6.1. Local heights of toric varieties [02W3]

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6.1. Local heights of toric varieties

Let $K$ be either $\mathbb{R}$ , $\mathbb{C}$ or a complete field with respect to an absolute value associated to a nontrivial discrete valuation. Let $N\simeq\mathbb{Z}^{n}$ be a lattice and $M=N^{\vee}$ the dual lattice. We will use the notations of §4 and we recall the definition of $\lambda_{K}$ in (5.3).

Let $\Sigma$ be a complete fan on $N_{\mathbb{R}}$ and $X_{\Sigma}$ the corresponding proper toric variety. In Definition 2.39 we recalled the definition of local heights. These local heights depend, not only on cycles and metrized line bundles, but also on the choice of sections of the involved line bundles. For toric line bundles, Proposition-Definition 5.20, provides us with a distinguished choice of a toric metric, the canonical metric. This metric is integrable and, if the line bundle is generated by global sections, it is approachable. By comparing any integrable metric to the canonical metric, we can define a local height for toric line bundles that is independent from the choice of sections.

Definition 6.1.

Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d$ , be a family of toric line bundles, with integrable toric metrics. Denote by ${\overline{L}}_{i}^{{\operatorname{can}}}$ the same line bundles equipped with the canonical metric. Let $Y$ be a $d$ -dimensional cycle of $X_{\Sigma}$ . Then the toric local height of $Y$ with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$ is

(6.2)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)=\operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0},\dots,\varphi^{\ast}{\overline{L}}_{d}}(Y^{\prime};s_{0},\dots,s_{d})-\\ \operatorname{h}_{\varphi^{\ast}{\overline{L}}_{0}^{{\operatorname{can}}},\dots,\varphi^{\ast}{\overline{L}}_{d}^{{\operatorname{can}}}}(Y^{\prime};s_{0},\dots,s_{d}),

where $\Sigma^{\prime}$ is a regular refinement of $\Sigma$ (hence $X_{\Sigma^{\prime}}$ is projective), $\varphi\colon X_{\Sigma^{\prime}}\to X_{\Sigma}$ is the corresponding proper toric morphism, $Y^{\prime}$ is a cycle of $X^{\prime}$ such that $\varphi_{\ast}Y^{\prime}=Y$ and $s_{0},\dots,s_{d}$ are sections meeting $Y^{\prime}$ properly. When ${\overline{L}}_{0}=\dots={\overline{L}}_{d}={\overline{L}}$ we will denote

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y).

Remark 6.3.

Even if the toric local height in the above definition differs from the local height of Definition 2.39, we will be able to use them to compute global heights because, for toric subvarieties and closures of orbits, the sum over all places of the local canonical heights is zero (see Proposition 6.35). This is the case, in particular, for the height of the total space $X_{\Sigma}$ .

By Theorem 2.46(4), the right-hand side of equation (6.2) does not depend on the choice of refinement nor on the choice of sections, but the toric local height depends on the toric structure of the line bundles (see Definition 4.19), because the canonical metric depends on the toric structure.

Proposition 6.4.

The toric local height is symmetric and multilinear with respect to tensor product of metrized toric line bundles. In particular, let $\Sigma$ be a complete fan, ${\overline{L}}_{i}$ a family of $d+1$ toric line bundles with integrable toric metrics and $Y$ an algebraic cycle of $X_{\Sigma}$ of dimension $d$ . Then

(6.5)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)=\sum_{j=0}^{d}(-1)^{d-j}\sum_{1\leq i_{0}<\cdots<i_{j}\leq d}\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{i_{0}}\otimes\cdots\otimes{\overline{L}}_{i_{j}}}(Y).

Proof.

It suffices to prove the statement for the case when $X_{\Sigma}$ is projective, as the general case reduces to this one by taking a suitable refinement of the fan.

