8. Variations on Fubini-Study metrics [02XZ]

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8. Variations on Fubini-Study metrics

8.1. Height of toric projective curves

In this section, we study the Arakelov invariants of curves which are the image of an equivariant map into a projective space. In the Archimedean case we equip the projective space with the Fubini-Study metric, while in the non-Archimedean case we equip it with the canonical metric. For each of these curves, the metric, measure and toric local height can be computed in terms of the roots of a univariate polynomial associated to the relevant equivariant map.

Let $K$ be either $\mathbb{R},\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. On $\mathbb{P}^{r}$ , we consider the universal line bundle ${\mathcal{O}}(1)$ equipped with the Fubini-Study metric in the Archimedean case, and with the canonical metric in the non-Archimedean case. We write ${\overline{{\mathcal{O}}(1)}}$ for the resulting metrized line bundle. We also consider the toric section $s_{\infty}$ of ${\mathcal{O}}(1)$ whose Weil divisor is the hyperplane at infinity. Next result gives the induced function $\psi$ for a subvariety of $\mathbb{P}^{r}$ which is the image of an equivariant map.

Proposition 8.1.

Let $H\colon N\to\mathbb{Z}^{r}$ be an injective map such that $H(N)$ is a saturated sublattice of $\mathbb{Z}^{r}$ , $p\in\mathbb{P}^{r}_{0}(K)$ . Consider the map $\varphi_{H,p}:\mathbb{T}\to\mathbb{P}^{r}$ , and set ${\overline{L}}=\varphi_{H,p}^{*}{\overline{{\mathcal{O}}(1)}}$ and $s=\varphi_{H,p}^{*}s_{\infty}$ . Let $\psi_{{\overline{L}},s}\colon N_{\mathbb{R}}\to\mathbb{R}$ be the associated concave function, $m_{i}=e_{i}^{\vee}\circ H\in M$ , $i=1,\dots,r$ , and $p=(1:p_{1}:\dots:p_{r})$ with $p_{i}\in K^{\times}$ . Then, for $u\in N_{\mathbb{R}}$ ,

\psi_{{\overline{L}},s}(u)=\begin{cases}-\frac{1}{2}\log(1+\sum_{i=1}^{r}|p_{i}|^{2}\operatorname{e}^{-2\langle m_{i},u\rangle}),&\text{in the Archimedean case},\\ \min_{1\leq i\leq r}\{0,\langle m_{i},u\rangle+{\operatorname{val}}_{K}(p_{i})\}&\text{in the non-Archimedean case}.\end{cases}

Proof.

In the Archimedean case, the expression for the concave function $\psi$ follows from that for $\mathbb{P}^{r}_{K}$ (Example 5.18(2)) and Proposition 5.24. The non-Archimedean case follows from Example 5.26. ∎

Let $Y\subset\mathbb{P}^{r}$ be the closure of the image of the map $\varphi_{H,p}$ . In this situation, the roof function seems difficult to calculate. Hence it is difficult to use it directly to compute the toric local height (see Example 3.57). A more promising approach is to apply the formula of Corollary 6.17. Writing $\psi=\psi_{{\overline{L}},s}$ this formula reads

(8.2)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\lambda_{K}(n+1)!\int_{N_{\mathbb{R}}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{M}(\psi).

To make this formula more explicit in the Archimedean case, we choose a basis of $N$ , hence coordinate systems in $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$ and we write

g=(g_{1},\dots,g_{n}):=\nabla\psi\colon N_{\mathbb{R}}\longrightarrow\Delta,

where $\Delta=\operatorname{stab}(\psi)$ is the associated polytope. Then, from Proposition 3.94 and Example 3.106(1), we derive

	$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)$	$\displaystyle=(n+1)!\int_{N_{\mathbb{R}}}(\langle\nabla\psi(u),u\rangle-\psi(u))\,(-1)^{n}\det(\operatorname{Hess}(\psi))\,\,\text{\rm d}\operatorname{vol}_{N}$
(8.3)			$\displaystyle=(n+1)!\int_{N_{\mathbb{R}}}(\left<g(u),u\right>-\psi(u))\,(-1)^{n}\,\text{\rm d}g_{1}\land\dots\land\,\text{\rm d}g_{n}.$

When $K$ is not Archimedean, we have ${\mathcal{M}}_{M}(\psi)=\sum_{v\in\Pi^{0}(\psi)}\delta_{v}$ and, for $v\in\Pi(\psi)^{0}$ ,

\psi^{\vee}\circ\partial\psi(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}\langle x,v\rangle\,\text{\rm d}\operatorname{vol}_{M}-\psi(v),

see Proposition 3.95 and Example 3.106(2). Thus, if now we denote by $g\colon N_{\mathbb{R}}\to M_{\mathbb{R}}$ the function that sends a point $u$ to the barycentre of $\partial\psi(u)$ , then

(8.4)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\lambda_{K}(n+1)!\sum_{v\in\Pi^{0}(\psi)}(\left<g(v),v\right>-\psi(v)).

In the case of curves, the integral of equation (8.2), can be transformed into another integral that will prove useful for explicit computations. We introduce a notation for derivatives of concave functions of one variable. Let $f\colon\mathbb{R}\to\mathbb{R}$ be a concave function. We write

(8.5)

f^{\prime}(u)=\frac{1}{2}(D_{+}f(u)+D_{-}f(u)),

where $D_{+}f$ and $D_{-}f$ denote the right and left derivatives of $f$ respectively, that exist always. Then $f^{\prime}$ is monotone and is continuous almost everywhere (with respect to the Lebesgue measure). The associated distribution agrees with the derivative of $f$ in the sense of distributions. This implies that, if $\{f_{n}\}_{n}$ is a sequence of concave functions converging uniformly to $f$ on compacts, then $\{f^{\prime}_{n}\}$ converges to $f^{\prime}$ almost everywhere.

Lemma 8.6.

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be a concave function whose stability set is an interval $[a,b]$ . Then

2\int_{\mathbb{R}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{\mathbb{Z}}(\psi)=(b-a)(\psi^{\vee}(a)+\psi^{\vee}(b))+\int_{\mathbb{R}}(\psi^{\prime}(u)-a)(b-\psi^{\prime}(u))\,\text{\rm d}u.

Proof.

By the properties of the Monge-Ampère measure (Proposition 3.93) and of the Legendre-Fenchel dual (Proposition 3.18) the left-hand side is continuous with respect to uniform convergence of functions. Again by Proposition 3.18 and the discussion before the lemma, the right-hand side is also continuous with respect to uniform convergence of functions. Therefore it is enough to treat the case when $\psi$ is smooth and strictly concave. Then

2\int_{\mathbb{R}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{\mathbb{Z}}(\psi)=2\int_{\mathbb{R}}(\psi(u)-u\psi^{\prime}(u))\psi^{\prime\prime}(u)\,\text{\rm d}u.

