Notation and conventions

Algebraic geometry (general). Denote by $k$ an arbitrary field. By a $k$-variety, we mean a reduced scheme of finite type over $k$. By a subvariety of a $k$-variety we mean a closed reduced subscheme. Note that a subvariety is determined by its underlying topological space and we frequently identify both. The tangent bundle of a $k$-variety $X$ is written $TX$ and its fiber over a point $x\in X(k)$ is denoted by $T_{x}X$. Furthermore, $X^{\text{sm}}$ denotes the smooth locus of $X$.

Meromorphic functions. For every $k$-variety $X$, we write ${\mathcal{K}}_{X}$ for the sheaf of its meromorphic functions (cf. [Reference LiuLiu02, Definition 7.1.13]). With each meromorphic function $f\in \unicode[STIX]{x1D6E4}(X,{\mathcal{K}}_{X})$, we associate the complement $D(f)$ of its zero set (i.e., those points $x\in X$ such that $f_{x}\notin \mathfrak{m}_{x}{\mathcal{O}}_{X,x}$).

Products and projections. For any product $Y_{1}\times _{k}\cdots \times _{k}Y_{m}$ of algebraic varieties, we write $\operatorname{pr}_{i}$ ($i=1,\ldots ,m$) for the projection $Y_{1}\times _{k}\cdots \times _{k}Y_{m}\rightarrow Y_{i}$ without further specification of the varieties $Y_{i}$. This leads to multiple different usages of the same notation $\operatorname{pr}_{i}$, sometimes close to each other. However, this should cause confusion nowhere if context is taken into account.

Algebraic groups. An algebraic $k$-group is a group scheme of finite type over $\text{Spec}(k)$. We refer to [SGA3, Exposé $\text{VI}_{A}$] for the basic properties of algebraic $k$-groups. An algebraic $k$-subgroup of an algebraic $k$-group $G$ is a $k$-subscheme $H$ such that the group law of $G$ induces a group law on $H$. Note that $H$ is necessarily Zariski closed in $G$ [SGA3, Corollaire $\text{VI}_{A}.0.5.2$] and of finite type over $\text{Spec}(k)$. Left-multiplication by an element $g\in G(k)$ is written $l_{g}:G\rightarrow G$. More generally, we use the same notation $l_{g}$ for the left-multiplication with respect to an action $G\times X\rightarrow X$.

A split $k$-torus is an algebraic $k$-group that is isomorphic to some direct product of copies of multiplicative groups $\mathbb{G}_{m}$. A $k$-torus is an algebraic group $G$ such that its base change $G_{k^{\text{sep}}}$ to the separable closure $k^{\text{sep}}$ of $k$ is a split torus. A linear $k$-algebraic group is an algebraic $k$-group whose underlying scheme is both affine and connected.

For fixed algebraic $k$-groups $G_{1}$ and $G_{3}$, the isomorphism classes of Yoneda extensions

(5)

form an abelian group $\operatorname{Ext}_{k}^{1}(G_{1},G_{3})$ with respect to Baer summation (cf. [Reference OortOor66, § I.3]).

We write $[n]_{G}$ for the multiplication-by-$n$ map on any commutative algebraic group $G$. The notation $\cdot _{G}:G\times G\rightarrow G$ is used for the group law of $G$ and $e_{G}:k\rightarrow G$ denotes the identity of $G$. We omit the reference to $G$ in these notations when this group can be inferred from context. We write $A^{\vee }$ for the dual abelian variety associated with an abelian variety $A$. Pulling back line bundles along a homomorphism $\unicode[STIX]{x1D711}:A\rightarrow B$ induces a homomorphism $\unicode[STIX]{x1D711}^{\vee }:B^{\vee }\rightarrow A^{\vee }$ of the associated dual abelian varieties.

Line bundles and linearizations. Line bundles are denoted by capital italic letters $L,M,\ldots \,$ whereas the corresponding calligraphic letters ${\mathcal{L}},{\mathcal{M}},\ldots \,$ are reserved for the invertible sheaves formed by their sections. The line bundle dual to $L$ is written $L^{\vee }$. In the situation where we have an algebraic group $G$ acting on a scheme $X$, we use Mumford’s definition of $G$-linearization [Reference Mumford, Fogarty and KirwanMFK94, Definition 1.6] for general ${\mathcal{O}}_{X}$-modules. For an invertible sheaf ${\mathcal{L}}$ on $X$, a $G$-linearization corresponds to an action $\unicode[STIX]{x1D71A}:G\times L\rightarrow L$ such that the projection $L\rightarrow X$ is $G$-equivariant. We refer to [Reference Mumford, Fogarty and KirwanMFK94, § 1.3] for details. Given a $G$-linearized line bundle $(L,\unicode[STIX]{x1D71A})$ we write $(L,\unicode[STIX]{x1D71A})^{\otimes n}$ for the line bundle $L^{\otimes n}$ with the $T$-linearization induced by $\unicode[STIX]{x1D71A}$. If $\unicode[STIX]{x1D711}:H\rightarrow G$ is a homomorphism from another algebraic group $H$, $Y$ a scheme with $H$-action and $f:Y\rightarrow X$ a $\unicode[STIX]{x1D711}$-equivariant algebraic map, we write $f^{\ast }(L,\unicode[STIX]{x1D71A})$ for the line bundle $f^{\ast }L$ with the induced $H$-linearization. For a $G$-equivariant closed immersion $\unicode[STIX]{x1D704}:Y{\hookrightarrow}X$, we also write $(L,\unicode[STIX]{x1D71A})|_{Y}$ instead of $\unicode[STIX]{x1D704}^{\ast }(L,\unicode[STIX]{x1D71A})$.

Chern classes. With a line bundle $L$ on a projective variety, we associate a first Chern class $c_{1}(L)$ in the sense of [Reference FultonFul98]; we refer the reader to there for an exposition on the basic properties of Chern classes and the basic intersection theory we are using. We denote by $[X]$ the $k$-cycle class associated with an irreducible algebraic variety $X$ of dimension $k$ (in some ambient projective variety).

Complex points and analytifications. Throughout this article, we choose once and for all an embedding $\overline{\mathbb{Q}}{\hookrightarrow}\mathbb{C}$. For every $\overline{\mathbb{Q}}$-variety $X$, we consider its complex points $X(\mathbb{C})$ as a complex (analytic) space (see [GPR94] for this notion), the analytification of $X$. By our above convention on varieties, $X(\mathbb{C})$ is in fact reduced.

Complex spaces, differential forms, and currents. Let $S$ be a reduced complex (analytic) space. Recall that this means that $S$ is locally biholomorphic to a closed analytic subvariety $V$ in a complex domain $U\subset \mathbb{C}^{n}$. A function $f$ on $S$ is smooth (respectively holomorphic, meromorphic) if, for each such sufficiently small local chart, it is the restriction of a smooth (respectively holomorphic, meromorphic) function on $U$. In the same way, we use local charts to define plurisubharmonic functions on $S$ as restrictions.

Similarly, a smooth differential form $\unicode[STIX]{x1D714}$ on $S$ is a differential form on the smooth locus $S^{\text{sm}}$ of $S$ with the following additional property: $S$ can be covered by local charts $V\subset U\subset \mathbb{C}^{n}$ as above such that for each such chart the differential form $\unicode[STIX]{x1D714}|_{V^{\text{sm}}}$ is the restriction of a smooth differential form on $U$. There are also well-defined linear operators $d$ and $d^{c}=i/2\unicode[STIX]{x1D70B}(\overline{\unicode[STIX]{x2202}}-\unicode[STIX]{x2202})$ on the smooth differential forms on $S$. For each local chart $V\subset U\subset \mathbb{C}^{n}$, these are simply the restrictions of the operators of the same name on $\mathbb{C}^{n}$. A differential form $\unicode[STIX]{x1D714}$ on $S$ is called closed (respectively exact) if $d\unicode[STIX]{x1D714}=0$ (respectively there exists a differential form $\unicode[STIX]{x1D714}^{\prime }$ on $S$ such that $d\unicode[STIX]{x1D714}^{\prime }=\unicode[STIX]{x1D714}$). Recall that, by a classical result of Lelong [Reference LelongLel57], we can locally integrate smooth differential forms of maximal degree on complex spaces.

A line bundle $L$ over $S$ is a complex analytic space over $S$ such that $S$ can be covered by open subsets $U$ with $L|_{U}=U\times \mathbb{C}$. A smooth hermitian metric on $L$ is a smooth (in the above sense) function $\Vert \cdot \Vert :L\rightarrow \mathbb{R}$ whose restriction to each fiber over $S$ is a hermitian metric. With such a metric, we can define a Chern form $c_{1}(L,\Vert \cdot \Vert )$ in the usual way; if $\mathbf{s}:U\rightarrow L$ is a non-zero holomorphic section over some open subset $U\subset S$, we set $c_{1}(L,\Vert \cdot \Vert )|_{U}=dd^{c}(-\log \Vert \mathbf{s}\Vert )$. This construction yields a smooth differential form $c_{1}(L,\Vert \cdot \Vert )$ on $S$.

1 Preliminaries on semiabelian varieties

1.1 Basics

Recall that a semiabelian variety $G$ over $k$ is a connected smooth algebraic $k$-group that is the extension

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of an abelian variety $A$ by a torus $T$. Any homomorphism from a smooth linear algebraic group to an abelian variety is the zero homomorphism (see, e.g., [Reference ConradCon02, Lemma 2.3]). Therefore, any smooth linear algebraic subgroup of $G$ must be contained in $T$. It follows that $T$ is the maximal smooth linear algebraic subgroup of $G$. We hence call $T$the toric part of $G$ and $G\rightarrow G/T=A$ (or just $A$) the abelian quotient of $G$. For a semiabelian variety $G$ over $k$, we write $\unicode[STIX]{x1D702}_{G}$ for the Yoneda extension class in $\operatorname{Ext}_{k}^{1}(A,T)$ described by (

6

). Each homomorphism $\unicode[STIX]{x1D711}:A\rightarrow B$ (respectively $\unicode[STIX]{x1D711}:T\rightarrow S$) of abelian varieties (respectively tori) induces a pullback $\unicode[STIX]{x1D711}^{\ast }:\operatorname{Ext}_{k}^{1}(B,T)\rightarrow \operatorname{Ext}_{k}^{1}(A,T)$ (respectively a pushforward $\unicode[STIX]{x1D711}_{\ast }:\operatorname{Ext}_{k}^{1}(A,T)\rightarrow \operatorname{Ext}_{k}^{1}(A,S)$).

