

Thus $b=a$ . Starting with the components $F_{i,j}$ that are terminal, we deduce that, for all $i$ and $j\in\Theta_{i}$ , $a_{i}=a_{i,j}$ . Therefore,

\operatorname{div}(t)=D_{0}-D_{\infty}+\sum_{i=0}^{k}a_{i}\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right).

In particular, the lemma is proved for $k=0$ . Assume now that $k>0$ .

It only remains to show that $a_{i}=a_{0}-i$ , that we prove by induction. For $i=1$ , we compute

0=\operatorname{div}(t)\cdot E_{0}=D_{0}\cdot E_{0}+a_{0}E_{0}\cdot E_{0}+a_{0}\sum_{j\in\Theta_{0}}F_{0,j}\cdot E_{0}+a_{1}E_{1}\cdot E_{0}=1-a_{0}+a_{1}.

Thus $a_{1}=a_{0}-1$ . For $1<i\leq k$ , by induction hypothesis, $a_{i-1}=a_{i-2}-1$ . Then

0=\operatorname{div}(t)\cdot E_{i-1}=a_{i-2}-2a_{i-1}+a_{i}=1-a_{i-1}+a_{i}.

Thus $a_{i}=a_{i-1}-1=a_{0}-i$ , proving the lemma. ∎

The determination of $\operatorname{div}(t)$ allows us to give a partial description of the map ${\operatorname{red}}\colon X_{\Sigma}^{{\text{\rm an}}}\to\mathcal{X}_{o}$ . For us, the most interesting points of $\mathcal{X}_{o}$ are the points $q_{0}:=D_{0}\cap E_{0}$ , $q_{i}:=E_{i-1}\cap E_{i}$ , $i=1,\dots,k$ , $q_{k+1}:=E_{k}\cap D_{\infty}$ and the generic points of the components $E_{i}$ that we denote $\eta_{i}$ , $i=0,\dots,k$ .

Lemma 5.57.

Let $p\in X_{\Sigma}^{{\text{\rm an}}}$ . Then

{\operatorname{red}}(p)=\begin{cases}q_{0},&\text{ if }|t(p)|<|\varpi|^{a}\\ q_{i},\ i=1\dots,k,&\text{ if }|\varpi|^{a-i+1}<|t(p)|<|\varpi|^{a-i}\\ q_{k+1},&\text{ if }|\varpi|^{a-k}<|t(p)|\\ \eta_{i},\ i=0\dots,k,&\text{ if }|t(p)|=|\varpi|^{a-i}\text{ and }p\in\operatorname{im}(\theta_{\Sigma}).\end{cases}

Proof.

Let $1\leq i\leq k$ . The rational function $x:=t\varpi^{-a+i}$ has a zero of order one along the component $E_{i-1}$ and the support of its divisor does not contain the component $E_{i}$ . On the other hand, the rational function $y:=t^{-1}\varpi^{a-i+1}$ has a zero of order one along the component $E_{i}$ and the support of its divisor does not contain the component $E_{i-1}$ . Thus $\{x,y\}$ is a system of parameters in a neighbourhood of $q_{i}$ . We denote

A=K^{\circ}[t\varpi^{-a+i},t^{-1}\varpi^{a-i+1}]\simeq K^{\circ}[x,y]/(xy-\varpi).

The local ring at the point $q_{i}$ is $A_{(x,y)}$ . Let $p$ be a point such that $|\varpi|^{a-i+1}<|t(p)|<|\varpi|^{a-i}$ . Therefore, for $f\in A$ we have $|f(p)|\leq 1$ . Moreover, if $f\in(x,y)$ , then $|f(p)|<1$ . Since the ideal $(x,y)$ is maximal, we deduce that, for $f\in A$ , the condition $|f(p)|<1$ is equivalent to the condition $f\in(x,y)$ . This implies that ${\operatorname{red}}(p)=q_{i}$ . A similar argument works for $q_{0}$ and $q_{k+1}$ .

Assume now that $p\in\operatorname{im}(\theta_{\Sigma})$ and that $|t(p)|=|\varpi|^{a-i}$ . If $i\not=0$ we consider again the ring $A$ , but in this case $|x(p)|=|t(p)\varpi^{-a+i}|=1$ . Let $I=\{f\in A\mid|f(p)|<1\}$ . It is clear that $(y,\varpi)\subset I$ . For $f=\sum_{m\in\mathbb{Z}}\beta_{m}t^{m}\in A$ , since $p\in\operatorname{im}(\theta_{\Sigma})$ , we have

|f(p)|=\sup_{m}(|\beta_{m}||t(p)|^{m}).