Consider the function

	$\displaystyle\gamma(u)$	$\displaystyle=(\psi^{\prime}(u)-\frac{a+b}{2})\psi(u)-u\frac{(\psi^{\prime})^{2}}{2}+u\frac{ab}{2}$
		$\displaystyle=-(\psi^{\prime}(u)-\frac{a+b}{2})\psi^{\vee}(\psi^{\prime}(u))-\frac{u}{2}(\psi^{\prime}(u)-a)(b-\psi^{\prime}(u)).$

Then

\lim_{u\to\infty}\gamma(u)=\frac{b-a}{2}\psi^{\vee}(a),\qquad\lim_{u\to-\infty}\gamma(u)=\frac{a-b}{2}\psi^{\vee}(b),

and

\,\text{\rm d}\gamma=(\psi-u\psi^{\prime})\psi^{\prime\prime}\,\text{\rm d}u-\frac{1}{2}(\psi^{\prime}-a)(b-\psi^{\prime})\,\text{\rm d}u,

from which the result follows. ∎

With the notation in Proposition 8.1, assume that $N=\mathbb{Z}$ . The elements $m_{j}\in N^{\vee}$ can be identified with integer numbers and the hypothesis that the image of $H$ is a saturated sublattice is equivalent to $\gcd(m_{1},\dots,m_{r})=1$ . Moreover, by reordering the variables of $\mathbb{P}^{r}$ and multiplying the expression of $\varphi_{H,p}$ by a monomial (which does not change the equivariant map), we may assume that $0\leq m_{1}\leq\dots\leq m_{r}$ . We make the further hypothesis that $0<m_{1}<\dots<m_{r}$ . With these conditions, we next obtain explicit expressions for the concave function $\psi$ and the associated measure and toric local height in terms of the roots of a univariate polynomial. We consider the absolute value $|\cdot|$ of the algebraic closure ${\overline{K}}$ extending the absolute value of $K$ . For $\xi\in{\overline{K}}^{\times}$ , we set ${\operatorname{val}}_{{\overline{K}}}(\xi)=-\frac{\log|\xi|}{\lambda_{K}}$ .

Theorem 8.7.

Let $0<m_{1}<\dots<m_{r}$ be integer numbers with $\gcd(m_{1},\dots,m_{r})=1$ , and $p_{1},\dots,p_{r}\in K^{\times}$ . Let $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ be the map given by $\varphi(t)=(1:p_{1}t^{m_{1}}:\dots:p_{r}t^{m_{r}})$ and let $Y$ be the closure of the image of $\varphi$ . Consider the polynomial $q\in K[z]$ defined as

\displaystyle q=\begin{cases}1+\sum_{j=1}^{r}|p_{j}|^{2}z^{m_{j}},&\text{ in the Archimedean case},\\ 1+\sum_{j=1}^{r}p_{j}z^{m_{j}},&\text{ in the non-Archimedean case}.\end{cases}

Let $\{\xi_{i}\}_{i}\subset{\overline{K}}^{\times}$ be the set of roots of $q$ and, for each $i$ , let $\ell_{i}\in\mathbb{N}$ be the multiplicity of $\xi_{i}$ . Let ${\overline{L}}$ and $s$ be as in Proposition 8.1. Then, in the Archimedean case,

(1)

$\displaystyle\psi_{{\overline{L}},s}(u)=-\log|p_{r}|-\frac{1}{2}\sum_{i}\ell_{i}\log|\operatorname{e}^{-2u}-\xi_{i}|$ for $u\in\mathbb{R}$ ,
(2)

$\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=-2\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}\,\,\text{\rm d}u$ ,
(3)

$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{\xi_{i}+\xi_{j}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j}))$ , where $\log$ is the principal determination of the logarithm.

While in the non-Archimedean case,

(4)

$\displaystyle\psi_{{\overline{L}},s}(u)={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{u,{\operatorname{val}}_{{\overline{K}}}(\xi_{i})\}$ for $u\in\mathbb{R}$ ,
(5)

$\displaystyle{\mathcal{M}}_{\mathbb{Z}}(\psi_{{\overline{L}},s})=\sum_{i}\ell_{i}\delta_{{\operatorname{val}}_{{\overline{K}}}(\xi_{i})}$ ,
(6)

$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\sum_{i<j}\ell_{i}\ell_{j}\log(\max\{1,|\xi_{i}|/|\xi_{j}|\})$ .

Remark 8.8.

The real roots of the polynomial $q$ are all negative, this allows the use of the principal determination of the logarithm in (3). Introducing the argument $\theta_{i}\in]-\pi,\pi[$ of $-\xi_{i}$ , the last sum in (3) can be rewritten

\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{(|\xi_{i}|^{2}-|\xi_{j}|^{2})\log|\xi_{i}/\xi_{j}|+2|\xi_{i}||\xi_{j}|(\theta_{i}-\theta_{j})\sin(\theta_{i}-\theta_{j})}{|\xi_{i}|^{2}+|\xi_{j}|^{2}-2|\xi_{i}||\xi_{j}|\cos(\theta_{i}-\theta_{j})}

showing that it is real.

Proof.

Write $\psi=\psi_{{\overline{L}},s}$ for short. First we consider the Archimedean case. We have that $q=|p_{r}|^{2}\prod_{i}(z-\xi_{i})^{\ell_{i}}$ . By Proposition 8.1,

\psi(u)=-\frac{1}{2}\log(q(\operatorname{e}^{-2u}))=-\log|p_{r}|-\frac{1}{2}\sum_{i}\ell_{i}\log|\operatorname{e}^{-2u}-\xi_{i}|,

which proves (1). Hence,

\psi^{\prime}(u)=\sum_{i}\ell_{i}\frac{1}{1-\xi_{i}\operatorname{e}^{2u}}\quad\text{and}\quad\psi^{\prime\prime}(u)=\sum_{i}2\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}.

The Monge-Ampère measure of $\psi$ is given by $-\psi^{\prime\prime}\,\text{\rm d}u$ , and so the above proves (2). To prove (3) we apply Lemma 8.6. We have that $\operatorname{stab}(\psi)=[0,m_{r}]$ , $\psi^{\vee}(0)=0$ , and $\psi^{\vee}(m_{r})=\log|p_{r}|$ . Thus,

(8.9)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u.

We have $\displaystyle m_{r}-\psi^{\prime}(u)=\sum_{i}\ell_{i}\bigg(1-\frac{1}{1-\xi_{i}\operatorname{e}^{2u}}\bigg)=-\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{1-\xi_{i}\operatorname{e}^{2u}}$ . Hence,

(m_{r}-\psi^{\prime}(u))\psi^{\prime}(u)=-\bigg(\sum_{i}\ell_{i}\frac{\xi_{i}\operatorname{e}^{2u}}{1-\xi_{i}\operatorname{e}^{2u}}\bigg)\bigg(\sum_{j}\ell_{j}\frac{1}{1-\xi_{j}\operatorname{e}^{2u}}\bigg)\\ =-\sum_{i}\ell_{i}^{2}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}-\sum_{i\neq j}\ell_{i}\ell_{j}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})(1-\xi_{j}\operatorname{e}^{2u})}.