The symmetry of the toric local height follows readily from the analogous property for the local height, see Theorem 2.46(1). For the multilinearity, let ${\overline{L}}_{d}^{\prime}$ be a further metrized line bundle. By the moving lemma, there are sections $s_{i}$ of $L_{i}$ , $0\leq i\leq d$ meeting properly on $Y$ and $s_{d}^{\prime}$ of $L_{d}^{\prime}$ such that $s_{0},\dots,s_{d-1},s_{d}^{\prime}$ meets properly on $Y$ too. By Theorem 2.46(1),

\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}\otimes{\overline{L}}_{d}^{\prime}}(Y;s_{0},\dots,s_{d-1},s_{d}\otimes s_{d}^{\prime})=\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y;s_{0},\dots,s_{d})\\ +\operatorname{h}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}^{\prime}}(Y;s_{0},\dots,s_{d-1},s_{d}^{\prime})

and a similar formula holds for the canonical metric. By the definition of the toric local height, $\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}\otimes{\overline{L}}_{d}^{\prime}}(Y)=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d}}(Y)+\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{d-1},{\overline{L}}_{d}^{\prime}}(Y)$ . The inclusion-exclusion formula follows readily from the symmetry and the multilinearity of the local toric height. ∎

Theorem 6.6.

Let $\Sigma$ be a complete fan on $N_{\mathbb{R}}$ . Let ${\overline{L}}=(L,\|\cdot\|)$ be a toric line bundle on $X_{\Sigma}$ , generated by global sections, and equipped with an approachable toric metric. Choose any toric section $s$ of $L$ ; let $\Psi$ be the associated support function on $\Sigma$ , and put $\Delta_{\Psi}=\operatorname{stab}(\Psi)$ for the associated polytope. Then, the toric local height of $X_{\Sigma}$ with respect to ${\overline{L}}$ is given by

(6.7)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(n+1)!\lambda_{K}\int_{\Delta_{\Psi}}\psi_{{\overline{L}},s}^{\vee}\,\text{\rm d}\operatorname{vol}_{M}.

where $\,\text{\rm d}\operatorname{vol}_{M}$ is the unique Haar measure of $M_{\mathbb{R}}$ such that the co-volume of $M$ is one and $\psi_{{\overline{L}},s}^{\vee}$ is the Legendre-Fenchel dual to the function $\psi_{{\overline{L}},s}$ associated to $({\overline{L}},s)$ in Definition 5.14.

We note that, by Theorem 5.73(2), the function $\psi_{{\overline{L}},s}$ is concave because the metric $\|\cdot\|$ on $L^{{\text{\rm an}}}$ is approachable. We also introduce the function

f_{{\overline{L}},s}(u)=(\psi_{{\overline{L}},s}\lambda_{K})(u)=\lambda_{K}\psi_{{\overline{L}},s}(u/\lambda_{K}).

Definition 6.8.

Let $({\overline{L}},s)$ be a metrized toric line bundle with a toric section as in the theorem above. Then the roof function associated to $({\overline{L}},s)$ is the concave function $\vartheta_{{\overline{L}},s}\colon\Delta_{\Psi}\to\mathbb{R}$ defined as

\vartheta_{{\overline{L}},s}=f_{{\overline{L}},s}^{\vee}=\lambda_{K}\psi_{{\overline{L}},s}^{\vee}.

The concave function $\psi_{{\overline{L}},s}^{\vee}$ will be called the rational roof function. When the toric section $s$ is clear form the context, we will denote $f_{{\overline{L}},s}$ and $\vartheta_{{\overline{L}},s}$ by $f_{\|\cdot\|}$ and $\vartheta_{\|\cdot\|}$ respectively.

The function $\psi_{\|\cdot\|}$ is not invariant under field extensions (see Proposition 5.53(3)) but it has the advantage that, if the metric $\|\cdot\|$ is algebraic, then it is rational with respect to the lattice $N$ . By contrast, the function $f_{\|\cdot\|}$ is invariant under field extensions. It is not rational, but it takes values in $\lambda_{K}\mathbb{Q}$ on $\lambda_{K}N_{\mathbb{Q}}$ . This is the function that appears in [BPS09].

In case $\psi_{\|\cdot\|}$ is a piecewise affine concave function, $\vartheta_{\|\cdot\|}$ and $\psi_{\|\cdot\|}^{\vee}$ parameterize the upper envelope of some extended polytope, as explained in Lemma 3.79, hence the terminology “roof function”. In case $K$ is non-Archimedean and $||\cdot||$ is algebraic, the function $\psi_{\|\cdot\|}^{\vee}$ is a rational concave function.