The Weil–Barsotti formula (see [Reference OortOor66, § III.18] or the appendix to [Reference Moret-Bailly and SzpiroMor81]) gives a canonical identification $\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m})=A^{\vee }(k)$. If $T$ is split (i.e., $T=\mathbb{G}_{m}^{t}$) we make frequent use of the identify $\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t})=\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m})^{t}=(A^{\vee })^{t}(k)$. The pullback

(7)$$\begin{eqnarray}\unicode[STIX]{x1D711}^{\ast \!}:\operatorname{Ext}_{k}^{1}(B,\mathbb{G}_{m}^{t})\longrightarrow \operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t})\end{eqnarray}$$

along a homomorphism $\unicode[STIX]{x1D711}:A\rightarrow B$ corresponds to the $t$-fold product $\unicode[STIX]{x1D711}^{\vee }\times \cdots \times \unicode[STIX]{x1D711}^{\vee }$ of the dual morphism $\unicode[STIX]{x1D711}^{\vee }:B^{\vee }\rightarrow A^{\vee }$. Pushforwards also allow a simple description. Indeed, let $\unicode[STIX]{x1D711}:\mathbb{G}_{m}^{t}\rightarrow \mathbb{G}_{m}^{t^{\prime }}$ be the homomorphism described by $\unicode[STIX]{x1D711}^{\ast }(Y_{v})=\prod _{u=1}^{t}X_{u}^{a_{uv}}$ in standard coordinates $X_{1},\ldots ,X_{t}$ (respectively $Y_{1},\ldots ,Y_{t_{2}}$) on $\mathbb{G}_{m}^{t}$ (respectively $\mathbb{G}_{m}^{t^{\prime }}$). Then, the pushforward

(8)$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\ast }:\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t})\longrightarrow \operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t^{\prime }})\end{eqnarray}$$

corresponds to the homomorphism $(A^{\vee })^{t}\rightarrow (A^{\vee })^{t^{\prime }}$ sending $(\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{t})$ to $(\sum _{u=1}^{t}a_{uv}\unicode[STIX]{x1D702}_{u})_{1\leqslant v\leqslant t^{\prime }}$.

As for abelian varieties, one calls two semiabelian varieties $G,G^{\prime }$ isogeneous if there exists an isogeny $G\rightarrow G^{\prime }$ (i.e., a surjective homomorphism $G\rightarrow G^{\prime }$ with finite kernel). Evidently, the multiplication-by-$n$ homomorphism $[n]$ of a semiabelian variety is an isogeny. As for abelian varieties, this yields an equivalence relation on semiabelian varieties.

Finally, we note that quotients as well as smooth algebraic subgroups of a semiabelian variety are themselves semiabelian varieties (cf. [Reference BrionBri17, Corollary 5.4.6]). In particular, the algebraic subgroups appearing in Theorem 2 are all semiabelian varieties because a well-known result of Cartier [SGA3, Corollaire $\text{VI}_{B}.1.6.1$] states that all algebraic $k$-groups are smooth if $k$ has characteristic 0.

1.2 Homomorphisms and quasi-homomorphisms

We recall the fundamental result on homomorphisms between semiabelian varieties.

Lemma 1. Let $G$ (respectively $G^{\prime }$) be a semiabelian variety over $k$ such that $A$ (respectively $A^{\prime }$) is the abelian quotient and $T$ (respectively $T^{\prime }$) is the toric part of $G$ (respectively $G^{\prime }$). For any homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ there exist unique homomorphisms $\unicode[STIX]{x1D711}_{\text{tor}}:T\rightarrow T^{\prime }$ and $\unicode[STIX]{x1D711}_{\text{ab}}:A\rightarrow A^{\prime }$ such that

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is a homomorphism of exact sequences. Furthermore, the induced map

(10)$$\begin{eqnarray}\operatorname{Hom}(G,G^{\prime })\longrightarrow \operatorname{Hom}(T,T^{\prime })\times \operatorname{Hom}(A,A^{\prime }),\quad \unicode[STIX]{x1D711}\longmapsto (\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}}),\end{eqnarray}$$

is an injective homomorphism with image

(11)$$\begin{eqnarray}\{(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}})\in \operatorname{Hom}(T,T^{\prime })\times \operatorname{Hom}(A,A^{\prime })\mid (\unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G}=(\unicode[STIX]{x1D711}_{a})^{\ast }\unicode[STIX]{x1D702}_{G^{\prime }}~\text{in}~\operatorname{Ext}_{k}^{1}(A,T^{\prime })\}.\end{eqnarray}$$

This lemma is well-known in the literature (see e.g. [Reference BertrandBer82] or [Reference VillarealVil08]). In fact, the existence of a pair $(\unicode[STIX]{x1D711}_{\text{ab}},\unicode[STIX]{x1D711}_{\text{tor}})$ follows directly from the fact that any map from a smooth linear algebraic group to an abelian variety is zero [Reference ConradCon02, Lemma 2.3] and its uniqueness is obvious. The remaining assertions can be shown by standard homological algebra in the category of commutative algebraic $k$-groups, which is abelian by a result of Grothendieck [SGA3, Théorème $\text{VI}_{A}.5.4.2$]. By the snake lemma, the homomorphism $\unicode[STIX]{x1D711}$ is surjective (respectively an isogeny) if and only if both $\unicode[STIX]{x1D711}_{\text{tor}}$ and $\unicode[STIX]{x1D711}_{\text{ab}}$ are surjective (respectively isogenies). All of this is contained in [Reference SerreSer88, ch. VII], described in a pre-schematic language.

In the situation of Lemma 1 we call $\unicode[STIX]{x1D711}_{\text{tor}}$ (respectively $\unicode[STIX]{x1D711}_{\text{ab}}$) the toric (respectively abelian) component of $\unicode[STIX]{x1D711}$. In addition, we say that $\unicode[STIX]{x1D711}$ is represented by the pair $(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}})$ and, conversely, that $(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}})$ represents $\unicode[STIX]{x1D711}$. We state an immediate consequence of Lemma 1 for later reference as a separate lemma.

Lemma 2. Assume that $k$ is algebraically closed. Let $G$ be a semiabelian variety over $k$ with abelian quotient $A$ and toric part $\mathbb{G}_{m}^{t}$. For every homomorphism $\unicode[STIX]{x1D711}_{\text{tor}}:\mathbb{G}_{m}^{t}\rightarrow \mathbb{G}_{m}^{t^{\prime }}$ and every isogeny $\unicode[STIX]{x1D711}_{\text{ab}}:A\rightarrow B$ there exists a semiabelian variety $G^{\prime }$ over $k$ and a homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ represented by $(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}})$.

Proof. Write $(\unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G}=(\unicode[STIX]{x1D702}_{1}^{\prime \prime },\ldots ,\unicode[STIX]{x1D702}_{t^{\prime }}^{\prime \prime })\in (A^{\vee })^{t^{\prime }}(k)$. Since $\unicode[STIX]{x1D711}_{\text{ab}}^{\vee }:B^{\vee }\rightarrow A^{\vee }$ is an isogeny (cf. [Reference MumfordMum70a, Remark (3) on p. 81]), there exist $\unicode[STIX]{x1D702}_{i}^{\prime }\in B^{\vee }(k)$ such that $\unicode[STIX]{x1D702}_{i}^{\prime \prime }=\unicode[STIX]{x1D711}_{\text{ab}}^{\vee }(\unicode[STIX]{x1D702}_{i}^{\prime })$. Let $G^{\prime }$ be the semiabelian variety described by $\unicode[STIX]{x1D702}_{G^{\prime }}=(\unicode[STIX]{x1D702}_{1}^{\prime },\ldots ,\unicode[STIX]{x1D702}_{t^{\prime }}^{\prime })\in (B^{\vee })^{t^{\prime }}(k)=\operatorname{Ext}_{k}^{1}(B,\mathbb{G}_{m}^{t^{\prime }})$. As $(\unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G}=(\unicode[STIX]{x1D711}_{\text{ab}})^{\ast }\unicode[STIX]{x1D702}_{G^{\prime }}$, there exists a homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ representing $(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{ab}})$ by Lemma 1.◻

We need to work also with quasi-homomorphisms of semiabelian varieties. First of all, note that for any semiabelian varieties $G$ and $G^{\prime }$ the $\mathbb{Z}$-module $\operatorname{Hom}(G,G^{\prime })$ of homomorphisms is torsion-free. Indeed, this is true for both tori and abelian varieties so that we may infer the general case from Lemma 1. By quasi-homomorphisms we mean the elements of $\operatorname{Hom}_{\mathbb{Q}}(G,G^{\prime })=\operatorname{Hom}(G,G^{\prime })\otimes _{\mathbb{Z}}\mathbb{Q}$. In analogy to actual homomorphisms, each quasi-homomorphism is denoted in the form $\unicode[STIX]{x1D719}:G\rightarrow _{\mathbb{Q}}G^{\prime }$. By tensoring (10) with $\mathbb{Q}$, we can also associate with each quasi-homomorphism $\unicode[STIX]{x1D719}:G\rightarrow _{\mathbb{Q}}G^{\prime }$ uniquely a toric component $\unicode[STIX]{x1D719}_{\text{tor}}:T\rightarrow _{\mathbb{Q}}T^{\prime }$ and an abelian component $\unicode[STIX]{x1D719}_{\text{ab}}:A\rightarrow _{\mathbb{Q}}A^{\prime }$. With each quasi-homomorphism $\unicode[STIX]{x1D719}:G\rightarrow _{\mathbb{Q}}G^{\prime }$ we can associate a ‘kernel up to torsion’ $\ker (\unicode[STIX]{x1D719})+\operatorname{Tors}(G)$ in the following way: let $n$ be a denominator of $\unicode[STIX]{x1D719}$ (i.e., $n\cdot \unicode[STIX]{x1D719}\in \operatorname{Hom}(G,G^{\prime })$) and set $\ker (\unicode[STIX]{x1D719})+\operatorname{Tors}(G)=\ker (n\cdot \unicode[STIX]{x1D719})+\operatorname{Tors}(G)$. Additionally, we say that $\unicode[STIX]{x1D719}$ is surjective if $n\cdot \unicode[STIX]{x1D719}$ is. These definitions are clearly independent of the chosen denominator $n$. Albeit a quasi-homomorphism $\unicode[STIX]{x1D719}_{\text{ab}}\in \operatorname{Hom}_{\mathbb{Q}}(A^{\prime },A)$ does not induce a pullback as in (7), it gives rise to a homomorphism

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D719}_{\text{ab}})^{\ast ,\mathbb{Q}}:\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t})_{\mathbb{Q}}=(A^{\vee }(k)\otimes _{\mathbb{Z}}\mathbb{Q})^{t}\longrightarrow ((A^{\prime })^{\vee }(k)\otimes _{\mathbb{Z}}\mathbb{Q})^{t}=\operatorname{Ext}_{k}^{1}(A^{\prime },\mathbb{G}_{m}^{t})_{\mathbb{Q}}. & & \displaystyle \nonumber\end{eqnarray}$$

Similarly, a quasi-homomorphism $\unicode[STIX]{x1D719}_{t}\in \operatorname{Hom}_{\mathbb{Q}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ induces a homomorphism

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D719}_{\text{tor}})_{\ast ,\mathbb{Q}}:\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t})_{\mathbb{Q}}=(A^{\vee }(k)\otimes _{\mathbb{Z}}\mathbb{Q})^{t}\longrightarrow (A^{\vee }(k)\otimes _{\mathbb{Z}}\mathbb{Q})^{t^{\prime }}=\operatorname{Ext}_{k}^{1}(A,\mathbb{G}_{m}^{t^{\prime }})_{\mathbb{Q}} & & \displaystyle \nonumber\end{eqnarray}$$

in place of (8).