Moreover $\displaystyle\int_{-\infty}^{\infty}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})^{2}}\,\text{\rm d}u=\bigg[\frac{1}{2(1-\xi_{i}\operatorname{e}^{2u})}\bigg]^{\infty}_{-\infty}=-\frac{1}{2}$ and

\int_{-\infty}^{\infty}\frac{\xi_{i}\operatorname{e}^{2u}}{(1-\xi_{i}\operatorname{e}^{2u})(1-\xi_{j}\operatorname{e}^{2u})}\,\text{\rm d}u=\bigg[\frac{\xi_{i}}{2(\xi_{i}-\xi_{j})}(\log(1-\xi_{j}\operatorname{e}^{2u}))-\log(1-\xi_{i}\operatorname{e}^{2u})\bigg]^{\infty}_{-\infty}\\ =\frac{\xi_{i}}{2(\xi_{i}-\xi_{j})}(\log(-\xi_{i})-\log(-\xi_{j})),

for the principal determination of $\log$ . These calculations together with equation (8.9) imply that

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i\neq j}\ell_{i}\ell_{j}\frac{\xi_{i}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j}))\\ =m_{r}\log|p_{r}|+\frac{1}{2}\sum_{i}\ell_{i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{i}\ell_{j}\frac{\xi_{i}+\xi_{j}}{\xi_{i}-\xi_{j}}(\log(-\xi_{i})-\log(-\xi_{j})),

which proves (3).

Next we consider the non-Archimedean case. Let $U\subset{\overline{K}}^{\times}$ be a sufficiently small open subset and $\zeta\in U$ . For short, write $v_{i}={\operatorname{val}}_{{\overline{K}}}(\xi_{i})$ . By Proposition 8.1, the genericity of $\zeta$ , and the condition $m_{i}\not=m_{j}$ for $i\not=j$ , imply

\psi({\operatorname{val}}_{{\overline{K}}}(\zeta))=\min_{i}\{0,m_{i}{\operatorname{val}}_{{\overline{K}}}(\zeta)+{\operatorname{val}}_{K}(p_{i})\}={\operatorname{val}}_{{\overline{K}}}(q(\zeta)).

By the factorization of $q$ ,

{\operatorname{val}}_{{\overline{K}}}(q(\zeta))={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}{\operatorname{val}}_{{\overline{K}}}(\zeta-\xi_{i})={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{{\operatorname{val}}_{{\overline{K}}}(\zeta),v_{i}\}.

The image of ${\operatorname{val}}_{{\overline{K}}}\colon{\overline{K}}^{\times}\to\mathbb{R}$ is a dense subset. We deduce that, $u\in\mathbb{R}$ ,

\psi(u)={\operatorname{val}}_{K}(p_{r})+\sum_{i}\ell_{i}\min\{u,{\operatorname{val}}_{K}(\xi_{i})\},

which proves (4). The gradient of this function is, for $u\in\mathbb{R}$ ,

\partial\psi(u)=\begin{cases}\Big[\sum_{j:v_{j}>v_{i}}\ell_{j},\sum_{j:v_{j}\geq v_{i}}\ell_{j}\Big]&\text{ if }u=v_{i}\text{ for some }i,\\ \sum_{j:v_{j}>x}\ell_{j}&\text{ otherwise.}\end{cases}

Hence, the associated Monge-Ampère measure is $\sum_{i}\ell_{i}\delta_{v_{i}},$ which proves (5). The derivative of $\psi$ in the sense of (8.5) is, for $u\in\mathbb{R}$ ,

\psi^{\prime}(u)=\begin{cases}\sum_{j:v_{j}>v_{i}}\ell_{j}+\frac{1}{2}\sum_{j:v_{j}=v_{i}}\ell_{j}&\text{ if }u=v_{i}\text{ for some }i,\\ \sum_{j:v_{j}>x}\ell_{j}&\text{ otherwise.}\end{cases}

Moreover, $\operatorname{stab}(\psi)=[0,m_{r}]$ , $\psi^{\vee}(0)=0$ and $\psi^{\vee}(m_{r})=-{\operatorname{val}}_{K}(p_{r})$ . By Lemma 8.6

(8.10)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=-m_{r}\lambda_{K}{\operatorname{val}}_{K}(p_{r})+\lambda_{K}\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u.

If we write

f_{i}(u)=\begin{cases}0,&\text{ if }x\leq v_{i}\\ \ell_{i},&\text{ if }x>v_{i},\end{cases}

then, we have that, almost everywhere $\psi^{\prime}(u)=\sum_{i}\ell_{i}-f_{i}(u)$ and $m_{r}-\psi^{\prime}(u)=\sum_{i}f_{i}$ . Therefore

(8.11)

\int_{-\infty}^{\infty}(m_{r}-\psi^{\prime})\psi^{\prime}\,\text{\rm d}u=\sum_{i,j}\int_{-\infty}^{\infty}f_{i}(\ell_{j}-f_{j})\,\text{\rm d}u=\sum_{i,j}\ell_{i}\ell_{j}\max\{0,v_{j}-v_{i}\}.

Thus, joining together (8.10), (8.11) and the relation $\log(|\zeta|)=-\lambda_{K}{\operatorname{val}}_{K}(\zeta)$ we deduce

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=m_{r}\log|p_{r}|+\sum_{i,j}\ell_{i}\ell_{j}\max\{0,\log(|\xi_{i}|/|\xi_{j}|)\},

finishing the proof of the theorem. ∎

We now treat the global case.

Corollary 8.12.

Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be a global field. Let $0<m_{1}<\dots<m_{r}$ be integer numbers with $\gcd(m_{1},\dots,m_{r})=1$ , and $p_{1},\dots,p_{r}\in\mathbb{K}^{\times}$ . Let $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ be the map given by $\varphi(t)=(1:p_{1}t^{m_{1}}:\dots:p_{r}t^{m_{r}})$ , $Y$ the closure of the image of $\varphi$ , and ${\overline{L}}=\varphi^{*}{\overline{{\mathcal{O}}(1)}}$ , where ${\overline{{\mathcal{O}}(1)}}$ is equipped with the Fubini-Study metric for the Archimedean places and with the canonical metric for the non-Archimedean places. For $v\in\mathfrak{M}_{K}$ , set

\displaystyle q_{v}=\begin{cases}1+\sum_{j=1}^{r}|p_{j}|_{v}^{2}z^{m_{j}},&\text{ if }v\text{ is Archimedean},\\ 1+\sum_{j=1}^{r}p_{j}z^{m_{j}},&\text{ if }v\text{ is not Archimedean}.\end{cases}

Let $\{\xi_{v,i}\}\subset{\overline{\mathbb{K}}}^{\times}$ be the set of roots of $q_{v}$ and, for each $i$ , let $\ell_{v,i}\in\mathbb{N}$ denote the multiplicity of $\xi_{v,i}$ . Then

\operatorname{h}_{{\overline{L}}}(Y)=\sum_{v|\infty}n_{v}\bigg(\frac{1}{2}\sum_{i}\ell_{v,i}^{2}+\frac{1}{2}\sum_{i<j}\ell_{v,i}\ell_{v,j}\frac{\xi_{v,i}+\xi_{v,j}}{\xi_{v,i}-\xi_{v,j}}(\log(-\xi_{v,i})-\log(-\xi_{v,j}))\bigg)\\ +\sum_{v\nmid\infty}n_{v}\bigg(\sum_{i<j}\ell_{v,i}\ell_{v,j}\log(\max\{1,|\xi_{v,i}|_{v}/|\xi_{v,j}|_{v}\})\bigg).

Proof.

This follows readily from Proposition 6.35, Theorem 8.7, and the product formula. ∎

Corollary 8.13.