Alternatively, we can express the toric height in terms of the roof function as

(6.9)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(n+1)!\int_{\Delta_{\Psi}}\vartheta_{{\overline{L}},s}\,\text{\rm d}\operatorname{vol}_{M}.

Proof of Theorem 6.6.

For short, we set $\Delta=\Delta_{\Psi}$ and $\psi=\psi_{\|\cdot\|}$ . Let $\Sigma_{\Delta}$ be the fan associated to $\Delta$ as in Remark 4.43. There is a toric morphism $\varphi\colon X_{\Sigma}\to X_{\Sigma_{\Delta}}$ . The function $\psi^{\vee}$ defines an approachable metric $\|\cdot\|^{\prime}$ on $\mathcal{O}(D_{\Psi_{\Delta}})^{{\text{\rm an}}}$ . We denote ${\overline{L}}^{\prime}=(\mathcal{O}(D_{\Psi_{\Delta}}),\|\cdot\|^{\prime})$ . Then there is an isometry $\varphi^{\ast}({\overline{L}}^{\prime})={\overline{L}}$ . By Corollary 5.25 there is an isometry $\varphi^{\ast}({\overline{L}}^{\prime}{}^{{\operatorname{can}}})={\overline{L}}^{{\operatorname{can}}}$ .

If the dimension of $\Delta$ is less than $n$ , then the right-hand side of equation (6.7) is zero. Moreover, $n=\dim(X_{\Sigma})>\dim(X_{\Sigma_{\Delta}})$ and the metrized line bundles ${\overline{L}}$ and ${\overline{L}}^{{\operatorname{can}}}$ come from a variety of smaller dimension. Therefore, by Theorem 2.46(2), the left-hand side of equation (6.7) is also zero, because $\varphi_{*}X_{\Sigma}$ is the cycle zero. If $\Delta$ has dimension $n$ then $\varphi$ is a birational morphism, so, by Theorem 2.46(2),

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}^{\prime}}(X_{\Sigma_{\Delta}}).

Therefore it is enough to prove the theorem for $X_{\Sigma_{\Delta}}$ . By construction, the fan $\Sigma_{\Delta}$ is regular; hence the variety $X_{\Sigma_{\Delta}}$ is projective and $L^{\prime}$ is ample. Thus we are reduced to prove the theorem in the case when $\Sigma$ is regular and $L$ is ample.

Now the proof is done by induction on $n$ , the dimension of $X_{\Sigma}$ . If $n=0$ then $X_{\Sigma}=\mathbb{P}^{0}$ , $\Psi=0$ , $\Delta=\{0\}$ and $L=\mathcal{O}(D_{0})=\mathcal{O}_{\mathbb{P}^{0}}$ . By equation (5.15), $\log\|s\|=\lambda_{K}\psi(0)$ and $\log\|s\|_{{\operatorname{can}}}=\lambda_{K}\Psi(0)=0$ . The Legendre-Fenchel dual of $\psi$ satisfies $\psi^{\vee}(0)=-\psi(0)$ . By equation (2.40), $\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s)=-\lambda_{K}\psi(0)$ and $\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s)=0$ . Therefore

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=-\lambda_{K}\psi(0)=\lambda_{K}\psi^{\vee}(0)=1!\lambda_{K}\int_{\Delta}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M}.

Let $n\geq 1$ and let $s_{0},\dots,s_{n-1}$ be rational sections of ${\mathcal{O}}(D_{\Psi})$ such that $s_{0},\dots,s_{n-1},s$ intersect $X_{\Sigma}$ properly. By the construction of local heights (Definition 2.39),

(6.10)		$\displaystyle\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s_{0},\dots,s_{n-1},s)=\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);$	$\displaystyle s_{0},\dots,s_{n-1})$
		$\displaystyle-\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\\|s\\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}$

and a similar formula holds for the canonical metric.