2 Compactifications

To compactify semiabelian varieties we use a well-known construction proposed by Serre (cf. [Reference SerreSer01, § 3.2] and Serre’s appendix in [Reference WaldschmidtWal79]). Let $G$ be a semiabelian variety over $k$ with split toric part $T=\mathbb{G}_{m}^{t}$ and abelian quotient $A$.Footnote ¹ Furthermore, let $\overline{T}$ be a $T$-equivariant compactification of $T$. This means that we are given a dense open immersion $T{\hookrightarrow}\overline{T}$ with $\overline{T}$ a proper $k$-variety and that there is an extension $\cdot _{\overline{T}}:T\times \overline{T}\rightarrow \overline{T}$ of the group law $\cdot _{T}:T\times T\rightarrow T$. We endow $G\times _{k}\overline{T}$ with the $T$-action given by

(12)$$\begin{eqnarray}t\cdot (g,\overline{t})=(t\cdot _{G}g,t^{-1}\cdot _{\overline{T}}\overline{t}),\quad t\in T(S),\overline{t}\in \overline{T}(S),g\in G(S),\end{eqnarray}$$

on $S$-points. It is well-known that the (categorical) quotient $G_{\overline{T}}=G\times _{k}\overline{T}/T$ in the category of $k$-schemes exists and is a proper $k$-variety (see, e.g., [Reference Faltings and WüstholzFW84, Reference Knop and LangeKL84]). In fact, there exists a (finite) Zariski covering $\{U_{i}\}$ of $A$ together with compatible $T$-equivariant trivializations $\unicode[STIX]{x1D719}_{i}:U_{i}\times _{A}G\rightarrow U_{i}\times _{k}T$ over each $U_{i}$. The isomorphisms

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{j}\circ \unicode[STIX]{x1D719}_{i}^{-1}|_{(U_{i}\cap U_{j})\times _{k}T}:(U_{i}\cap U_{j})\times _{k}T\longrightarrow (U_{i}\cap U_{j})\times _{k}T & & \displaystyle \nonumber\end{eqnarray}$$

determine sections $t_{ij}\in \unicode[STIX]{x1D6E4}(U_{i}\cap U_{j},T)$. The variety $G$ can be described as a gluing of these trivial $T$-torsors by means of the Čech cocycle $\{t_{ij}\}\in \check{\text{H}}^{1}(\{U_{i}\},T)$. (In fact, $\{t_{ij}\}$ is also the cocycle describing $\unicode[STIX]{x1D702}_{G}\in (A^{\vee })^{t}(k)=\operatorname{Ext}_{k}^{1}(A,T)$ in the Barsotti–Weil formula.) Via the extension $\cdot _{\overline{T}}$ of the group law $\cdot _{T}$, the same Čech cocycle $\{t_{ij}\}$ determines also a gluing of the $k$-varieties $U_{i}\times _{k}\overline{T}$, yielding a proper $k$-variety $X$ and a projection $\overline{\unicode[STIX]{x1D70B}}:G_{\overline{T}}\rightarrow A$. There is a canonical map $p:G\times _{k}\overline{T}\rightarrow G_{\overline{T}}$ over $A$ such that its base change

$$\begin{eqnarray}\displaystyle p\times _{G_{\overline{T}}}(U_{i}\times _{k}\overline{T}):(U_{i}\times _{A}G)\times _{k}\overline{T}\longrightarrow U_{i}\times _{k}\overline{T} & & \displaystyle \nonumber\end{eqnarray}$$

coincides with the action

$$\begin{eqnarray}\displaystyle U_{i}{\times _{k}\cdot }_{\overline{T}}:U_{i}\times _{k}(T\times _{k}\overline{T})\longrightarrow U_{i}\times _{k}\overline{T} & & \displaystyle \nonumber\end{eqnarray}$$

under the identification $U_{i}\times _{A}G=U_{i}\times _{k}T$ described by $\unicode[STIX]{x1D719}_{i}$. Neither $G_{\overline{T}}$ nor $p$ depends on the Zariski covering $\{U_{i}\}$ as the above is compatible with any further refinement. In addition, the $G$-action given by the group law $+_{G}:G\times G\rightarrow G$ extends uniquely to an action $G\times G_{\overline{T}}\rightarrow G_{\overline{T}}$.

If $(M,\unicode[STIX]{x1D71A}:T\times _{k}M\rightarrow M)$ is a $T$-linearized line bundle on $\overline{T}$, we endow $G\times _{k}M$ with a $T$-action in a way similar to (12) and form the quotient $G(M,\unicode[STIX]{x1D71A})=G\times _{k}M/T$. Repeating the above procedure, it is easy to infer that $G(M,\unicode[STIX]{x1D71A})$ is a line bundle over $G_{\overline{T}}$. One checks also a compatibility $G(M\otimes M^{\prime },\unicode[STIX]{x1D71A}\otimes \unicode[STIX]{x1D71A}^{\prime })\approx G(M,\unicode[STIX]{x1D71A})\otimes G(M^{\prime },\unicode[STIX]{x1D71A}^{\prime })$ with tensor products.

Proof. By [Reference LazarsfeldLaz04, Example 1.2.22], the line bundle $M^{\otimes 3k}$ is normally generated for sufficiently large integers $k$. This allows us to apply [Reference Knop and LangeKL84, Theorem 3.5], which yields that

$$\begin{eqnarray}\displaystyle G(M,\unicode[STIX]{x1D71A})^{\otimes 3k}\otimes \overline{\unicode[STIX]{x1D70B}}^{\ast }N^{\otimes 3k}=(G(M,\unicode[STIX]{x1D71A})\otimes \overline{\unicode[STIX]{x1D70B}}^{\ast }N)^{\otimes 3k} & & \displaystyle \nonumber\end{eqnarray}$$

is normally generatedFootnote ² and hence very ample (cf. [Reference MumfordMum70b, p. 38] for this final implication).Footnote ³

For nefness, let $C$ be a proper curve in $G_{\overline{T}}$. We already know that

$$\begin{eqnarray}\displaystyle G(M,\unicode[STIX]{x1D71A})^{\otimes k}\otimes (\overline{\unicode[STIX]{x1D70B}}^{\ast }N)=G(M^{\otimes k},\unicode[STIX]{x1D71A}^{\otimes k})\otimes (\overline{\unicode[STIX]{x1D70B}}^{\ast }N) & & \displaystyle \nonumber\end{eqnarray}$$

is ample for any integer $k\geqslant 1$. Hence, the degree of the 0-cycle class

$$\begin{eqnarray}\displaystyle kc_{1}(G(M,\unicode[STIX]{x1D71A}))\cap [C]+c_{1}(\overline{\unicode[STIX]{x1D70B}}^{\ast }N)\cap [C] & & \displaystyle \nonumber\end{eqnarray}$$

is positive for any $k$ (see e.g. [Reference FultonFul98, Lemma 12.1]). Dividing by $k$ and taking the limit $k\rightarrow \infty$, we obtain

$$\begin{eqnarray}\displaystyle \deg (c_{1}(G(M,\unicode[STIX]{x1D71A}))\cap [C])\geqslant 0, & & \displaystyle \nonumber\end{eqnarray}$$

which means that $G(M,\unicode[STIX]{x1D71A})$ is nef.◻

We are interested in the behavior of the above constructions with regard to homomorphisms. For this, let $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ be a homomorphism of semiabelian varieties with toric component $\unicode[STIX]{x1D711}_{\text{tor}}:T\rightarrow T^{\prime }$ as in (9). In addition, let $\overline{T}$ (respectively $\overline{T}^{\prime }$) be a $T$-equivariant (respectively $T^{\prime }$-equivariant) compactification of $T$ (respectively $T^{\prime }$) so that $\unicode[STIX]{x1D711}_{\text{tor}}$ extends to a $\unicode[STIX]{x1D711}_{\text{tor}}$-equivariant map $\overline{\unicode[STIX]{x1D711}}_{\text{tor}}:\overline{T}\rightarrow \overline{T}^{\prime }$. Endowing $G\times _{k}\overline{T}$ (respectively $G^{\prime }\times _{k}\overline{T}^{\prime }$) with a $T$-action (respectively $T^{\prime }$-action) as in (12), the $\unicode[STIX]{x1D711}_{\text{tor}}$-equivariant map $\unicode[STIX]{x1D711}\times _{k}\overline{\unicode[STIX]{x1D711}}_{\text{tor}}:G\times _{k}\overline{T}\rightarrow G^{\prime }\times _{k}\overline{T}^{\prime }$ induces a map $\overline{\unicode[STIX]{x1D711}}:G_{\overline{T}}\rightarrow G_{\overline{T}^{\prime }}^{\prime }$. Now let $(M,\unicode[STIX]{x1D71A})$ be a $T^{\prime }$-linearized line bundle on $\overline{T}^{\prime }$. We have $\overline{\unicode[STIX]{x1D711}}^{\ast }G^{\prime }(M,\unicode[STIX]{x1D71A})\approx G(\overline{\unicode[STIX]{x1D711}}_{\text{tor}}^{\ast }(M,\unicode[STIX]{x1D71A}))$; for the line bundle $G\times _{k}\overline{\unicode[STIX]{x1D711}}_{\text{tor}}^{\ast }M$ is the pullback of $G^{\prime }\times _{k}M$ along $\unicode[STIX]{x1D711}\times _{k}\overline{\unicode[STIX]{x1D711}}_{\text{tor}}$ and the induced map $G\times _{k}\overline{\unicode[STIX]{x1D711}}_{\text{tor}}^{\ast }M\rightarrow G^{\prime }\times _{k}M$ is $\unicode[STIX]{x1D711}_{\text{tor}}$-equivariant.