Let $C_{r}\subset\mathbb{P}^{r}_{\mathbb{Q}}$ be the Veronese curve of degree $r$ and ${\overline{{\mathcal{O}}(1)}}$ the universal line bundle on $\mathbb{P}^{r}_{\mathbb{Q}}$ equipped with the Fubini-Study metric at the Archimedean place and with the canonical metric at the non-Archimedean ones. Then

(8.14)

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=\frac{r}{2}+\pi\sum_{j=1}^{\lfloor r/2\rfloor}\bigg(1-\frac{2\,j}{r+1}\bigg)\,\cot\bigg(\frac{\pi\,j}{r+1}\bigg)\in\frac{r}{2}+\pi\,{\overline{\mathbb{Q}}}.

Proof.

The curve $C_{r}$ coincides with the closure of the image of the map $\varphi\colon\mathbb{T}\to\mathbb{P}^{r}$ given by $\varphi(t)=(1:t:t^{2}:\dots:t^{r})$ . With the notation in Corollary 8.12, this map correspond to $m_{i}=i$ and $p_{i}=1$ , for $i=1,\dots,r$ . Then $q_{v}=\sum_{j=0}^{r}z^{j}$ for all $v\in\mathfrak{M}_{\mathbb{Q}}$ . Consider the primitive $(r+1)$ -th root of unity $\omega=\operatorname{e}^{\frac{2\pi i}{r+1}}$ . The polynomial $q_{v}$ is separable and its set of roots is $\{\omega^{l}\}_{l=1,\dots,r}$ . Since $|\omega^{l}|_{v}=1$ for all $v$ , Corollary 8.12 implies that

(8.15)

\operatorname{h}_{{\overline{L}}}(Y)=\frac{r}{2}+\frac{1}{2}\sum_{l<j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}(\log(-\omega^{l})-\log(-\omega^{j}))\\ =\frac{r}{2}+\frac{1}{2}\sum_{l\neq j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}\log(-\omega^{l}).

We have that

\sum_{j=1}^{r}\frac{\omega^{j}+1}{\omega^{j}-1}=\sum_{j=1}^{r}\frac{\omega^{j}}{\omega^{j}-1}+\sum_{j=1}^{r}\frac{1}{\omega^{j}-1}=\sum_{j=1}^{r}\frac{1}{1-\omega^{-j}}+\sum_{j=1}^{r}\frac{1}{\omega^{j}-1}=0.

This implies that, for $l=1,\dots,r$ ,

\sum_{1\leq j\leq r,j\neq l}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}=-\frac{\omega^{l}+1}{\omega^{l}-1}=i\cot\Big(\frac{\pi l}{r+1}\Big)

Hence,

\frac{1}{2}\sum_{l\neq j}\frac{\omega^{l}+\omega^{j}}{\omega^{l}-\omega^{j}}\log(-\omega^{l})=-\frac{i}{2}\sum_{l=1}^{r}\cot\Big(\frac{\pi l}{r+1}\Big)\log(-\omega^{l})\\ =\pi\sum_{l=1}^{\lfloor r/2\rfloor}\cot\Big(\frac{\pi l}{r+1}\Big)\Big(1-\frac{2l}{r+1}\Big),

since $\cot(\frac{\pi(r+1-l)}{r+1})\log(-\omega^{r+1-l})=\cot(\frac{\pi l}{r+1})\log(-\omega^{l})$ for $l=1,\dots,\lfloor r/2\rfloor$ and $\log(-\omega^{\frac{r+1}{2}})=0$ whenever $r$ is odd. The statement follows from this calculations together with (8.15). ∎

Here follow some special values:

$r$	1	2	3	5	7
$\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})$	$\displaystyle\frac{1}{2}$	$\displaystyle 1+\frac{1}{3\,\sqrt{3}}\,\pi$	$\displaystyle\frac{3}{2}+\frac{1}{2}\,\pi$	$\displaystyle\frac{5}{2}+\frac{7}{3\,\sqrt{3}}\,\pi$	$\displaystyle\frac{7}{2}+(1+\sqrt{2})\,\pi$

Corollary 8.16.

With the notation of Corollary 8.13, $\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=r\log r+O(r)$ for $r\to\infty$ .

Proof.

We have that $\displaystyle\pi\cot(\pi x)=\frac{1}{x}+O(1)$ for $x\to 0$ . Hence,

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=\sum_{j=1}^{\lfloor r/2\rfloor}\bigg(1-\frac{2\,j}{r+1}\bigg)\,\frac{j}{r+1}+O(r)=r\bigg(\sum_{j=1}^{\lfloor r/2\rfloor}\frac{1}{j}\bigg)+O(r)=r\log r+O(r).

∎

By the theorem of algebraic successive minima [Zha95a],

\mu^{\operatorname{ess}}(C_{r})\leq\frac{\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})}{\deg_{{{\mathcal{O}}(1)}}(C_{r})}\leq 2\mu^{\operatorname{ess}}(C_{r})

The essential minimum of $C_{r}$ is $\mu^{\operatorname{ess}}(C_{r})=\frac{1}{2}\log(r+1)$ [Som05]. Hence, the quotient $\frac{\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})}{\deg_{{{\mathcal{O}}(1)}}(C_{r})}$ is asymptotically closer to the upper bound than to the lower bound.

8.2. Height of toric bundles

Let $n\geq 0$ and write $\mathbb{P}^{n}=\mathbb{P}^{n}_{\mathbb{Q}}$ for short. Given $a_{r}\geq\dots\geq a_{0}\geq 1$ , consider the bundle $\mathbb{P}(E)\rightarrow\mathbb{P}^{n}$ of hyperplanes of the vector bundle

E={\mathcal{O}}(a_{0})\oplus{\mathcal{O}}(a_{1})\oplus\dots\oplus{\mathcal{O}}(a_{r})\longrightarrow\mathbb{P}^{n},

where ${\mathcal{O}}(a_{j})$ denotes the $a_{j}$ -th power of the universal line bundle of $\mathbb{P}^{n}$ . Equivalently, $\mathbb{P}(E)$ can be defined as the bundle of lines of the dual vector bundle $E^{\vee}$ . The fibre of the map $\pi\colon\mathbb{P}(E)\to\mathbb{P}^{n}$ over each point $p\in\mathbb{P}^{n}({\overline{\mathbb{Q}}})$ is a projective space of dimension $r$ . This bundle is a smooth toric variety over $\mathbb{Q}$ of dimension $n+r$ , see [Oda88, pp. 58-59], [Ful93, p. 42]. The particular case $n=r=1$ corresponds to Hirzebruch surfaces: for $b\geq 0$ , we have $\mathbb{F}_{b}=\mathbb{P}({\mathcal{O}}(0)\oplus{\mathcal{O}}(b))\simeq\mathbb{P}({\mathcal{O}}(a_{0})\oplus{\mathcal{O}}(a_{0}+b))$ for any $a_{0}\geq 1$ .