For each facet $F$ of $\Delta$ let $v_{F}$ be as in Notation 3.103. Since $L$ is ample, Proposition 4.46 implies

(6.11)

\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);s_{0},\dots,s_{n-1})=\sum_{F}-\langle v_{F},F\rangle\operatorname{h}_{{\overline{L}}}(V(\tau_{F});s_{0},\dots,s_{n-1}),

where the sum is over the facets $F$ of $\Delta$ . Observe that the local height of $V(\tau_{F})$ with respect to the metrized line bundle ${\overline{L}}$ coincides with the local height associated to the restriction of ${\overline{L}}$ to this subvariety. Moreover by Corollary 5.23, the restriction of the canonical metric of $L^{{\text{\rm an}}}$ to this subvariety agrees with the canonical metric of $L^{{\text{\rm an}}}|_{V(\tau_{F})}$ . Hence, by substracting from equation (6.11) the analogous formula for the canonical metric, we obtain

(6.12)		$\displaystyle\sum_{F}-\langle m_{F},v_{F}\rangle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}\|_{V(\tau_{F})}}(V(\tau_{F}))=\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);$	$\displaystyle s_{0},\dots,s_{n-1})$
		$\displaystyle-\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(\operatorname{div}(s);s_{0},\dots,s_{n-1}).$

Moreover, Proposition 2.37 implies that

\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=\int_{X_{\Sigma,0}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}.

By equation (5.15), $\log\|s\|=({\operatorname{val}}_{K})^{*}(\lambda_{K}\psi)$ . Moreover

\displaystyle\int_{X_{\Sigma,0}^{{\text{\rm an}}}}({\operatorname{val}}_{K})^{*}(\lambda_{K}\psi)c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=

\displaystyle\int_{N_{\mathbb{R}}}\lambda_{K}\psi({\operatorname{val}}_{K})_{*}(c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}})

and by Theorem 5.81, $({\operatorname{val}}_{K})_{*}(c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}})=n!{\mathcal{M}}_{M}(\psi)$ . Hence

(6.13)

\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi).

By Example 3.96, ${\mathcal{M}}_{M}(\Psi)=\operatorname{vol}_{M}(\Delta)\delta_{0}$ . Therefore, in the case of the canonical metric, equation (6.13) reads as

(6.14)

\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|_{{\operatorname{can}}}c_{1}({\overline{L}}^{{\operatorname{can}}})^{n}\wedge\delta_{X_{\Sigma}}=n!\lambda_{K}\operatorname{vol}_{M}(\Delta)\Psi(0)=0.

Thus, substracting from equation (6.10) the analogous formula for the canonical metric and using equations (6.12), (6.13) and (6.14), we obtain

(6.15)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\sum_{F}-\langle v_{F},F\rangle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}|_{V(\tau_{F})}}(V(\tau_{F}))-n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi).

By the inductive hypothesis and equation (5.77)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}|_{V(\tau_{F})}}(V(\tau_{F}))=n!\lambda_{K}\int_{F}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M({F})}.

Hence, by Corollary 3.104,

	$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})$	$\displaystyle=-n!\lambda_{K}\sum_{F}\langle v_{F},F\rangle\int_{F}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M({F})}-n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi)$
		$\displaystyle=(n+1)!\lambda_{K}\int_{\Delta}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M},$

proving the theorem ∎

Remark 6.16.

The left-hand side of equation (6.7) only depends on the structure of toric line bundle of $L$ and not on a particular choice of toric section, while the right-hand side seems to depend on the section $s$ . We can see directly that the right hand side actually does not depend on the section. If we pick a different toric section, say $s^{\prime}$ , then the corresponding support function $\Psi^{\prime}$ differs from $\Psi$ by a linear functional. The polytope $\Delta_{\Psi^{\prime}}$ is the translated of $\Delta_{\Psi^{\prime}}$ by the corresponding element of $M$ . The function $\psi_{{\overline{L}},s^{\prime}}$ differs from $\psi_{{\overline{L}},s}$ by the same linear functional and $\psi_{{\overline{L}},s^{\prime}}^{\vee}$ is the translated of $\psi_{{\overline{L}},s}^{\vee}$ by the same element of $M$ . Thus the integral on the right has the same value whether we use the section $s$ of the section $s^{\prime}$ .