In these considerations, the case where $\unicode[STIX]{x1D711}$ is the multiplication-by-$n$ homomorphism $[n]_{G}$ for a semiabelian variety $G$ with toric part $T$ is of particular importance. To avoid pathologies, some further technical requirements on both the $T$-equivariant compactification $\overline{T}$and the $T$-linearizable line bundle $M$should be met. First, an extension of $[n]_{T}$ to a map $[n]_{\overline{T}}:\overline{T}\rightarrow \overline{T}$ should exist for each integer $n$. (Such an extension is unique by separatedness.) Under this condition, there is an extension $\overline{\unicode[STIX]{x1D711}}=[n]_{\overline{G}}:G_{\overline{T}}\rightarrow G_{\overline{T}}$ of $[n]_{G}$ by the last paragraph. Second, there should be a $T$-equivariant isomorphism $[n]_{\overline{T}}^{\ast }M\approx M^{\otimes |n|}$. If this is satisfied, the last assertion of the preceding paragraph specializes to $[n]_{\overline{G}}^{\ast }G(M,\unicode[STIX]{x1D71A})\approx G([n]_{\overline{T}}^{\ast }(M,\unicode[STIX]{x1D71A}))\approx G((M,\unicode[STIX]{x1D71A})^{\otimes |n|})\approx G(M,\unicode[STIX]{x1D71A})^{\otimes |n|}$.

Before introducing the two types of compactifications to be employed in our proof of Theorem 2, we recall a further notion. Let $T$ be a torus with $T$-equivariant compactification $\overline{T}$. Pulling meromorphic functions back fabricates a $T$-linearization of ${\mathcal{K}}_{\overline{T}}$. Denote by $\operatorname{pr}_{2}:T\times \overline{T}\rightarrow \overline{T}$ the projection and by $\unicode[STIX]{x1D70E}:T\times \overline{T}\rightarrow \overline{T}$ the $T$-action on $\overline{T}$. A Cartier divisor $D$ on $\overline{T}$ is called $T$-invariant if the pullbacks $\operatorname{pr}_{2}^{\ast }D$ and $\unicode[STIX]{x1D70E}^{\ast }D$ are equal. In this case, $D$ gives rise to a $T$-invariant invertible subsheaf ${\mathcal{O}}(D)$ of ${\mathcal{K}}_{\overline{T}}$. Hence, there is an induced $T$-linearization on ${\mathcal{O}}(D)$. We always mean this linearization when associating a $T$-linearized line bundle $(L(D),\unicode[STIX]{x1D71A}_{D})$ with a $T$-invariant Cartier divisor $D$. Note that this $T$-linearization on ${\mathcal{O}}(D)$ is uniquely characterized by the fact that its rational section $1\in {\mathcal{K}}_{\overline{T}}(\overline{T})$ is $T$-invariant.

Any $T$-invariant Cartier divisor $D$ on $\overline{T}$ yields naturally a Cartier divisor on $G_{\overline{T}}$. Indeed, assume that $D$ is represented by $(V_{j},f_{j})$ with Zariski opens $V_{j}\subset \overline{T}$. For each Zariski open $U_{i}\subset A$ this gives a Cartier divisor on $U_{i}\times _{A}G_{\overline{T}}=U_{i}\times _{k}\overline{T}$ that is represented by $(U_{i}\times _{k}V_{j},f_{j}\circ \operatorname{pr}_{2})$. By $T$-invariance, these Cartier divisors glue together to a Cartier divisor $G(D)$ on $G_{\overline{T}}$. Furthermore, it is easy to see that $L(G(D))$ is isomorphic to $G(L(D),\unicode[STIX]{x1D71A}_{D})$.

Construction 4 $(D_{t},(M_{t},\unicode[STIX]{x1D71A}_{t}))$.

The torus $\mathbb{G}_{m}=\text{Spec}(k[X,X^{-1}])$ has a $\mathbb{G}_{m}$-equivariant compactification $\unicode[STIX]{x1D704}_{1}:\mathbb{G}_{m}{\hookrightarrow}\mathbb{P}^{1}=\text{Proj}(k[Z_{1},Z_{2}])$ with $\unicode[STIX]{x1D704}_{1}^{\ast }(Z_{2}/Z_{1})=X$. There is an extension $[n]_{\mathbb{P}^{1}}:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ of $[n]_{\mathbb{G}_{m}}:\mathbb{G}_{m}\rightarrow \mathbb{G}_{m}$. Let $E_{0}$ (respectively $E_{\infty }$) be the $\mathbb{G}_{m}$-invariant Cartier divisor on $\mathbb{P}^{1}$ represented by

$$\begin{eqnarray}\displaystyle (D(Z_{1}),Z_{2}/Z_{1})\quad \text{and}\quad (D(Z_{2}),1)\quad (\text{respectively}~(D(Z_{1}),1)~\text{and}~(D(Z_{2}),Z_{1}/Z_{2})). & & \displaystyle \nonumber\end{eqnarray}$$

For the torus $T=\mathbb{G}_{m}^{t}$, the map $\unicode[STIX]{x1D704}_{t}=\unicode[STIX]{x1D704}_{1}\times \cdots \times \unicode[STIX]{x1D704}_{1}:\mathbb{G}_{m}^{t}{\hookrightarrow}\overline{T}=(\mathbb{P}^{1})^{t}$ gives a $T$-equivariant compactification. Denoting by $\operatorname{pr}_{i}:\overline{T}=(\mathbb{P}^{1})^{t}\rightarrow \mathbb{P}^{1}$ the projection to the $i\text{th}$ component, we set $D_{t}=\sum _{1\leqslant i\leqslant t}\operatorname{pr}_{i}^{\ast }(E_{0}+E_{\infty })$ and ${\mathcal{M}}_{t}={\mathcal{O}}(D_{t})$. By the above, there is a natural $T$-linearization $\unicode[STIX]{x1D71A}_{t}=\unicode[STIX]{x1D71A}_{D_{t}}$ on the associated line bundle $M_{t}$ that acts trivially on its global section $1\in {\mathcal{M}}_{t}(\overline{T})$. Furthermore, from the evident identity $[n]_{\overline{T}}^{\ast }D_{t}=|n|\cdot D_{t}$ of Cartier divisors we obtain an identity $[n]_{\overline{T}}^{\ast }{\mathcal{M}}_{t}={\mathcal{M}}_{t}^{\otimes |n|}$ of ${\mathcal{O}}_{X}$-submodules of ${\mathcal{K}}_{\overline{T}}$ so that $[n]_{\overline{T}}^{\ast }(M_{t},\unicode[STIX]{x1D71A}_{t})=(M_{t},\unicode[STIX]{x1D71A}_{t})^{\otimes |n|}$.

Construction 6 $(G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}},M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}},\unicode[STIX]{x1D70B}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}})$.

Assume given a semiabelian variety $G$ with split toric part $\mathbb{G}_{m}^{t}$ and abelian quotient $\unicode[STIX]{x1D70B}:G\rightarrow A$ as well as a homomorphism $\unicode[STIX]{x1D711}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$. Let $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})\subset \mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }}$ be the graph of $\unicode[STIX]{x1D711}_{\text{tor}}$ and $\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$ its Zariski closure in the $(\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }})$-equivariant compactification $(\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t^{\prime }}$.Footnote ⁴ The projection to $\mathbb{G}_{m}^{t}$ induces an identification $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})=\mathbb{G}_{m}^{t}$. In this way, $\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$ can be considered as a $\mathbb{G}_{m}^{t}$-equivariant compactification of $\mathbb{G}_{m}^{t}$. As $[n]_{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$ is just the restriction of $[n]_{\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }}}$, it clearly extends to $\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$ because $[n]_{\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }}}$ extends to $(\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t^{\prime }}$. Therefore, there is an extension of $[n]_{G}$ to the ‘graph compactification’ $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$. To fix notations, we record the following self-explanatory commutative diagram.

(13)

Construction 4 gives a $\mathbb{G}_{m}^{t^{\prime }}$-linearized line bundle $(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})$ on $(\mathbb{P}^{1})^{t^{\prime }}$. Its $(\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }})$-linearized pullback $\operatorname{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})$ yields a line bundle $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}=G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\operatorname{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}})$ on $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$. Setting $\unicode[STIX]{x1D711}_{\text{tor}}=\operatorname{id}_{\mathbb{G}_{m}^{t}}$, this construction specializes to Construction 5 above (i.e., $\overline{G}\approx G_{\overline{\unicode[STIX]{x1D6E4}(\operatorname{id}_{\mathbb{ G}_{m}^{t}})}}$ with compatible $M_{\overline{G}}\approx M_{\overline{\unicode[STIX]{x1D6E4}(\operatorname{id}_{\mathbb{ G}_{m}^{t}})}}$).

For any non-zero integer $n$, we can relate $(G_{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})},M_{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})})$ with $(G_{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})},M_{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})})$. For this, we define $G^{\prime }$ and $G^{\prime \prime }$ to be the semiabelian varieties such that $\unicode[STIX]{x1D702}_{G^{\prime }}=(\unicode[STIX]{x1D702}_{G},(\unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G})$ and $\unicode[STIX]{x1D702}_{G^{\prime \prime }}=(\unicode[STIX]{x1D702}_{G},(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G})$ in $\operatorname{Ext}^{1}(A,\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t^{\prime }})$. The equivariant closed immersions $\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$, $\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}\subset (\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t^{\prime }}$ yield closed immersions $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\subset \overline{G}^{\prime }$ and $G_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}\subset \overline{G}^{\prime \prime }$. In addition, the finite morphism $[1]_{(\mathbb{P}^{1})^{t}}\times [n]_{(\mathbb{P}^{1})^{t^{\prime }}}$ yields a finite map $\unicode[STIX]{x1D717}_{n}:\overline{G}^{\prime }\rightarrow \overline{G}^{\prime \prime }$. As $[1]_{(\mathbb{P}^{1})^{t}}\times [n]_{(\mathbb{P}^{1})^{t^{\prime }}}$ restricts to a $\mathbb{G}_{m}^{t}$-equivariant birational map $\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}\rightarrow \overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}$, $\unicode[STIX]{x1D717}_{n}$ restricts to a birational map $\unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow G_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}$. Furthermore,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}^{\ast }M_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}\approx \unicode[STIX]{x1D717}_{n}^{\ast }\overline{G}^{\prime \prime }(\operatorname{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }}))|_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}\approx \overline{G}^{\prime }(\operatorname{pr}_{2}^{\ast }(M_{t},\unicode[STIX]{x1D71A}_{t})^{\otimes |n|})|_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\approx M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}^{\otimes |n|}. & & \displaystyle \nonumber\end{eqnarray}$$

In addition, there are the evident relations $\unicode[STIX]{x1D70B}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}=\unicode[STIX]{x1D70B}_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}\circ \unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}$, $q_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}=q_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}\circ \unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}$ and $\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}=\unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}\circ \unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$.