The tautological line bundle of $\mathbb{P}(E)$ , denoted ${\mathcal{O}}_{\mathbb{P}(E)}(-1)$ , is defined as a subbundle of $\pi^{*}E^{\vee}$ . Its fibre over a point of $\mathbb{P}(E)$ is the inverse image under $\pi$ of the line in $E^{\vee}$ which is dual to the hyperplane of $E$ defining the given point. The universal line bundle ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ of $\mathbb{P}(E)$ is defined as the dual of the tautological one. Since ${\mathcal{O}}(a_{j})$ , $j=0,\dots,r$ , is ample, the universal line bundle is also ample [Har66]. This is the line bundle corresponding to the Cartier divisor $a_{0}D_{0}+D_{1}$ , where $D_{0}$ denotes the inverse image in $\mathbb{P}(E)$ of the hyperplane at infinity of $\mathbb{P}^{n}$ and $D_{1}=\mathbb{P}(0\oplus{\mathcal{O}}(a_{1})\oplus\dots\oplus{\mathcal{O}}(a_{r}))$ . Observe that, although $\mathbb{P}(E)$ is isomorphic to the bundle associated to the family of integers $a_{i}+c$ for any $c\in\mathbb{N}$ , this is not the case for the associated universal line bundle, that depends on the choice of $c$ .

Following Example 4.3, we regard $\mathbb{P}^{n}$ as a toric variety over $\mathbb{Q}$ equipped with the action of the split torus $\mathbb{G}_{m}^{n}$ . Let $s$ be the toric section of ${\mathcal{O}}(1)$ which corresponds to the hyperplane at infinity $H_{0}$ and let $s_{j}=s^{\otimes-a_{j}}$ , which is a section of ${\mathcal{O}}(-a_{j})$ . Let $U=\mathbb{P}^{n}\setminus H_{0}$ . The restriction of $\mathbb{P}(E)$ to $U$ is isomorphic to $U\times\mathbb{P}^{r}$ through the map $\varphi$ defined, for $p\in U$ and $q\in\mathbb{P}^{r}$ , as

(p,q)\longmapsto(p,q_{0}s_{0}(p)\oplus\dots\oplus q_{r}s_{r}(p)).

The torus $\mathbb{T}:=\mathbb{G}_{m}^{n+r}$ can then be included as an open subvariety of $\mathbb{P}(E)$ through the map $\varphi$ composed with the standard inclusion of $\mathbb{G}_{m}^{n+r}$ into $U\times\mathbb{P}^{r}$ . The action of $\mathbb{T}$ on itself by translation extends to an action of the torus on the whole of $\mathbb{P}(E)$ . Hence $\mathbb{P}(E)$ is a toric variety over $\mathbb{Q}$ . With this action the divisor $a_{0}D_{0}+D_{1}$ is a $\mathbb{T}$ -Cartier divisor.

By abuse of notation, we also denote $E^{\vee}$ the total space associated to the vector bundle $E^{\vee}$ . The map $\mathbb{G}_{m}^{n+r}\to E^{\vee}$ defined as

(z,w)\longmapsto((1:z),(s_{0}(1:z)\oplus w_{1}s_{1}(1:z)\oplus\dots\oplus w_{r}s_{r}(1:z)))

induces a no-where vanishing section of the tautological line bundle of $\mathbb{P}(E)$ over the open subset $\mathbb{T}$ . Its inverse, denoted $s$ , is a no-where vanishing section of ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ over $\mathbb{T}$ . In particular, this section induces a structure of toric line bundle on ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ . The divisor of the section $s$ is precisely the $\mathbb{T}$ -Cartier divisor $a_{0}D_{0}+D_{1}$ considered above.

We now introduce an adelic toric metric on ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ . For $v=\infty$ , we consider the complex vector bundle $E(\mathbb{C})$ that can be naturally metrized by the direct sum of the Fubiny-Study metric on each factor ${\mathcal{O}}(a_{j})(\mathbb{C})$ . By duality, this gives a metric on $E^{\vee}(\mathbb{C})$ , which induces by restriction a metric on the tautological line bundle. Applying duality once more time, we obtain a smooth metric, denoted $\|\cdot\|_{\infty}$ , on $O_{\mathbb{P}(E)(\mathbb{C})}(1)$ . For $v\in M_{\mathbb{Q}}\setminus\{\infty\}$ , we equip $O_{\mathbb{P}(E)}(1)$ with the canonical metric (Proposition-Definition 5.20). We write $\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}=({\mathcal{O}}_{\mathbb{P}(E)}(1),(\|\cdot\|_{v})_{v\in M_{\mathbb{Q}}})$ for the obtained adelic metrized toric line bundle.

We have made a choice of splitting of $\mathbb{T}$ and therefore a choice of an identification $N=\mathbb{Z}^{n+r}$ . Thus we obtain a system of coordinates in the real vector space associated to the toric variety $\mathbb{P}(E)$ , $N_{\mathbb{R}}=\mathbb{R}^{n+r}=\mathbb{R}^{n}\times\mathbb{R}^{r}$ . Since the metric considered at each non-Archimedean place is the canonical one, the only nontrivial contribution to the global height will come from the Archimedean place. The restriction to the principal open subset $\mathbb{P}(E)_{0}(\mathbb{C})\simeq(\mathbb{C}^{\times})^{n+r}=(\mathbb{C}^{\times})^{n}\times(\mathbb{C}^{\times})^{r}$ of the valuation map is expressed, in these coordinates, as the map ${\operatorname{val}}\colon(\mathbb{C}^{\times})^{n+r}\to N_{\mathbb{R}}$ defined by

{\operatorname{val}}(z,w)=(-\log|z_{1}|,\dots,-\log|z_{n}|,-\log|w_{1}|,\dots,-\log|w_{r}|).

Let $\theta_{0}$ be the natural inclusion of real variety $\mathbb{P}(E)_{0}(\mathbb{R}_{\geq 0})\simeq(\mathbb{R}_{>0})^{n+r}$ in $\mathbb{P}(E)_{0}(\mathbb{C})$ and let ${\operatorname{\mathbf{e}}}_{\mathbb{C}}$ be the homeomorphism $N_{\mathbb{R}}\to\mathbb{P}(E)_{0}(\mathbb{R}_{\geq 0})$ , both defined in §5.1. In these coordinates, the composition map $\theta_{0}\circ{\operatorname{\mathbf{e}}}_{\mathbb{C}}\colon N_{\mathbb{R}}\to\mathbb{P}(E)_{0}(\mathbb{C})$ is given by $(u,v)\mapsto(\operatorname{e}^{-u_{1}},\dots,\operatorname{e}^{-u_{n}},\operatorname{e}^{-v_{1}},\dots,\operatorname{e}^{-v_{r}})$ .

Write $\psi_{\infty}\colon N_{\mathbb{R}}\to\mathbb{R}$ for the function corresponding to the metric $\|\cdot\|_{\infty}$ and the toric section $s$ defined above.

Lemma 8.17.

The function $\psi_{\infty}$ is defined, for $u\in\mathbb{R}^{n}$ and $v\in\mathbb{R}^{r}$ , as

\psi_{\infty}(u,v)=-\frac{1}{2}\log\left(\sum_{j=0}^{r}{\operatorname{e}}^{-2v_{j}}\left(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}}\right)^{a_{j}}\right),

with the convention $u_{0}=v_{0}=0$ . It is a strictly concave function.

Proof.