This implies that $I\subset(y,\varpi)$ . Hence $I$ is the ideal that defines the component $E_{i}$ and this is equivalent to ${\operatorname{red}}(p)=\eta_{i}$ . The case $i=0$ is analogous. ∎

The image by ${\operatorname{red}}$ of the remaining points of $X_{\Sigma}^{{\text{\rm an}}}$ is not characterized only by the value of $|t(p)|$ . Using a proof similar to that of the lemma, one can show that, if $|t(p)|=|\varpi|^{a-i}$ then ${\operatorname{red}}(p)$ belongs either to $E_{i}$ or to any of the components $F_{i,j}$ , $j\in\Theta_{i}$ .

We denote by $\xi_{i}$ (resp. $\xi_{i,j}$ ) the point of $X_{\Sigma}^{{\text{\rm an}}}$ corresponding to the component $E_{i}$ (resp. $F_{i,j}$ ). That is, ${\operatorname{red}}(\xi_{i})=\eta_{i}$ and ${\operatorname{red}}(\xi_{i,j})=\eta_{i,j}$ , where $\eta_{i,j}$ is the generic point of $F_{i,j}$ (see (2.15) and (2.14)).

Lemma 5.58.

Let $0\leq i\leq k$ . Then, for every $j\in\Theta_{i}$ ,

{\operatorname{val}}_{K}(\xi_{i})={\operatorname{val}}_{K}(\xi_{i,j})=a-i,

where $a$ is the integer of Lemma 5.56.

Proof.

We consider the rational function $\varpi^{-a+i}t$ . Since the support of $\operatorname{div}(\varpi^{-a+i}t)$ does not contain the component $E_{i}$ nor any of the components $F_{i,j}$ , we have that

|\varpi^{-a+i}t(\xi_{i})|=|\varpi^{-a+i}t(\xi_{i,j})|=1.

Since $t=\chi^{1}$ , we deduce, using equation (5.4), that

{\operatorname{val}}_{K}(\xi_{i})=\frac{-\log|\chi^{1}(x_{i})|}{\lambda_{K}}=\frac{-\log|\varpi^{a-i}|}{-\log|\varpi|}=a-i.

∎

Let now $\Psi$ be a virtual support function on $\Sigma$ . It can be written as

\Psi(u)=\begin{cases}m_{\infty}u,&\text{ if }u\leq 0,\\ m_{0}u,&\text{ if }u\geq 0.\end{cases}

for some $m_{0},m_{\infty}\in\mathbb{Z}$ . Then, $L=\mathcal{O}(D_{\Psi})\simeq\mathcal{O}(m_{\infty}-m_{0})$ , and $\operatorname{div}(s_{\Psi})=-m_{0}[0]+m_{\infty}[\infty]$ . Let $\mathcal{L}$ be a model over $\mathcal{X}$ of $L^{\otimes e}$ . If we consider $s_{\Psi}^{\otimes e}$ as a rational section of $\mathcal{L}$ , then

(5.59)

\operatorname{div}(s_{\Psi}^{\otimes e})=-em_{0}D_{0}+em_{\infty}D_{\infty}+\sum_{i=0}^{k}\left(\alpha_{i}E_{i}+\sum_{j\in\Theta_{i}}\alpha_{i,j}F_{i,j}\right)

for certain coefficients $\alpha_{i}$ and $\alpha_{i,j}$ . Let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ determined by this model.

Lemma 5.60.

The function $\psi_{\|\cdot\|}$ is given by

\psi_{\|\cdot\|}(u)=\begin{cases}m_{0}u-m_{0}a-\frac{\alpha_{0}}{e},&\text{ if }u\geq a,\\ \frac{(\alpha_{i+1}-\alpha_{i})u-(\alpha_{i+1}-\alpha_{i})(a-i)-\alpha_{i}}{e},&\text{ if }a-i\geq u\geq a-i-1,\\ m_{\infty}u-m_{\infty}(a-k)-\frac{\alpha_{k}}{e},&\text{ if }a-k\geq u.\end{cases}

In other words, if $\Pi$ is the polyhedral complex in $N_{\mathbb{R}}$ given by the intervals

(-\infty,a-k],\quad[a-i,a-i+1],\ i=1,\dots,k,\quad[a,\infty),

then $\psi_{\|\cdot\|}$ is the rational piecewise affine function on $\Pi$ characterized by the conditions

(1)

$\operatorname{rec}(\psi_{\|\cdot\|})=\Psi$ ,
(2)

the value of $\psi_{\|\cdot\|}$ at the point $a-i$ is $-\alpha_{i}/e$ .

Proof.