Construction 7 $(\overline{\unicode[STIX]{x1D711}}:G_{\overline{\unicode[STIX]{x1D711}_{\text{tor}}}}\rightarrow \overline{G}^{\prime })$.

We describe a subcase of Construction 6 for later reference, enlarging also the commutative diagram (13). In this case, we start with a homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ of semiabelian varieties with split toric parts $T=\mathbb{G}_{m}^{t}$ and $T^{\prime }=\mathbb{G}_{m}^{t^{\prime }}$. We obtain a compactification $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ from Construction 6. Furthermore, the homomorphism $\unicode[STIX]{x1D711}$ induces now an even larger commutative diagram

(14)

such that there is a decomposition $\unicode[STIX]{x1D711}=\overline{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}$; the map $\overline{\unicode[STIX]{x1D711}}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow \overline{G}^{\prime }$ here arises naturally as follows: the toric part $\unicode[STIX]{x1D711}_{\text{tor}}:\mathbb{G}_{m}^{t}\rightarrow \mathbb{G}_{m}^{t^{\prime }}$ of $\unicode[STIX]{x1D711}$ extends to a $\unicode[STIX]{x1D711}_{\text{tor}}$-equivariant map $\overline{\unicode[STIX]{x1D711}}_{\text{tor}}:\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}\rightarrow (\mathbb{P}^{1})^{t^{\prime }}$, which is just a restriction of $\text{pr}_{2}:(\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t^{\prime }}\rightarrow (\mathbb{P}^{1})^{t^{\prime }}$. As described above, this induces a corresponding extension $\overline{\unicode[STIX]{x1D711}}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow \overline{G}^{\prime }$ of $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$.

In addition, each line bundle $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ is a pullback $\overline{\unicode[STIX]{x1D711}}^{\ast }M_{\overline{G}^{\prime }}$ for some homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ of semiabelian varieties. In fact, we can take $\unicode[STIX]{x1D702}_{G^{\prime }}=(\unicode[STIX]{x1D711}_{\text{tor}})_{\ast }\unicode[STIX]{x1D702}_{G}\in \operatorname{Ext}^{1}(A,\mathbb{G}_{m}^{t^{\prime }})$ and the homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ represented by $(\unicode[STIX]{x1D711}_{\text{tor}},\operatorname{id}_{A})$.

3 Heights

We consistently work with (logarithmic) Weil heights and refer to [Reference Hindry and SilvermanHS00, Theorem B.3.6] for the main features of Weil’s height machinery. In short, it provides for each line bundle $L$ on a projective $\overline{\mathbb{Q}}$-variety $X$ a class of height functions $h_{L}:X(\overline{\mathbb{Q}})\rightarrow \mathbb{R}$ such that any two height functions attached to $(X,L)$ differ by a globally bounded function on $X(\overline{\mathbb{Q}})$.

Let $G$ be a semiabelian variety over $\overline{\mathbb{Q}}$ with toric part $T$ and abelian quotient $\unicode[STIX]{x1D70B}:G\rightarrow A$. Assume also given a $T$-equivariant compactification $\overline{T}$of the torus $T$and a $T$-linearized line bundle $(M,\unicode[STIX]{x1D71A})$on $\overline{T}$such that $[n]_{T}$extends to $[n]_{\overline{T}}:\overline{T}\rightarrow \overline{T}$and that there is an isomorphism $[n]_{\overline{T}}^{\ast }(M,\unicode[STIX]{x1D71A})\approx (M,\unicode[STIX]{x1D71A})^{\otimes n}$. For our purposes, these conditions on $\overline{T}$ and $(M,\unicode[STIX]{x1D71A})$ are always satisfied. Additionally, we choose a symmetric line bundle $N$ on $A$. We furnish $G_{\overline{T}}$ with the line bundle $L=G(M,\unicode[STIX]{x1D71A})\otimes \overline{\unicode[STIX]{x1D70B}}^{\ast }N$, which is ample if both $M$ and $N$ are ample (Lemma 3). Weil’s height machinery supplies us with some height function $h_{L}:G_{\overline{T}}(\overline{\mathbb{Q}})\rightarrow \mathbb{R}$. The function $h_{L}$ is neither unique nor does it enjoy homogeneity properties like the Néron–Tate height of a symmetric line bundle on an abelian variety. However, the following lemma remedies this partially. We call a $T$-linearized line bundle $T$-effective if it has a $T$-invariant non-zero global section.

It is natural to work with the unique $\widehat{h}_{L}=\widehat{h}_{G(M,\unicode[STIX]{x1D71A})}+\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}$ instead of a non-canonical Weil height $h_{L}$. By (a) of the above theorem, their difference is globally bounded on $G(\overline{\mathbb{Q}})$. As for abelian varieties, the zero set of $\widehat{h}_{L}$ coincides with the torsion points of $G$ if both $M$ and $N$ are ample and $(M,\unicode[STIX]{x1D71A})$ is $T$-effective.

The assumption of $T$-effectivity in (d) cannot be relaxed to mere effectivity. In fact, assume that $T=\mathbb{G}_{m}^{t}$ and let $Q_{i}$, $1\leqslant i\leqslant t$, be the line bundles on $A$ such that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{G}=(Q_{1},\ldots ,Q_{t})\in A^{\vee }(k)^{t}=\operatorname{Ext}_{k}^{1}(A,T). & & \displaystyle \nonumber\end{eqnarray}$$

For $\unicode[STIX]{x1D71A}$ running through all possible $T$-linearizations of the trivial line bundle $\mathbb{A}_{\overline{T}}^{1}$, the line bundle $G(\mathbb{A}_{\overline{T}}^{1},\unicode[STIX]{x1D71A})$ runs through $\overline{\unicode[STIX]{x1D70B}}^{\ast }(Q_{1}^{k_{i}}\otimes \cdots \otimes Q_{t}^{k_{t}})$ for arbitrary integers $k_{i}$, as a comparison of Čech cocycles shows. Except for this caveat, we do not need this and leave the verification to the interested reader.

Proof. (a, b) The first two assertions of the lemma as well as uniqueness can be inferred directly from [Reference Hindry and SilvermanHS00, Theorem B.4.1] applied to $G(M,\unicode[STIX]{x1D71A})$ (respectively $N$). Indeed, $[n]_{\overline{G}}^{\ast }G(M,\unicode[STIX]{x1D71A})\approx G(M,\unicode[STIX]{x1D71A})^{\otimes n}$ by our assumption (compare § 2) and $[n]_{\overline{G}}^{\ast }\overline{\unicode[STIX]{x1D70B}}^{\ast }N\approx \overline{\unicode[STIX]{x1D70B}}^{\ast }[n]_{A}^{\ast }N\approx \overline{\unicode[STIX]{x1D70B}}^{\ast }N^{\otimes n^{2}}$ since $N$ is symmetric. (The result in [Reference Hindry and SilvermanHS00] is stated for divisor classes on smooth varieties but it is also true for line bundles on general varieties with exactly the same proof. The reader may compare also [Reference Bombieri and GublerBG06, Lemma 9.2.4].)

(c) As $G(M\otimes M^{\prime },\unicode[STIX]{x1D71A}\otimes \unicode[STIX]{x1D71A}^{\prime })\approx G(M,\unicode[STIX]{x1D71A})\otimes G(M^{\prime },\unicode[STIX]{x1D71A}^{\prime })$, the global boundedness of

$$\begin{eqnarray}\displaystyle |h_{G(M\otimes M^{\prime },\unicode[STIX]{x1D71A}\otimes \unicode[STIX]{x1D71A}^{\prime })}-h_{G(M,\unicode[STIX]{x1D71A})}-h_{G(M^{\prime },\unicode[STIX]{x1D71A}^{\prime })}| & & \displaystyle \nonumber\end{eqnarray}$$

is a standard property of the Weil height. As (a) and (b) already characterize $\widehat{h}_{G(M\otimes M^{\prime },\unicode[STIX]{x1D71A}\otimes \unicode[STIX]{x1D71A}^{\prime })}$ uniquely, we infer the first equality in (c). The second one follows similarly.

(d) Similarly, one observes that $\widehat{h}_{G(M,\unicode[STIX]{x1D71A})}$ (respectively $\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}$) is non-negative if $h_{G(M,\unicode[STIX]{x1D71A})}$ (respectively $h_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}$) is bounded from below on $G(\overline{\mathbb{Q}})$ (respectively $G_{\overline{T}}(\overline{\mathbb{Q}})$). For the height $h_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}$, this is true because the ampleness of $N$ implies that $N$ and hence $\overline{\unicode[STIX]{x1D70B}}^{\ast }N$ has empty base locus. By assumption, we have a $T$-invariant non-zero global section $s:\overline{T}\rightarrow M$. This gives rise to local sections

$$\begin{eqnarray}\displaystyle s_{i}^{\prime }=U_{i}\times _{k}s:U_{i}\times _{A}G_{\overline{T}}=U_{i}\times _{k}\overline{T}\rightarrow U_{i}\times _{k}M=U_{i}\times _{A}G(M,\unicode[STIX]{x1D71A}). & & \displaystyle \nonumber\end{eqnarray}$$

Due to the $T$-invariance of $s$, the sections $s_{i}^{\prime }$ glue together to a non-zero global section $s^{\prime }$ of $G(M,\unicode[STIX]{x1D71A})$. Furthermore, $T$-invariance guarantees that $s_{x}$ generates $M_{x}$ for every $x\in T(\overline{\mathbb{Q}})$. We infer that $s_{x}^{\prime }$ generates $G(M,\unicode[STIX]{x1D71A})$ for every $x\in G(\overline{\mathbb{Q}})$. Therefore the base locus of $G(M,\unicode[STIX]{x1D71A})$ is contained in $G_{\overline{T}}\setminus G$ and $h_{G(M,\unicode[STIX]{x1D71A})}|_{G(\overline{\mathbb{Q}})}$ is bounded from below.◻

In addition, we have a good functorial behavior of the heights $\widehat{h}_{G(M,\unicode[STIX]{x1D71A})}$ and $\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}$. To state precisely what this means, let $G$ (respectively $G^{\prime }$) be a semiabelian variety over $\overline{\mathbb{Q}}$ with toric part $T$ (respectively $T^{\prime }$) and abelian quotient $A$ (respectively $A^{\prime }$). Take furthermore equivariant compactifications $\overline{T}$ and $\overline{T}^{\prime }$ so that $\unicode[STIX]{x1D711}_{\text{tor}}:T\rightarrow T^{\prime }$ extends to a $\unicode[STIX]{x1D711}_{\text{tor}}$-equivariant map $\overline{\unicode[STIX]{x1D711}}_{\text{tor}}:\overline{T}\rightarrow \overline{T}^{\prime }$. In this situation, we consider a $T^{\prime }$-linearized line bundle $(M,\unicode[STIX]{x1D71A})$ on $\overline{T}^{\prime }$ such that there is a $T^{\prime }$-equivariant isomorphism $[n]_{\overline{T}^{\prime }}^{\ast }(M,\unicode[STIX]{x1D71A})\approx (M,\unicode[STIX]{x1D71A})^{\otimes n}$. We also take a symmetric ample line bundle $N$ on $A^{\prime }$.