Theorem 6.6 can be reformulated in terms of an integral over $N_{\mathbb{R}}$ .

Corollary 6.17.

Let notation be as in Theorem 6.6 and write $\psi=\psi_{{\overline{L}},s}$ for short. Then

(6.18)

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

where $\psi^{\vee}\circ\partial\psi$ is the integrable function defined by (3.105). When $\psi\in{\mathcal{C}}^{2}(N_{\mathbb{R}})$ ,

(6.19)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(-1)^{n}(n+1)!\int_{N_{\mathbb{R}}}(\langle\nabla\psi(u),u\rangle-\psi(u))\det(\operatorname{Hess}(\psi))\,\,\text{\rm d}\operatorname{vol}_{N}.

When $\psi$ is piecewise affine,

(6.20)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=(n+1)!\sum_{v\in\Pi(\psi)^{0}}\int_{v^{*}}(\langle x,v\rangle-\psi(v))\,\,\text{\rm d}\operatorname{vol}_{M}(x).

Proof.

Equation (6.18) follows readily from Theorem 6.6 and (3.105). The second statement follows from Proposition 3.94 and Example 3.106(1) while the third one follows from Proposition 3.95 and Example 3.106(2). ∎

Theorem 6.6 can be extended to compute the local toric height associated to distinct line bundles in term of the mixed integral of the associated roof functions.

Corollary 6.21.

Let $\Sigma$ be a complete fan on $N_{\mathbb{R}}$ and ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,n$ , be toric line bundles on $X_{\Sigma}$ generated by global sections and equipped with approachable toric metrics. Choose toric sections $s_{i}$ of $L_{i}$ and let $\Psi_{i}$ be the corresponding support functions. Then the toric height of $X_{\Sigma}$ with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{n}$ is given by

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{n}}(X_{\Sigma})=\operatorname{MI}_{M}(\vartheta_{\|\cdot\|_{0}},\dots,\vartheta_{\|\cdot\|_{n}})=\lambda_{K}\operatorname{MI}_{M}(\psi_{\|\cdot\|_{0}}^{\vee},\dots,\psi_{\|\cdot\|_{n}}^{\vee}).

Proof.

Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=1,2$ , be toric line bundles equipped with toric metrics and let $s_{i}$ be a toric section of $L_{i}$ . By propositions 5.19 (1) and 3.38 (3)

(6.22)

(\psi_{{\overline{L}}_{1}\otimes{\overline{L}}_{2},s_{1}\otimes s_{2}})^{\vee}=\psi_{{\overline{L}}_{1},s_{1}}^{\vee}\boxplus\psi_{{\overline{L}}_{2},s_{2}}^{\vee}.

The result then follows from (6.5), the definition of the mixed integral (Definition 3.113) and Theorem 6.6. ∎

Remark 6.23.

In the integrable case, the toric height can be expressed as an alternating sum of mixed integrals as follows. Let ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,n$ , be toric line bundles on $X_{\Sigma}$ equipped with integrable toric metrics and set ${\overline{L}}_{i}={\overline{L}}_{i,+}\otimes({\overline{L}}_{i,-})^{-1}$ for some approachable metrized toric line bundles ${\overline{L}}_{i,+}$ , ${\overline{L}}_{i,-}$ . Choose a toric section for each line bundle and write $\vartheta_{i,+}$ and $\vartheta_{i,-}$ for the corresponding roof functions. Then

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{0},\dots,{\overline{L}}_{n}}(X_{\Sigma})=\sum_{\varepsilon_{0},\dots,\varepsilon_{n}\in\{\pm 1\}}\varepsilon_{0}\dots\varepsilon_{n}\operatorname{MI}_{M}(\vartheta_{{0},\varepsilon_{0}},\dots,\vartheta_{n,\varepsilon_{n}}).

We have defined and computed the local height of a toric variety. We now will compute the toric height of toric subvarieties. We start with the case of orbits.

Proposition 6.24.