Let $p\in\operatorname{im}\theta_{\Sigma}$ be such that ${\operatorname{val}}_{K}(p)>a$ , hence $|t(p)|<|\varpi|^{a}$ . By Lemma 5.57, this implies that ${\operatorname{red}}(p)=q_{0}$ . In a neighbourhood of $q_{0}$ , the divisor of the rational section $s_{\Psi}^{\otimes e}t^{em_{0}}\varpi^{-\alpha_{0}-em_{0}a}$ is zero, and so

\|s_{\Psi}^{\otimes e}(p)t^{em_{0}}(p)\varpi^{-\alpha_{0}-em_{0}a}\|=1.

Set $u={\operatorname{val}}(p)$ . Then,

	$\displaystyle\psi_{\\|\cdot\\|}(u)$	$\displaystyle=\frac{\log\\|s_{\Psi}^{\otimes e}(p)\\|}{e\lambda_{K}}$
		$\displaystyle=\frac{-em_{0}\log\|t(p)\|+(\alpha_{0}+em_{0}a)\log\|\varpi\|}{-e\log\|\varpi\|}$
		$\displaystyle=m_{0}(u-a)-\frac{\alpha_{0}}{e}.$

The other cases are proved in a similar way. ∎

Since $\operatorname{rec}(\Pi)=\Sigma$ , this polyhedral complex defines a toric model $\mathcal{X}_{\Pi}$ of $X_{\Sigma}$ .

Proposition 5.61.

The identity map of $X_{\Sigma}$ extend to an isomorphism of models $\mathcal{X}_{\mathbb{S}}\to\mathcal{X}_{\Pi}$ .

Proof.

The special fibre of $\mathcal{X}_{\Pi}$ is a chain of rational curves $E_{i}$ , $i=0,\dots,k$ , corresponding to the points $a-i$ . The monomial $\chi^{1}$ is a section of the trivial line bundle and corresponds to the function $\psi(u)=-u$ . Using Proposition 4.84 we obtain that

\operatorname{div}(\chi^{1})=D_{0}-D_{\infty}+\sum_{i=0}^{k}(a-i)E_{i},

where $D_{0}$ and $D_{\infty}$ are again the horizontal divisors determined by the points $0$ and $\infty$ .

Since the vertices of the polyhedral complex $\Pi$ are integral, by equation (4.87), we deduce that $\operatorname{div}(\varpi)$ is reduced.

Then the result follows from [Lic68, Corollary 1.13] using an explicit description of the local rings at the points of the special fibre as in the proof of Lemma 5.57. ∎

From Proposition 5.61 we obtain a proper morphism $\pi\colon\mathcal{X}\to\mathcal{X}_{\Pi}$ . On $\mathcal{X}$ we had a line bundle $\mathcal{L}$ and $s_{\Psi}^{\otimes e}$ was considered as a rational section of this line bundle. Let $D=\operatorname{div}(s_{\Psi}^{\otimes e})$ be the divisor given by equation (5.59). We denote

(5.62)

D_{\mathbb{S}}=\pi_{\ast}D=-em_{0}D_{0}+em_{\infty}D_{\infty}+\sum_{i=0}^{k}\alpha_{i}E_{i}.

By Proposition 4.84 and Lemma 5.60 we see that $D_{\mathbb{S}}=D_{e\psi_{h}}$ . Thus $\mathcal{O}(D_{\mathbb{S}})$ is a toric model of $L^{\otimes e}$ . Recall that $\|\cdot\|$ denoted the metric associated to the model $\mathcal{O}(D)$ . Let $\|\cdot\|_{\mathbb{S}}$ be the toric metric obtained from $\|\cdot\|$ as in Proposition 5.51. By this proposition and equation (5.62), the metric $\|\cdot\|_{\mathbb{S}}$ agrees with the metric defined by the model $\mathcal{O}(D_{\mathbb{S}})$ . Thus, we have identified a toric model that corresponds to the metric $\|\cdot\|_{\mathbb{S}}$ . This allows us to compute directly the associated measure.

Proposition 5.63.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $L\simeq\mathcal{O}(D_{\Psi})$ be a toric line bundle and let $\|\cdot\|$ be an algebraic metric defined by a semi-stable model and let $\|\cdot\|_{\mathbb{S}}$ be the associated toric metric. Then

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\left(c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}\right).

Proof.

Since the special fibre is reduced, by equation (2.29)

c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}=\frac{1}{e}\sum_{i=0}^{k}\left(\deg_{\mathcal{L}}E_{i}\delta_{\xi_{i}}+\sum_{j\in\Theta_{i}}\deg_{\mathcal{L}}F_{i,j}\delta_{\xi_{i,j}}\right).