Lemma 9. In the situation described in the above paragraph, let $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ be a homomorphism with toric (respectively abelian) component $\unicode[STIX]{x1D711}_{\text{tor}}$ (respectively $\unicode[STIX]{x1D711}_{\text{ab}}$). For every $x\in G(\overline{\mathbb{Q}})$, we have then

$$\begin{eqnarray}\displaystyle \widehat{h}_{G^{\prime }(M,\unicode[STIX]{x1D71A})}(\unicode[STIX]{x1D711}(x))=\widehat{h}_{G(\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }(M,\unicode[STIX]{x1D71A}))}(x)\quad \text{and}\quad \widehat{h}_{(\overline{\unicode[STIX]{x1D70B}}^{\prime })^{\ast }N}(\unicode[STIX]{x1D711}(x))=\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }(\unicode[STIX]{x1D711}_{\text{ab}}^{\ast }N)}(x). & & \displaystyle \nonumber\end{eqnarray}$$

We note a further addendum to Lemma 8, which is specifically related to the line bundles $M_{\overline{G}}$ and $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$.

Fix again a semiabelian variety $G$ over $\overline{\mathbb{Q}}$ with toric part $\mathbb{G}_{m}^{t}$ and abelian quotient $\unicode[STIX]{x1D70B}:G\rightarrow A$. Furthermore, let $\overline{G}$ be a compactification of $G$ and $\overline{\unicode[STIX]{x1D70B}}:\overline{G}\rightarrow A$ its abelian quotient as in Construction 5. We want to estimate the difference between $\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}$ and $\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}}}$ for two ‘close’ homomorphisms $\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{tor}}^{\prime }\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$. Simultaneously, we examine the corresponding ‘abelian’ analog. For this purpose, let $A^{\prime }$ be a second abelian variety and $N$ (respectively $N^{\prime }$) an ample symmetric line bundle on $A$ (respectively $A^{\prime }$). We choose some linear norms $|\cdot |$ on $\operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ and $\operatorname{Hom}(A,A^{\prime })$ for quantification.Footnote ⁵

Lemma 11. In the above situation, there exist constants $c_{1}$ and $c_{2}$ depending only on $G$, $N$, $t^{\prime }$, $A^{\prime }$, $N^{\prime }$ and the linear norms $|\cdot |$ on $\operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ and $\operatorname{Hom}(A,A^{\prime })$ such that the following assertions are true: for any pair $(\unicode[STIX]{x1D711}_{\text{tor}},\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})^{2}$ and any $x\in G(\overline{\mathbb{Q}})$, we have

(15)$$\begin{eqnarray}|\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}(x)-\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}}}(x)|\leqslant c_{1}|\unicode[STIX]{x1D711}_{\text{tor}}-\unicode[STIX]{x1D711}_{\text{tor}}^{\prime }|\cdot \widehat{h}_{M_{\overline{G}}}(x).\end{eqnarray}$$

Similarly, we have

(16)$$\begin{eqnarray}|\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D711}_{\text{ab}}^{\ast }N^{\prime }}(x)-\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }(\unicode[STIX]{x1D711}_{\text{ab}}^{\prime })^{\ast }N^{\prime }}(x)|\leqslant c_{2}|\unicode[STIX]{x1D711}_{\text{ab}}-\unicode[STIX]{x1D711}_{\text{ab}}^{\prime }|^{2}\cdot \widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(x)\end{eqnarray}$$

for any pair $(\unicode[STIX]{x1D711}_{\text{ab}},\unicode[STIX]{x1D711}_{\text{ab}}^{\prime })\in \operatorname{Hom}(A,A^{\prime })^{2}$.

Proof. We prove first the inequality (15). The proof takes place on the ‘graph compactification’ $G_{\overline{\unicode[STIX]{x1D6E4}}}$ of $G$ where $\overline{\unicode[STIX]{x1D6E4}}=\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}\times \unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}\subset (\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t^{\prime }}\times (\mathbb{P}^{1})^{t^{\prime }}$. We denote the projections corresponding to these three factors by $\text{pr}_{i}$ ($i=1,2,3$). The projections $(\operatorname{pr}_{1}\times \operatorname{pr}_{2})|_{\overline{\unicode[STIX]{x1D6E4}}}:\overline{\unicode[STIX]{x1D6E4}}\rightarrow \overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}$ and $(\operatorname{pr}_{1}\times \operatorname{pr}_{3})|_{\overline{\unicode[STIX]{x1D6E4}}}:\overline{\unicode[STIX]{x1D6E4}}\rightarrow \overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}$ are $\mathbb{G}_{m}^{t}$-equivariant and hence induce maps $G_{\overline{\unicode[STIX]{x1D6E4}}}\rightarrow G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ and $G_{\overline{\unicode[STIX]{x1D6E4}}}\rightarrow G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}}$, which both restrict to the identity on $G$. By Lemma 9, we obtain

(17)$$\begin{eqnarray}\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}(x)=\widehat{h}_{G(\operatorname{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)\quad \text{and}\quad \widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })}}}(x)=\widehat{h}_{G(\operatorname{pr}_{3}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)\end{eqnarray}$$

for any $x\in G(\overline{\mathbb{Q}})$. Similarly, we have

(18)$$\begin{eqnarray}\widehat{h}_{M_{\overline{G}}}(x)=\widehat{h}_{G(\operatorname{pr}_{1}^{\ast }(M_{t},\unicode[STIX]{x1D71A}_{t})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)\end{eqnarray}$$

for every $x\in G(\overline{\mathbb{Q}})$.

Using standard coordinates $X_{u}$, $1\leqslant u\leqslant t$, (respectively $Y_{v}$, $1\leqslant v\leqslant t^{\prime }$) on $\mathbb{G}_{m}^{t}$ (respectively $\mathbb{G}_{m}^{t^{\prime }}$), we write

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{tor}}^{\ast }(Y_{v})=X_{1}^{a_{1v}}X_{2}^{a_{2v}}\cdots X_{t}^{a_{tv}}\quad (\text{respectively}~(\unicode[STIX]{x1D711}_{\text{tor}}^{\prime })^{\ast }(Y_{v})=X_{1}^{a_{1v}^{\prime }}X_{2}^{a_{2v}^{\prime }}\cdots X_{t}^{a_{tv}^{\prime }}),\,1\leqslant v\leqslant t^{\prime }, & & \displaystyle \nonumber\end{eqnarray}$$

with integers $a_{uv}$ (respectively $a_{uv}^{\prime }$). Our strategy is to compare the restriction of the line bundles $\text{pr}_{1}^{\ast }(M_{t},\unicode[STIX]{x1D71A}_{t})^{\otimes l}$, $l$ sufficiently large, and $\text{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})\otimes \text{pr}_{3}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})^{\otimes -1}$ on $\overline{\unicode[STIX]{x1D6E4}}$. In fact, we claim that both

(19)$$\begin{eqnarray}(\text{pr}_{1}^{\ast }M_{t}^{\otimes l}\otimes \text{pr}_{2}^{\ast }M_{t^{\prime }}\otimes \text{pr}_{3}^{\ast }M_{t^{\prime }}^{\otimes -1})|_{\overline{\unicode[STIX]{x1D6E4}}}\quad \text{and}\quad (\text{pr}_{1}^{\ast }M_{t}^{\otimes l}\otimes \text{pr}_{2}^{\ast }M_{t^{\prime }}^{\otimes -1}\otimes \text{pr}_{3}^{\ast }M_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}}\end{eqnarray}$$

are $\mathbb{G}_{m}^{t}$-effective with respect to the induced linearizations. In this case, Lemma 8(c, d) implies that

(20)$$\begin{eqnarray}|\widehat{h}_{G(\operatorname{pr}_{2}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)-\widehat{h}_{G(\operatorname{pr}_{3}^{\ast }(M_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)|\leqslant l\cdot \widehat{h}_{G(\operatorname{pr}_{1}^{\ast }(M_{t},\unicode[STIX]{x1D71A}_{t})|_{\overline{\unicode[STIX]{x1D6E4}}})}(x)\end{eqnarray}$$

for each $x\in G(\overline{\mathbb{Q}})\subset G_{\overline{\unicode[STIX]{x1D6E4}}}(\overline{\mathbb{Q}})$. Using the equalities (17) and (18), the inequality (15) can be derived from (20) if we have adequate control on $l$. For this, we note that $\operatorname{pr}_{1}^{\ast }(L_{t},\unicode[STIX]{x1D71A}_{t})|_{\overline{\unicode[STIX]{x1D6E4}}}$ (respectively $\operatorname{pr}_{2}^{\ast }(L_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}}$, $\operatorname{pr}_{3}^{\ast }(L_{t^{\prime }},\unicode[STIX]{x1D71A}_{t^{\prime }})|_{\overline{\unicode[STIX]{x1D6E4}}}$) can be defined by means of the $\mathbb{G}_{m}^{t}$-invariant Cartier divisor $\operatorname{pr}_{1}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}}$ (respectively $\operatorname{pr}_{2}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}$, $\operatorname{pr}_{3}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}$). We next describe these divisors explicitly and start with giving a covering of $\overline{\unicode[STIX]{x1D6E4}}$. With each $(t+2t^{\prime })$-tuple $\text{}\underline{m}$ of numbers $m_{r}\in \{-1,1\}$, $1\leqslant r\leqslant t+2t^{\prime }$, we associate a Zariski open

$$\begin{eqnarray}\displaystyle U_{\text{}\underline{m}}=\overline{\unicode[STIX]{x1D6E4}}\cap \mathop{\bigcap }_{1\leqslant u\leqslant t}D(\operatorname{pr}_{1}^{\ast }X_{u}^{m_{u}})\cap \mathop{\bigcap }_{1\leqslant v\leqslant t^{\prime }}D(\operatorname{pr}_{2}^{\ast }Y_{v}^{m_{v+t}})\cap \mathop{\bigcap }_{1\leqslant v\leqslant t^{\prime }}D(\operatorname{pr}_{3}^{\ast }Y_{v}^{m_{v+t+t^{\prime }}}). & & \displaystyle \nonumber\end{eqnarray}$$