The metric on $E^{\vee}$ is given, for $p\in\mathbb{P}^{n}(\mathbb{C})$ and $q_{0},\dots,q_{r}\in\mathbb{C}$ , by

||q_{0}s_{0}(p)\oplus\dots\oplus q_{r}s_{r}(p)||_{\infty}^{2}=|q_{0}|^{2}||s_{0}(p)||^{2}+\dots+|q_{r}|^{2}||s_{r}(p)||^{2},

where $||s_{j}(p)||$ is the norm of $s_{j}(p)$ with respect to the Fubini-Study metric on ${\mathcal{O}}(-a_{j})^{{\text{\rm an}}}$ . By Example 2.2,

||s_{j}(p)||^{2}=\bigg(\frac{|p_{0}|^{2}}{|p_{0}|^{2}+\dots+|p_{n}|^{2}}\bigg)^{-a_{j}}.

Let $s^{\otimes-1}$ be the monomial section of the tautological line bundle defined by $s$ . Then

(8.18)

\|s^{\otimes-1}\circ\theta_{0}\circ{\operatorname{\mathbf{e}}}_{\mathbb{C}}(u,v)\|^{2}=\sum_{j=0}^{r}{\operatorname{e}}^{-2v_{j}}\left(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}}\right)^{a_{j}}.

By Proposition 5.19(2), $\psi_{\infty}$ is $-1/2$ times the logarithm of the above expression.

For the last statement, observe that the functions ${\operatorname{e}}^{-2v_{j}}(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}})^{a_{j}}$ are log-strictly convex, because $-1/2$ times their logarithm is the function associated to the Fubini-Study metric on ${\mathcal{O}}(a_{j})^{{\text{\rm an}}}$ , which is a strictly concave function. Their sum is also log-strictly convex [BV04, §3.5.2]. Hence, $\psi_{\infty}$ is strictly concave. ∎

Corollary 8.19.

The metric $\|\cdot\|_{\infty}$ is a semipositive smooth toric metric.

The following result summarizes the toric structure of $\mathbb{P}(E)$ and of ${\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}$ .

Proposition 8.20.

(1)

Let $e_{i}$ , $1\leq i\leq n$ , and $f_{j}$ , $1\leq j\leq r$ , be the $i$ -th and $(n+j)$ -th vectors of the standard basis of $N=\mathbb{Z}^{n+r}$ . Set $f_{0}=-f_{1}-\cdots-f_{r}$ and $e_{0}=a_{0}f_{0}+\cdots+a_{r}f_{r}-e_{1}-\cdots-e_{n}$ . The fan $\Sigma$ corresponding to $\mathbb{P}(E)$ is the fan in $N_{\mathbb{R}}$ whose maximal cones are the convex hull of the rays generated by the vectors

$e_{0},\cdots,e_{k-1},e_{k+1},\cdots,e_{n},f_{0},\cdots,f_{\ell-1},f_{\ell+1},\cdots,f_{r}$

for $0\leq k\leq n,0\leq\ell\leq r$ . This is a complete regular fan.
(2)

The support function $\Psi\colon N_{\mathbb{R}}\to{\mathbb{R}}$ corresponding to the universal line bundle ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ is defined, for $u\in\mathbb{R}^{n}$ and $v\in\mathbb{R}^{r}$ , as

$\Psi(u,v)=\mathop{\min_{0\leq k\leq n}}_{0\leq\ell\leq r}(a_{\ell}u_{k}+v_{\ell}),$

where, for short, we have set $u_{0}=v_{0}=0$ .

(3)

The polytope $\Delta$ in $M_{\mathbb{R}}=\mathbb{R}^{n}\times\mathbb{R}^{r}$ associated to $(\Sigma,\Psi)$ is

\Big\{(x,y)|y_{1},\dots,y_{r}\geq 0,\ \sum_{\ell=1}^{r}y_{\ell}\leq 1,\ x_{1},\dots,x_{n}\geq 0,\ \sum_{k=1}^{n}x_{k}\leq L(y)\Big\}

with $L(y)=a_{0}+\sum_{\ell=1}^{r}(a_{\ell}-a_{0})y_{\ell}$ . Using the convention $y_{0}=1-\sum_{\ell=1}^{r}y_{\ell}$ and $x_{0}=L(y)-\sum_{k=1}^{n}x_{k}$ , then $L(y)=\sum_{\ell=0}^{r}a_{\ell}y_{\ell}$ and the polytope $\Delta$ can be written as

\Big\{(x,y)|y_{0},\dots,y_{r}\geq 0,\ x_{0},\dots,x_{n}\geq 0\Big\}.

(4)

The Legendre-Fenchel dual of $\psi_{\infty}$ is the concave function $\psi_{\infty}^{\vee}\colon\Delta\to\mathbb{R}$ defined, for $(x,y)\in\Delta$ , as

\psi^{\vee}_{\infty}(x,y)=-\frac{1}{2}\left(\varepsilon_{r}(y_{1},\dots,y_{r})+L(y)\cdot\varepsilon_{n}\left(\frac{x_{1}}{L(y)},\dots,\frac{x_{n}}{L(y)}\right)\right),

where, for $k\geq 0$ , $\varepsilon_{k}$ is the function defined in (3.54). For $v\neq\infty$ , the concave function $\psi^{\vee}_{v}$ is the indicator function of $\Delta$ .

Proof.

By Corollary 5.17, we have $\Psi=\operatorname{rec}(\psi_{\infty})$ . By equation (3.49), we have $\operatorname{rec}(\psi_{\infty})=\lim_{\lambda\to\infty}\lambda^{-1}\psi_{\infty}(\lambda(u,v))$ . Statement (2) follows readily from this and from the expression for $\psi_{\infty}$ in Lemma 8.17.

The function $\Psi$ is strictly concave on $\Sigma$ , because ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ is an ample line bundle. Hence $\Sigma=\Pi(\Psi)$ and this is the fan described in statement (1).

Let $(e_{1}^{\vee},\dots,e_{n}^{\vee},f_{1}^{\vee},\dots,f_{r}^{\vee})$ be the dual basis of $M$ induced by the basis of $N$ . By Proposition 3.64 and statement (2), we have

\Delta=\operatorname{conv}\bigg(0,(a_{0}e^{\vee}_{k})_{1\leq k\leq n},(f^{\vee}_{\ell})_{1\leq\ell\leq r},(a_{\ell}e^{\vee}_{k}+f^{\vee}_{\ell})_{\stackrel{{\scriptstyle 1\leq k\leq n}}{{\scriptscriptstyle 1\leq\ell\leq r}}}\bigg).

Statement (3) follows readily from this.

For the first part of statement (4), it suffices to compute the Legendre-Fenchel dual of $\psi_{\infty}$ at a point $(x,y)$ in the interior of the polytope. Lemma 8.17 shows that $\psi_{\infty}$ is strictly concave. Hence, by Theorem 3.52(3), $\nabla\psi_{\infty}$ is a homeomorphism between $N_{\mathbb{R}}$ and $\Delta^{\circ}$ . Thus, there exist a unique $(u,v)\in N_{\mathbb{R}}$ such that, for $i=1,\dots,n$ and $j=1,\dots,r$ ,

x_{i}=\frac{\partial\psi_{\infty}}{\partial u_{i}}(u,v),\quad y_{j}=\frac{\partial\psi_{\infty}}{\partial v_{j}}(u,v).