Let $\Sigma$ be a complete fan on $N_{\mathbb{R}}$ and $\sigma\in\Sigma$ a cone of codimension $d$ . To it, we have associated the dimension $d$ closed subvariety $V(\sigma)$ and the closed immersion $\iota_{\sigma}\colon X_{\Sigma(\sigma)}\to X_{\Sigma}$ whose image is $V(\sigma)$ . Let $L$ be a toric line on $X_{\Sigma}$ generated by global sections, $s$ a toric section, $\Psi$ the corresponding support function, and $\|\cdot\|$ an approachable toric metric on $L^{{\text{\rm an}}}$ . As usual write ${\overline{L}}=(L,\|\cdot\|)$ . Then

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(V(\sigma))=\operatorname{h}^{\operatorname{tor}}_{\iota_{\sigma}^{\ast}{\overline{L}}}(X_{\Sigma(\sigma)})=(d+1)!\int_{F_{\sigma}}\vartheta_{{\overline{L}},s}\,\text{\rm d}\operatorname{vol}_{M(F_{\sigma})},

where $F_{\sigma}$ is the face of $\Delta_{\Psi}$ corresponding to $\sigma$ , $M(F_{\sigma})$ is the lattice induced by $M$ on the linear space associated to $F_{\sigma}$ and $\iota_{\sigma}^{\ast}L$ has the structure of toric line bundle of Proposition 4.34.

Proof.

By Corollary 5.23 the restriction of the canonical metric of $L^{{\text{\rm an}}}$ is the canonical metric of $\iota_{\sigma}^{\ast}L^{{\text{\rm an}}}$ . Therefore, the equality $\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(V(\sigma))=\operatorname{h}^{\operatorname{tor}}_{\iota_{\sigma}^{\ast}{\overline{L}}}(X_{\Sigma(\sigma)})$ follows from Theorem 2.46(2).

To prove the second equality, choose $m_{\sigma}\in F_{\sigma}\cap M$ . We will follow the notation of Proposition 5.75. By Theorem 6.6,

\operatorname{h}^{\operatorname{tor}}_{\iota_{\sigma}^{\ast}{\overline{L}}}(X_{\Sigma(\sigma)})=(d+1)!\int_{\Delta_{(\Psi-m_{\sigma})(\sigma)}}\vartheta_{\|\cdot\|_{\sigma}}\,\text{\rm d}\operatorname{vol}_{M(\sigma)}.

By Proposition 4.47, $\Delta_{(\Psi-m_{\sigma})(\sigma)}=(\pi_{\sigma}^{\vee}+m_{\sigma})^{-1}F_{\sigma}.$ By Proposition 5.75

\vartheta_{\|\cdot\|_{\sigma}}=\lambda_{K}\varphi^{\vee}_{\|\cdot\|_{\sigma}}=\lambda_{K}(\pi_{\sigma}^{\vee}+m_{\sigma})^{\ast}\varphi^{\vee}_{\|\cdot\|}=(\pi_{\sigma}^{\vee}+m_{\sigma})^{\ast}\vartheta_{\|\cdot\|}.

Since $M(F_{\sigma})=M(\sigma)$ , we obtain

\int_{\Delta_{(\Psi-m_{\sigma})(\sigma)}}\vartheta_{\|\cdot\|_{\sigma}}\,\text{\rm d}\operatorname{vol}_{M(\sigma)}=\int_{F_{\sigma}}\vartheta_{\|\cdot\|}\,\text{\rm d}\operatorname{vol}_{M(F_{\sigma})},

proving the result. ∎

We now study the behaviour of the toric local height with respect to toric morphisms. Let $N_{1}$ be a lattice of rank $d$ and $M_{1}$ the dual lattice. Let $H\colon N_{1}\to N$ be a linear map and $\Sigma_{1}$ a fan on $N_{1,\mathbb{R}}$ such that, for each cone $\sigma\in\Sigma_{1}$ , $H(\sigma)$ is contained in a cone of $\Sigma$ . Let $\varphi\colon X_{\Sigma_{1}}\to X_{\Sigma}$ be the associated morphism. Denote $Q=H(N_{1})^{\operatorname{sat}}$ the saturated sublattice of $N$ and let $Y_{Q}$ be the image of $X_{\Sigma_{1}}$ under $\varphi$ . Then $Y_{Q}$ is equal to the toric subvariety $Y_{\Sigma,Q}=Y_{\Sigma,Q,x_{0}}$ of Definition 4.12, where we recall that $x_{0}$ denote the distinguished point of the principal orbit of $X_{\Sigma}$ .