Evidently, $\operatorname{pr}_{1}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}}$ is represented by $(U_{\text{}\underline{m}},f_{\text{}\underline{m}})$ with

$$\begin{eqnarray}\displaystyle f_{\text{}\underline{m}}=\operatorname{pr}_{1}^{\ast }(X_{1}^{-m_{1}}X_{2}^{-m_{2}}\cdots X_{t}^{-m_{t}}) & & \displaystyle \nonumber\end{eqnarray}$$

and $\operatorname{pr}_{2}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}$ (respectively $\operatorname{pr}_{3}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}$) is represented by $(U_{\text{}\underline{m}},g_{\text{}\underline{m}})$ (respectively $(U_{\text{}\underline{m}},g_{\text{}\underline{m}}^{\prime })$) with

$$\begin{eqnarray}\displaystyle g_{\text{}\underline{m}}=\mathop{\prod }_{1\leqslant v\leqslant t^{\prime }}\operatorname{pr}_{2}^{\ast }(Y_{v}^{-m_{v+t}})\quad \biggl(\text{respectively}~g_{\text{}\underline{m}}^{\prime }=\mathop{\prod }_{1\leqslant v\leqslant t^{\prime }}\operatorname{pr}_{3}^{\ast }(Y_{v}^{-m_{v+t+t^{\prime }}})\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

The meromorphic function $1\in {\mathcal{K}}_{\overline{\unicode[STIX]{x1D6E4}}}(\overline{\unicode[STIX]{x1D6E4}})$ gives a $\mathbb{G}_{m}^{t}$-invariant rational section of ${\mathcal{O}}(\operatorname{pr}_{1}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}})$, ${\mathcal{O}}(\operatorname{pr}_{2}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}})$ and ${\mathcal{O}}(\operatorname{pr}_{3}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}})$ by our choice of linearizations. It thus also gives a $\mathbb{G}_{m}^{t}$-invariant rational section of ${\mathcal{O}}(l\cdot \operatorname{pr}_{1}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}}+\operatorname{pr}_{2}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}-\operatorname{pr}_{3}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}})$ and ${\mathcal{O}}(l\cdot \operatorname{pr}_{1}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}}-\operatorname{pr}_{2}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}}+\operatorname{pr}_{3}^{\ast }D_{t^{\prime }}|_{\overline{\unicode[STIX]{x1D6E4}}})$, to which the line bundles in (19) are associated. For $\mathbb{G}_{m}^{t}$-effectivity, we may hence prove that it is actually a global section. In other words, we have to prove that both $f_{\text{}\underline{m}}^{l}\cdot g_{\text{}\underline{m}}\cdot (g_{\text{}\underline{m}}^{\prime })^{-1}$ and $f_{\text{}\underline{m}}^{l}\cdot (g_{\text{}\underline{m}})^{-1}\cdot g_{\text{}\underline{m}}^{\prime }$ are regular on $U_{\text{}\underline{m}}$. Let us remark first that for $l_{v}=\max _{1\leqslant u\leqslant t}\{|a_{uv}-a_{uv}^{\prime }|\}$ the meromorphic function

$$\begin{eqnarray}\displaystyle f_{\text{}\underline{m}}^{l_{v}}\cdot \operatorname{pr}_{2}^{\ast }(Y_{v})\operatorname{pr}_{3}^{\ast }(Y_{v})^{-1}=\mathop{\prod }_{1\leqslant u\leqslant t}\operatorname{pr}_{1}^{\ast }(X_{u}^{s_{u}}),\quad s_{u}=-m_{u}l_{v}+(a_{uv}-a_{uv}^{\prime }), & & \displaystyle \nonumber\end{eqnarray}$$

is regular on $U_{\text{}\underline{m}}$; for $\operatorname{pr}_{1}^{\ast }(X_{u}^{-m_{u}})$ is regular on $U_{\text{}\underline{m}}\subset D(\operatorname{pr}_{1}^{\ast }(X_{u}^{m_{u}}))$. Similarly, the meromorphic function $f_{\text{}\underline{m}}^{l_{v}}\cdot \operatorname{pr}_{2}^{\ast }(Y_{v})^{-1}\operatorname{pr}_{3}^{\ast }(Y_{v})$ is regular on $U_{\text{}\underline{m}}$. We write $g_{\text{}\underline{m}}\cdot (g_{\text{}\underline{m}}^{\prime })^{-1}=\prod _{v=1}^{t^{\prime }}h_{\text{}\underline{m}}$ with

$$\begin{eqnarray}\displaystyle h_{\text{}\underline{m}}=\operatorname{pr}_{2}^{\ast }(Y_{v}^{-m_{v+t}})\operatorname{pr}_{3}^{\ast }(Y_{v}^{m_{v+t+t^{\prime }}}) & & \displaystyle \nonumber\end{eqnarray}$$

and claim that $f_{\text{}\underline{m}}^{l_{v}}\cdot h_{\text{}\underline{m}}$ is regular on $U_{\text{}\underline{m}}$. If $m_{v+t}=m_{v+t+t^{\prime }}=1$ or $m_{v+t}=m_{v+t+t^{\prime }}=-1$, this follows directly from our previous remark. In case $m_{v+t}=-1$ and $m_{v+t+t^{\prime }}=1$, the function

$$\begin{eqnarray}\displaystyle f_{\text{}\underline{m}}^{l_{v}}\cdot h_{\text{}\underline{m}}=\operatorname{pr}_{2}^{\ast }(Y_{v})^{2}\cdot (f_{\text{}\underline{m}}^{l_{v}}\cdot \operatorname{pr}_{2}^{\ast }(Y_{v})^{-1}\operatorname{pr}_{3}^{\ast }(Y_{v})) & & \displaystyle \nonumber\end{eqnarray}$$

is regular by our remark and the fact that $\operatorname{pr}_{2}^{\ast }(Y_{v})$ is regular on $U_{\text{}\underline{m}}\subset D(\operatorname{pr}_{2}^{\ast }(Y_{v})^{-1})$. The case $m_{v+t}=1$ and $m_{v+t+t^{\prime }}=-1$ can be handled in the same way, establishing our claim. In conclusion, the condition

(21)$$\begin{eqnarray}l\geqslant \mathop{\sum }_{1\leqslant v\leqslant t^{\prime }}l_{v}=\mathop{\sum }_{1\leqslant v\leqslant t^{\prime }}\Bigl(\max _{1\leqslant u\leqslant t}\{|a_{uv}-a_{uv}^{\prime }|\}\Bigr)\end{eqnarray}$$

suffices to ensure the regularity of $f_{\text{}\underline{m}}^{l}\cdot g_{\text{}\underline{m}}\cdot (g_{\text{}\underline{m}}^{\prime })^{-1}$. The same argument shows that each meromorphic function $f_{\text{}\underline{m}}^{l}\cdot (g_{\text{}\underline{m}})^{-1}\cdot g_{\text{}\underline{m}}^{\prime }$ is regular on $U_{\text{}\underline{m}}$. Combining (20) and (21), we obtain (15).

The inequality (16) boils down to

$$\begin{eqnarray}\displaystyle |\widehat{h}_{N^{\prime }}(\unicode[STIX]{x1D711}_{\text{ab}}(y))-\widehat{h}_{N^{\prime }}(\unicode[STIX]{x1D711}_{\text{ab}}^{\prime }(y))|\leqslant c_{2}|\unicode[STIX]{x1D711}_{\text{ab}}-\unicode[STIX]{x1D711}_{\text{ab}}^{\prime }|^{2}\cdot \widehat{h}_{N}(y),y=\unicode[STIX]{x1D70B}(x), & & \displaystyle \nonumber\end{eqnarray}$$

where $\widehat{h}_{N}$ and $\widehat{h}_{N^{\prime }}$ are now just the Néron–Tate heights on the abelian varieties $A$ and $A^{\prime }$. This follows straightforwardly from the fact that the map

$$\begin{eqnarray}\displaystyle \operatorname{Hom}(A,A^{\prime })\longrightarrow \operatorname{Pic}(A),\quad \unicode[STIX]{x1D711}_{\text{ab}}\longmapsto \unicode[STIX]{x1D711}_{\text{ab}}^{\ast }N^{\prime }, & & \displaystyle \nonumber\end{eqnarray}$$

is quadratic, which is a direct consequence of the theorem of the cube [Reference MumfordMum70a, Corollary II.6.2]. The reader may refer to [Reference HabeggerHab09a, p. 417] for details. ◻

Finally, we state a lemma on the behavior of the heights $\widehat{h}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ with respect to the group law. Again, there is an ‘abelian’ analog and we mention this also for later reference.

Lemma 12. For any $\unicode[STIX]{x1D711}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ and any points $x,y\in G(\overline{\mathbb{Q}})$, we have

$$\begin{eqnarray}\displaystyle \widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}(xy)\leqslant \widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}(x)+\widehat{h}_{M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}}(y). & & \displaystyle \nonumber\end{eqnarray}$$

Similarly, we have

$$\begin{eqnarray}\displaystyle \widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(xy)\leqslant 2\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(x)+2\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(y). & & \displaystyle \nonumber\end{eqnarray}$$

Note that this statement includes the fact that

(22)$$\begin{eqnarray}\widehat{h}_{M_{\overline{G}}}(xy)\leqslant \widehat{h}_{M_{\overline{G}}}(x)+\widehat{h}_{M_{\overline{G}}}(y)\end{eqnarray}$$

for all $x,y\in G(\overline{\mathbb{Q}})$ (set $\unicode[STIX]{x1D711}_{\text{tor}}=\operatorname{id}_{\mathbb{G}_{m}^{t}}$). Most of our proof is actually about establishing this inequality, which has already been provided in the literature (see, e.g., [Reference RémondRém03, Corollaire 3.1]). Nevertheless, we give a proof here both for completeness and because it is very close to the proof of Lemma 11 above.