We use the conventions $x_{0}=L(y)-\sum_{i=1}^{n}x_{i}$ , $y_{0}=1-\sum_{j=1}^{r}y_{j}$ , and $u_{0}=v_{0}=0$ as before, and also $\eta=\sum_{i=0}^{n}\operatorname{e}^{-2u_{i}}$ and $\psi=\psi_{\infty}$ , so that $-2\psi=\log\big(\sum_{j=0}^{r}\operatorname{e}^{-2v_{j}}\eta^{a_{j}}\big)$ . Computing the gradient of $\psi$ , we obtain, for $i=1,\dots,n$ and $j=1,\dots,r$ ,

x_{i}\operatorname{e}^{-2\psi}=\Big(\sum_{j=0}^{r}a_{j}\eta^{a_{j}-1}\operatorname{e}^{-2v_{j}}\Big)\operatorname{e}^{-2u_{i}},\quad y_{j}\operatorname{e}^{-2\psi}=\eta^{a_{j}}\operatorname{e}^{-2v_{j}}.

Combining these expressions, we obtain, for $i=0,\dots,n$ and $j=0,\dots,r$ ,

\frac{x_{i}}{L(y)}=\frac{\operatorname{e}^{-2u_{i}}}{\eta},\quad y_{j}=\frac{\operatorname{e}^{-2v_{j}+2\psi}}{\eta^{a_{j}}}.

From the case $i=0$ we deduce $\eta=L(y)/x_{0}$ and from the case $j=0$ it results $2\psi=\log(y_{0})+a_{0}\log(x_{0}/L(y))$ . From this, one can verify

u_{i}=\frac{1}{2}\log\Big(\frac{x_{0}}{x_{i}}\Big),\quad v_{j}=\frac{1}{2}\log\Big(\frac{y_{0}}{y_{j}}\Big)+\frac{a_{0}-a_{j}}{2}\log\Big(\frac{x_{0}}{L(y)}\Big).

From Theorem 3.52(4), we have $\psi^{\vee}(x,y)=\langle x,u\rangle+\langle y,v\rangle-\psi(u,v)$ . Inserting the expressions above for $\psi$ , $u_{i}$ and $v_{j}$ in terms of $x,y$ , we obtain the stated formula.

For $v\neq\infty$ , we have $\psi_{v}=\Psi$ . The last statement follows from Example 3.16. ∎

Proposition 4.37 and Theorem 6.37 imply

(8.21)		$\displaystyle\deg_{{\mathcal{O}}_{\mathbb{P}(E)}(1)}(\mathbb{P}(E))$	$\displaystyle=(n+r)!\operatorname{vol}(\Delta),$
(8.21)		$\displaystyle\operatorname{h}_{\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}(\mathbb{P}(E))$	$\displaystyle=(n+r+1)!\int_{\Delta}\psi^{\vee}_{\infty}\ \,\text{\rm d}x\,\,\text{\rm d}y,$

where, for short, $\,\text{\rm d}x$ and $\,\text{\rm d}y$ stand for $\,\text{\rm d}x_{1}\dots\,\text{\rm d}x_{n}$ and $\,\text{\rm d}y_{1}\dots\,\text{\rm d}y_{r}$ , respectively.

We now compute these volume and integral giving the degree and the height of $\mathbb{P}(E)$ . We show, in particular, that the height is a rational number. Recall that $\Delta^{r}$ and $\Delta^{n}$ are the standard simplexes of $\mathbb{R}^{r}$ and $\mathbb{R}^{n}$ , respectively.

Lemma 8.22.

With the above notation, we have

(8.23)		$\displaystyle\deg_{{\mathcal{O}}_{\mathbb{P}(E)}(1)}(\mathbb{P}(E))=\frac{(n+r)!}{n!}\int_{\Delta^{r}}L(y)^{n}\,\text{\rm d}y$
(8.24)		$\displaystyle\operatorname{h}_{\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}(\mathbb{P}(E))=\frac{(n+r+1)!}{(n+1)!}\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})\int_{\Delta^{r}}L(y)^{n+1}\,\text{\rm d}y$
	$\displaystyle\hskip 160.0pt-\frac{(n+r+1)!}{2\,n!}\int_{\Delta^{r}}L(y)^{n}\varepsilon_{r}(y)\,\text{\rm d}y,$

where $\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})=\sum_{h=1}^{n}\sum_{j=1}^{h}\frac{1}{2j}$ is the height of the projective space relative to the Fubini-Study metric.

Proof.

Equation (8.21) shows that the degree of $\mathbb{P}(E)$ is equal to $(n+r)!\operatorname{vol}(\Delta)$ . The same equation together with Proposition 8.20(4) gives that the height of $\mathbb{P}(E)$ is equal to :

(8.25)

-\frac{(n+r+1)!}{2}\left(\int_{\Delta}\varepsilon_{r}(y)\,\text{\rm d}x\,\text{\rm d}y+\int_{\Delta}L(y)\cdot\varepsilon_{n}(L(y)^{-1}x)\,\text{\rm d}x\,\text{\rm d}y\right).

Let $I_{1}$ and $I_{2}$ be the two above integrals. Observe $\Delta=\bigcup_{y\in\Delta^{r}}(\{y\}\times L(y)\cdot\Delta^{n})$ . Then

	$\displaystyle\operatorname{vol}(\Delta)$	$\displaystyle=\int_{\Delta^{r}}\left(\int_{L(y)\cdot\Delta^{n}}dx\right)\,\text{\rm d}y=\frac{1}{n!}\int_{\Delta^{r}}L(y)^{n}\,\text{\rm d}y,$
	$\displaystyle I_{1}$	$\displaystyle=\int_{\Delta^{r}}\left(\int_{L(y)\cdot\Delta^{n}}\,\text{\rm d}x\right)\varepsilon_{r}(y)\,\text{\rm d}y=\frac{1}{n!}\int_{\Delta^{r}}L(y)^{n}\varepsilon_{r}(y)\,\text{\rm d}y,$

since $\int_{L(y)\cdot\Delta^{n}}\,\text{\rm d}x=L(y)^{n}/n!$ . And, for the second integral,

	$\displaystyle I_{2}$	$\displaystyle=\int_{\Delta^{r}}L(y)\left(\int_{L(y)\cdot\Delta^{n}}\varepsilon_{n}(L(y)^{-1}x)\,\text{\rm d}x\right)\,\text{\rm d}y$
		$\displaystyle=\left(\int_{\Delta^{r}}L(y)^{n+1}\,\text{\rm d}y\right)\cdot\left(\int_{\Delta^{n}}\varepsilon_{n}(x)\,\text{\rm d}x\right)=-\frac{2\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})}{(n+1)!}\int_{\Delta^{r}}L(y)^{n+1}\,\text{\rm d}y.$

since $\int_{L(y)\cdot\Delta^{n}}\varepsilon_{n}(L(y)^{-1}x)\,\text{\rm d}x=L(y)^{n}\int_{\Delta^{n}}\varepsilon_{n}(x)\,\text{\rm d}x$ and

\int_{\Delta^{n}}\varepsilon_{n}(x)\,\text{\rm d}x=\frac{-1}{(n+1)!}\cdot\sum_{h=1}^{n}\sum_{j=1}^{h}\frac{1}{j}=-\frac{2\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})}{(n+1)!}.