Denote this measure temporarily by $\mu$ . Then

	$\displaystyle(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\mu$	$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(\deg_{\mathcal{L}}E_{i}+\sum_{j\in\Theta_{i}}\deg_{\mathcal{L}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(D\cdot E_{i}+\sum_{j\in\Theta_{i}}D\cdot F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\sum_{l=0}^{k}\left(\alpha_{l}E_{l}+\sum_{s\in\Theta_{l}}\alpha_{l,s}F_{l,s}\right)\cdot\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}\left(\alpha_{i-1}E_{i-1}+\alpha_{i}E_{i}+\alpha_{i+1}E_{i+1}\right)\cdot\left(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j}\right)\delta_{\xi_{i}}$
		$\displaystyle=\frac{1}{e}\sum_{i=0}^{k}(\alpha_{i-1}-2\alpha_{i}+\alpha_{i+1})\delta_{\xi_{i}}.$

In the previous computation, we have used that, since $E_{l}\cdot\operatorname{div}(\varpi)=F_{l,s}\cdot\operatorname{div}(\varpi)=0$ , then

	$\displaystyle F_{l,s}\cdot(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j})$	$\displaystyle=0,\text{ for all }i,j,l,s,$
	$\displaystyle E_{l}\cdot(E_{i}+\sum_{j\in\Theta_{i}}F_{i,j})$	$\displaystyle=\begin{cases}0,&\text{ if }l\not=i-1,i,i+1,\\ 1,&\text{ if }l=i-1,i+1,\\ -2,&\text{ if }l=i.\end{cases}$

An analogous computation shows that

(5.64)

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=\frac{1}{e}\sum_{i=0}^{k}(\alpha_{i-1}-2\alpha_{i}+\alpha_{i+1})\delta_{\xi_{i}}.

∎

Using Proposition 5.55 we can extend the above result to the case when the model is not semi-stable.

Corollary 5.65.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $L\simeq\mathcal{O}(D_{\Psi})$ be a toric line bundle, $\|\cdot\|$ an algebraic metric, and $\|\cdot\|_{\mathbb{S}}$ the associated toric metric. Then

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\left(c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}\right).

Proof.

Let $(\mathcal{X},\mathcal{L})$ be a model of $(X_{\Sigma},L^{\otimes e})$ that realizes the algebraic metric $\|\cdot\|$ . For short, denote $\mu=c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ and $\mu_{\mathbb{S}}=c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ . By Proposition 5.55 there is a non-Archimedean field $H$ over $K$ and a semi-stable model ${\mathcal{X}}^{\prime}$ of $X_{\Sigma,H}$ . We may further assume that all the components of the special fibre of ${\mathcal{X}}^{\prime}$ are defined over $H^{\circ}/H^{\circ\circ}$ . Let $(L^{\prime},\|\cdot\|^{\prime})$ be the metrized line bundle obtained by base change to $H$ . Then $(\|\cdot\|^{\prime})_{\mathbb{S}}$ is obtained from $\|\cdot\|_{\mathbb{S}}$ by base change. We denote by $\pi\colon X^{{\text{\rm an}}}_{\Sigma,H}\to X^{{\text{\rm an}}}_{\Sigma,K}$ the map of analytic spaces. Be will denote by $\mu^{\prime}$ , $\mu^{\prime}_{\mathbb{S}}$ , $\theta_{\Sigma}^{\prime}$ and $\rho_{\Sigma}^{\prime}$ the corresponding objects for $X_{\Sigma,H}$ . Then, by Proposition 2.35 and Proposition 5.53,

\mu_{\mathbb{S}}=\pi_{\ast}\mu^{\prime}_{\mathbb{S}}=\pi_{\ast}(\theta^{\prime}_{\Sigma})_{\ast}(\rho^{\prime}_{\Sigma})_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\pi_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\mu.

∎

We can now relate semipositivity of the metric with concavity of the associated function on the one-dimensional case.

Corollary 5.66.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $(L,s)$ be a toric line bundle with a toric section and let $\|\cdot\|$ be a semipositive algebraic metric. Then $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric algebraic metric and $\psi_{\|\cdot\|}$ is concave.

Proof.

Since $\|\cdot\|$ is semipositive, $c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ is a positive measure. By Corollary 5.65, $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ is a positive measure. Hence $\|\cdot\|_{\mathbb{S}}$ is a semipositive toric metric. By equation (5.64) and Lemma 5.60, the positivity of $c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$ implies that the function $\psi_{\|\cdot\|}=\psi_{\|\cdot\|_{\mathbb{S}}}$ is concave. ∎

	$\displaystyle\psi_{\\|\cdot\\|}(u)$	$\displaystyle=\frac{\log\\|s_{\Psi}^{\otimes e}(p)\\|}{e\lambda_{K}}$
		$\displaystyle=\frac{-em_{0}\log\|t(p)\|+(\alpha_{0}+em_{0}a)\log\|\varpi\|}{-e\log\|\varpi\|}$
		$\displaystyle=m_{0}(u-a)-\frac{\alpha_{0}}{e}.$