Proposition 6.25.

With the previous notation, let ${\overline{L}}$ be a toric line bundle on $X_{\Sigma}$ generated by global sections, equipped with an approachable toric metric. We put on $\varphi^{\ast}L$ the structure of toric line bundle of Remark 4.36. Choose a toric section $s$ of $L$ and let $\Psi$ be the associated support function.

(1)

If $H$ is not injective, then $\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=0$ .

(2)

If $H$ is injective, then $\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=[Q:H(N_{1})]\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y_{Q})$ . Moreover

(6.26)

\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=(d+1)!\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\vartheta_{\|\cdot\|})\,\text{\rm d}\operatorname{vol}_{M_{1}}.

Proof.

By Corollary 5.25, the inverse image of the canonical metric by a toric morphism is the canonical metric. Thus (1) and the first statement of (2) follow from 2.46 (2).

By Proposition 5.24 and Theorem 6.6 we deduce

\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=(d+1)!\lambda_{K}\int_{\Delta_{\Psi\circ H}}(H^{\ast}\psi_{\|\cdot\|})^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\vartheta_{\|\cdot\|})\,\text{\rm d}\operatorname{vol}_{M_{1}},

proving the result. ∎

We now study the case of an equivariant morphism. Let $N$ , $N_{1}$ , $d$ , $H$ , $\Sigma$ and $\Sigma_{1}$ as before. For simplicity, we assume that $H\colon N_{1}\to N$ is injective and that $Q=H(N_{1})$ is a saturated sublattice, because the effect of a non-injective map or a non-saturated sublattice is explained in Proposition 6.25. Let $p\in X_{\Sigma,0}(K)$ be a point of the principal open subset and $u={\operatorname{val}}_{K}(p)\in N_{\mathbb{R}}$ . Then, in the non-Archimedean case, $u\in N$ . Denote $\varphi=\varphi_{p,H}$ the equivariant morphism determined by $H$ and $p$ , also denote $Y=Y_{\Sigma,Q,p}$ the image of $X_{\Sigma_{1}}$ by $\varphi$ , and $A=H+u$ the associated affine map.

Let ${\overline{L}}$ be a toric line bundle generated by global sections, equipped with an approachable toric metric. Recall that there is no natural structure of toric line bundle in the inverse image $\varphi^{\ast}L$ . Therefore we have to choose a toric section $s$ of $L$ . Let ${\overline{L}}_{1}$ denote the line bundle $\varphi^{\ast}L$ with the metric induced by $\|\cdot\|$ and the toric structure induced by the section $s$ . We denote by $\Psi$ the support function associated to $(L,s)$ .

Proposition 6.27.

With the previous hypothesis and notations, the equality

(6.28)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{1}}(X_{\Sigma_{1}})=(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}(A^{\ast}\psi_{{\overline{L}},s})^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\psi^{\vee}_{{\overline{L}},s}-u)\,\text{\rm d}\operatorname{vol}_{M_{1}}

holds. Moreover

(6.29)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{1}}(X_{\Sigma_{1}})-\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}(A^{\ast}\Psi)^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\iota_{\Delta_{\Psi}}-\lambda_{K}u)\,\text{\rm d}\operatorname{vol}_{M_{1}},

where $\iota_{\Delta_{\Psi}}$ is the indicator function of $\Delta_{\Psi}$ (Example 3.16).

Proof.

By Proposition 5.24, $\psi_{{\overline{L}}_{1},\varphi^{\ast}s}=A^{\ast}\psi_{{\overline{L}},s}$ . By Proposition 3.46(3) we obtain that $\operatorname{stab}(A^{\ast}\psi_{{\overline{L}},s})=H^{\vee}(\Delta_{\Psi})$ and that

(A^{\ast}\psi_{{\overline{L}},s})^{\vee}=H^{\vee}_{\ast}(\psi_{{\overline{L}},s}-u),

from which equation (6.28) follows.