The expression for $\operatorname{vol}(\Delta)$ gives the formula for the degree. Carrying the expressions of $I_{1}$ and $I_{2}$ in (8.25) concludes the proof of Lemma 8.22. ∎

Proposition 8.26.

In the above setting, one has :

	$\displaystyle\deg_{{\mathcal{O}}_{\mathbb{P}(E)}(1)}(\mathbb{P}(E))$	$\displaystyle=\sum_{{i_{0},\dots,i_{r}\in{\mathbb{N}}}\atop{i_{0}+\dots+i_{r}=n}}a_{0}^{i_{0}}\dots a_{r}^{i_{r}}$
	$\displaystyle\operatorname{h}_{{\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}}(\mathbb{P}(E))$	$\displaystyle=\left(\sum_{{i_{0},\dots,i_{r}\in{\mathbb{N}}}\atop{i_{0}+\dots+i_{r}=n+1}}a_{0}^{i_{0}}\dots a_{r}^{i_{r}}\right)\operatorname{h}_{{\overline{{\mathcal{O}}_{\mathbb{P}^{n}}(1)}}}(\mathbb{P}^{n})$
		$\displaystyle\kern 99.58464pt+\sum_{{i_{0},\dots,i_{r}\in{\mathbb{N}}}\atop{i_{0}+\dots+i_{r}=n}}a_{0}^{i_{0}}\dots a_{r}^{i_{r}}A_{n,r}(i_{0},\dots,i_{r}),$

where $A_{n,r}(i_{0},\dots,i_{r})=\sum_{m=0}^{r}(i_{m}+1)\sum_{j=i_{m}+2}^{n+r+1}\frac{1}{2j}$ . In particular, the height of $\mathbb{P}(E)$ is a positive rational number.

Proof.

To prove this result it suffices to compute the two integrals appearing in Lemma 8.22. However

L(y)=a_{0}+\sum_{\ell=1}^{r}(a_{\ell}-a_{0})y_{\ell}=a_{0}y_{0}+\dots+a_{r}y_{r},

with $y_{0}=1-y_{1}-\dots-y_{r}$ , and therefore

L(y)^{n}=\sum_{\stackrel{{\scriptstyle\alpha\in\mathbb{N}^{r+1}}}{{\scriptscriptstyle|\alpha|=n}}}\binom{n}{\alpha_{0},\dots,\alpha_{r}}\prod_{\ell=0}^{r}(a_{\ell}y_{\ell})^{\alpha_{\ell}}

and similarly for $L(y)^{n+1}$ . Now, Corollary 7.19 gives :

	$\displaystyle\int_{\Delta^{r}}y_{0}^{\alpha_{0}}y_{1}^{\alpha_{1}}\dots y_{r}^{\alpha_{r}}\,\text{\rm d}y$	$\displaystyle=\frac{\alpha_{0}!\dots\alpha_{r}!}{(\|\alpha\|+r)!},$
	$\displaystyle\int_{\Delta^{r}}y_{0}^{\alpha_{0}}y_{1}^{\alpha_{1}}\dots y_{r}^{\alpha_{r}}\log(y_{j})\,\text{\rm d}y$	$\displaystyle=-\frac{\alpha_{0}!\dots\alpha_{r}!}{(\|\alpha\|+r)!}\sum_{\ell=\alpha_{j}+1}^{\|\alpha\|+r}\frac{1}{\ell},$

which, combined with the above expression for $L(y)^{n}$ and $L(y)^{n+1}$ , gives

	$\displaystyle\int_{\Delta^{r}}L(y)^{n}dy$	$\displaystyle=\sum_{\stackrel{{\scriptstyle\alpha\in\mathbb{N}^{r+1}}}{{\scriptscriptstyle\|\alpha\|=n}}}\frac{n!}{(n+r)!}\prod_{\ell=0}^{r}a_{\ell}^{\alpha_{\ell}}=\sum_{\stackrel{{\scriptstyle i_{0},\dots,i_{r}\in\mathbb{N}}}{{\scriptscriptstyle i_{0}+\dots+i_{r}=n}}}\prod_{\ell=0}^{r}a_{\ell}^{i_{\ell}}$
	$\displaystyle\int_{\Delta^{r}}L(y)^{n+1}dy$	$\displaystyle=\sum_{\stackrel{{\scriptstyle\alpha\in\mathbb{N}^{r+1}}}{{\scriptscriptstyle\|\alpha\|=n+1}}}\frac{(n+1)!}{(n+1+r)!}\prod_{\ell=0}^{r}a_{\ell}^{\alpha_{\ell}}=\sum_{\stackrel{{\scriptstyle i_{0},\dots,i_{r}\in\mathbb{N}}}{{\scriptscriptstyle i_{0}+\dots+i_{r}=n+1}}}\prod_{\ell=0}^{r}a_{\ell}^{i_{\ell}}$
	$\displaystyle\int_{\Delta^{r}}L(y)^{n}\varepsilon_{r}(y)dy$	$\displaystyle=-\sum_{m=0}^{r}\sum_{\stackrel{{\scriptstyle\alpha\in\mathbb{N}^{r+1}}}{{\scriptscriptstyle\|\alpha\|=n}}}\frac{n!(\alpha_{m}+1)}{(n+1+r)!}\bigg(\prod_{\ell=0}^{r}a_{\ell}^{\alpha_{\ell}}\bigg)\sum_{\ell=\alpha_{m}+2}^{n+1+r}\frac{1}{\ell}$
		$\displaystyle=-\frac{n!}{(n+1+r)!}\sum_{\stackrel{{\scriptstyle i_{0},\dots,i_{r}\in\mathbb{N}}}{{\scriptscriptstyle i_{0}+\dots+i_{r}=n}}}\bigg(\prod_{\ell=0}^{r}a_{\ell}^{i_{\ell}}\bigg)\sum_{m=0}^{r}(i_{m}+1)\sum_{\ell=i_{m}+2}^{n+1+r}\frac{1}{\ell}$
		$\displaystyle=-\frac{2\,n!}{(n+1+r)!}\sum_{\stackrel{{\scriptstyle i_{0},\dots,i_{r}\in\mathbb{N}}}{{\scriptscriptstyle i_{0}+\dots+i_{r}=n}}}\bigg(\prod_{\ell=0}^{r}a_{\ell}^{i_{\ell}}\bigg)A_{n,r}(i_{0},\dots,i_{r}).$

The statement follows from these expressions together with Lemma 8.22. ∎

Remark 8.27.

We check $A_{1,1}(0,1)=A_{1,1}(1,0)={3}/{4}$ . Let $b\geq 0$ and let ${\overline{{\mathcal{O}}_{\mathbb{F}_{b}}(1)}}$ the adelic line bundle on $\mathbb{F}_{b}$ associated to $a_{0}=1$ and $a_{1}=b+1$ . Putting $n=r=1$ , $a_{0}=1$ and $a_{1}=b+1$ in Proposition 8.26, we recover the expression for the height of Hirzebruch surfaces established in [Mou06]: $\operatorname{h}_{{\overline{{\mathcal{O}}_{\mathbb{F}_{b}}(1)}}}(\mathbb{F}_{b})=\frac{1}{2}b^{2}+\frac{9}{4}b+3$ .