To prove equation (6.29), we observe that, by the definition of $\operatorname{h}^{\operatorname{tor}}$ ,

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{1}}(X_{\Sigma_{1}})-\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}({\overline{L}}^{{\operatorname{can}}})}(X_{\Sigma_{1}}),

where $\varphi^{\ast}({\overline{L}}^{{\operatorname{can}}})$ has the toric structure induced by $s$ and the metric induced by the canonical metric of $L^{{\text{\rm an}}}$ . We remark here that this metric differs from the canonical metric of $\varphi^{\ast}L^{{\text{\rm an}}}$ . Now equation (6.29) follows from equation (6.28) and the definition of the canonical metric. ∎

Corollary 6.30.

With the previous hypothesis

\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}({\overline{L}}^{{\operatorname{can}}})}(X_{\Sigma_{1}})=(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}(A^{\ast}\Psi)^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}.

Example 6.31.

We continue with Example 5.26. Let $\mathbb{Z}^{r}$ be the standard lattice of rank $r$ , $\Delta^{r}$ the standard simplex of dimension $r$ and $\Sigma_{\Delta^{r}}$ the fan of $\mathbb{R}^{r}$ associated to $\Delta^{r}$ . The corresponding toric variety is $\mathbb{P}^{r}$ . Let $H\colon N\to\mathbb{Z}^{r}$ be an injective linear morphism such that $H(N)$ is a saturated sublattice. Denote $m_{i}=e_{i}^{\vee}\circ H\in M$ , $i=1,\dots,r$ . Let $\Sigma$ the regular fan on $N$ defined by $H$ and $\Sigma_{\Delta^{r}}$ . Let $\Psi_{\Delta^{r}}$ be the support function of $\Delta^{r}$ and let $\Psi=\Psi_{\Delta^{r}}\circ H$ . Explicitly,

\Psi(v)=\min(0,m_{1}(v),\dots,m_{r}(v)).

Let $p\in\mathbb{P}^{r}_{0}(K)$ and $u={\operatorname{val}}_{K}(p)\in\mathbb{R}^{r}$ . Write $u=(u_{1},\dots,u_{r})$ . If $p=(1:\alpha_{1}:\dots:\alpha_{r})$ , then $u_{i}=\frac{-\log(|\alpha_{i}|)}{\lambda_{K}}$ . There is an equivariant morphism $\varphi:=\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$ . Consider the toric line bundle with toric section determined by $\Psi_{\Delta^{r}}$ with the canonical metric and denote by $({\overline{L}},s)$ the induced toric line bundle with toric section on $X_{\Sigma}$ equipped with the induced metric. Then

\psi_{{\overline{L}},s}(v)=\min(0,m_{1}(v)+u_{1},\dots,m_{r}(v)+u_{r}).

Thus $\Delta=\operatorname{stab}(\psi_{{\overline{L}},s})=\operatorname{conv}(0,m_{1},\dots,m_{r})=H^{\vee}(\Delta^{r})$ . By Proposition 3.64 the Legendre-Fenchel dual $\psi_{{\overline{L}},s}^{\vee}\colon\Delta\to\mathbb{R}$ is given by

\psi_{{\overline{L}},s}^{\vee}(x)=\sup\bigg\{\sum_{j=1}^{r}-\lambda_{j}u_{j}\bigg|\ \lambda_{j}\geq 0,\sum_{j=1}^{r}\lambda_{j}\leq 1,\ \sum_{j=1}^{r}\lambda_{j}a_{j}=x\bigg\}\ \text{ for }x\in\Delta.

This function is the upper envelope of the extended polytope of $M_{\mathbb{R}}\times\mathbb{R}$ ,

\operatorname{conv}\left((0,0),(m_{1},-u_{1}),\dots,(m_{r},-u_{r})\right),

Similarly, the roof function $\vartheta_{{\overline{L}},s}=\lambda_{K}\psi_{{\overline{L}},s}^{\vee}$ is the upper envelope of the extended polytope

\operatorname{conv}\left((0,0),(m_{1},\log|\alpha_{1}|),\dots,(m_{r},\log|\alpha_{r}|)\right).