Proof. For the first assertion, it suffices to prove (22). In fact, each $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ is isomorphic to some pullback $\overline{\unicode[STIX]{x1D711}}^{\ast }M_{\overline{G}^{\prime }}$ along a homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ to another semiabelian variety $G^{\prime }$ (see Construction 7). In order to prove (22), we use the same strategy as for Lemma 11. This means we consider the Zariski closure $\overline{\unicode[STIX]{x1D6E4}}\subset ((\mathbb{P}^{1})^{t}\times (\mathbb{P}^{1})^{t})\times (\mathbb{P}^{1})^{t}$ of the graph of the group law $\cdot _{T}:\mathbb{G}_{m}^{t}\times \mathbb{G}_{m}^{t}\rightarrow \mathbb{G}_{m}^{t}$. Again, we denote the projection to the $i\text{th}$ component by $\operatorname{pr}_{i}$ ($i=1,2,3$). For this, we use standard coordinates $X_{u}$, $1\leqslant u\leqslant t$, on $\mathbb{G}_{m}^{t}$ (and on its extension to $(\mathbb{P}^{1})^{t}$). Note that $(\operatorname{pr}_{3}^{\ast }X_{u})=(\operatorname{pr}_{1}^{\ast }X_{u})(\operatorname{pr}_{2}^{\ast }X_{u})$ on $\overline{\unicode[STIX]{x1D6E4}}$.

With each $(3t)$-tuple $\text{}\underline{m}\in \{-1,1\}^{3t}$ of numbers $m_{r}\in \{-1,1\}$, $1\leqslant r\leqslant 3t$, we associate a Zariski open. To wit, we define

$$\begin{eqnarray}\displaystyle U_{\text{}\underline{m}}=\overline{\unicode[STIX]{x1D6E4}}\cap \mathop{\bigcap }_{1\leqslant u\leqslant t}D(\operatorname{pr}_{1}^{\ast }X_{u}^{m_{u}})\cap \mathop{\bigcap }_{1\leqslant u\leqslant t}D(\operatorname{pr}_{2}^{\ast }X_{u}^{m_{u+t}})\cap \mathop{\bigcap }_{1\leqslant u\leqslant t}D(\operatorname{pr}_{3}^{\ast }X_{u}^{m_{u+2t}}). & & \displaystyle \nonumber\end{eqnarray}$$

It is easy to see that each $\operatorname{pr}_{i}^{\ast }D_{t}|_{\overline{\unicode[STIX]{x1D6E4}}}$ ($i=1,2,3$) is represented by $(U_{\text{}\underline{m}},f_{\text{}\underline{m}}^{(i)})$, where

$$\begin{eqnarray}\displaystyle f_{\text{}\underline{m}}^{(i)}=\operatorname{pr}_{i}^{\ast }(X_{1}^{-m_{1+(i-1)t}}X_{2}^{-m_{2+(i-1)t}}\cdots X_{t}^{-m_{t+(i-1)t}}). & & \displaystyle \nonumber\end{eqnarray}$$

Consequently, the restriction of $\operatorname{pr}_{1}^{\ast }D_{t}+\operatorname{pr}_{2}^{\ast }D_{t}-\operatorname{pr}_{3}^{\ast }D_{t}$ to $\overline{\unicode[STIX]{x1D6E4}}$ is represented by

$$\begin{eqnarray}\displaystyle (U_{\text{}\underline{m}},f_{\text{}\underline{m}}^{(1)}\cdot f_{\text{}\underline{m}}^{(2)}\cdot (f_{\text{}\underline{m}}^{(3)})^{-1}). & & \displaystyle \nonumber\end{eqnarray}$$

The meromorphic function $f_{\text{}\underline{m}}^{(1)}\cdot f_{\text{}\underline{m}}^{(2)}\cdot (f_{\text{}\underline{m}}^{(3)})^{-1}$ equals

$$\begin{eqnarray}\displaystyle \operatorname{pr}_{1}^{\ast }(X_{1})^{-m_{1}+m_{2t+1}}\cdots \operatorname{pr}_{1}^{\ast }(X_{t})^{-m_{t}+m_{3t}}\operatorname{pr}_{2}^{\ast }(X_{1})^{-m_{t+1}+m_{2t+1}}\cdots \operatorname{pr}_{2}^{\ast }(X_{t})^{-m_{2t}+m_{3t}}. & & \displaystyle \nonumber\end{eqnarray}$$

By definition, each $\operatorname{pr}_{i}^{\ast }(X_{u})^{-m_{u+(i-1)t}}$, $i\in \{1,2,3\}$, $1\leqslant u\leqslant t$, is regular on $U_{\text{}\underline{m}}$. Since

$$\begin{eqnarray}\displaystyle -m_{u}+m_{2t+u}\in \{0,-2m_{u}\}\quad \text{and}\quad -m_{t+u}+m_{2t+u}\in \{0,-2m_{t+u}\}, & & \displaystyle \nonumber\end{eqnarray}$$

we infer the regularity of $f_{\text{}\underline{m}}^{(1)}\cdot f_{\text{}\underline{m}}^{(2)}\cdot (f_{\text{}\underline{m}}^{(3)})^{-1}$ on $U_{\text{}\underline{m}}$. As in the proof of Lemma 11, we see that this implies that $1\in {\mathcal{K}}_{\overline{\unicode[STIX]{x1D6E4}}}(\overline{\unicode[STIX]{x1D6E4}})$ is a $T$-invariant global section of ${\mathcal{O}}(\operatorname{pr}_{1}^{\ast }D_{t}+\operatorname{pr}_{2}^{\ast }D_{t}-\operatorname{pr}_{3}^{\ast }D_{t})$. Thus, the first assertion follows from Lemma 8(d). For the second assertion, it suffices to note the equivalence of the assertion with

$$\begin{eqnarray}\displaystyle \widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(\unicode[STIX]{x1D70B}(x)+\unicode[STIX]{x1D70B}(y))\leqslant 2\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(\unicode[STIX]{x1D70B}(x))+2\widehat{h}_{\overline{\unicode[STIX]{x1D70B}}^{\ast }N}(\unicode[STIX]{x1D70B}(y)). & & \displaystyle \nonumber\end{eqnarray}$$

Indeed, this inequality follows directly from the parallelogram law for the Néron–Tate height [Reference Hindry and SilvermanHS00, Theorem B.5.1(c)] and its non-negativity for symmetric line bundles. ◻

4 Hermitian differential geometry

In the next two sections, we make extensive use of hermitian differential geometry at the level of rather explicit computations on semiabelian varieties. To avoid permanent interruptions in these, we recall here the necessary abstract framework separately. The reader is referred to [Reference VoisinVoi07, § 3.1] as well as [Reference Griffiths and HarrisGH94, § 0.2] and [Reference HuybrechtsHuy05, § 1.2] for details.

Let $Y$ be a complex manifold (e.g., $X^{\text{sm}}(\mathbb{C})$ for a complex algebraic variety $X$). To $Y$ is associated its real tangent bundle $T_{\mathbb{R}}Y$, its holomorphic tangent bundle $T_{\mathbb{C}}^{1,0}Y$ (e.g., $T_{x}X(\mathbb{C})$ for a smooth complex algebraic variety $X$) and its anti-holomorphic tangent bundle $T_{\mathbb{C}}^{0,1}Y$. As real vector bundles, all three can be canonically identified (cf. [Reference Griffiths and HarrisGH94, p. 17]) and we do so from now on. In this way, we obtain an almost complex structure $I:T_{\mathbb{R}}Y\rightarrow T_{\mathbb{R}}Y$ (i.e., a linear map $I:T_{\mathbb{R}}Y\rightarrow T_{\mathbb{R}}Y$ such that $I^{2}=-\text{id}_{T_{\mathbb{R}}Y}$) from the multiplication-by-$i$ (respectively multiplication-by-$(-i)$) homomorphism on the complex vector bundle $T_{\mathbb{C}}^{1,0}Y$ (respectively $T_{\mathbb{C}}^{0,1}Y$). A $(1,1)$-form of real type on $Y$ is an alternating $\mathbb{R}$-bilinear pairing

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}:T_{\mathbb{R}}Y\times _{Y}T_{\mathbb{R}}Y\longrightarrow \mathbb{R}\times Y & & \displaystyle \nonumber\end{eqnarray}$$

such that $\unicode[STIX]{x1D714}(I(\cdot ),I(\cdot ))=\unicode[STIX]{x1D714}(\cdot ,\cdot )$. Under the identification $T_{\mathbb{C}}^{1,0}Y=T_{\mathbb{R}}Y$, this corresponds to an alternating $\mathbb{R}$-bilinear pairing

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}:T_{\mathbb{C}}^{1,0}Y\times _{Y}T_{\mathbb{C}}^{0,1}Y\longrightarrow \mathbb{R}\times Y & & \displaystyle \nonumber\end{eqnarray}$$

such that $\unicode[STIX]{x1D714}(i(\cdot ),i(\cdot ))=-\unicode[STIX]{x1D714}(\cdot ,\cdot )$.Footnote ⁶ The Chern forms of hermitian line bundles are the basic examples of such $(1,1)$-forms. More generally, for any smooth function $\unicode[STIX]{x1D706}:Y\rightarrow \mathbb{R}$ the $(1,1)$-form $dd^{c}\unicode[STIX]{x1D706}$ ($d^{c}=i/2\unicode[STIX]{x1D70B}(\overline{\unicode[STIX]{x2202}}-\unicode[STIX]{x2202})$) is always of real type. To such a $(1,1)$-form $\unicode[STIX]{x1D714}$ is associated a symmetric $\mathbb{R}$-bilinear pairing

$$\begin{eqnarray}\displaystyle g_{\unicode[STIX]{x1D714}}:T_{\mathbb{R}}Y\times _{Y}T_{\mathbb{R}}Y\longrightarrow \mathbb{R}\times Y,\quad (v,w)\longmapsto \unicode[STIX]{x1D714}(v,Iw). & & \displaystyle \nonumber\end{eqnarray}$$

In fact, this establishes a one-to-one correspondence between $(1,1)$-forms of real type and symmetric $\mathbb{R}$-bilinear forms on $T_{\mathbb{R}}Y$. Using our identification of $T_{\mathbb{R}}Y$ with $T_{\mathbb{C}}^{1,0}Y$ and $T_{\mathbb{C}}^{0,1}Y$, the $(1,1)$-form $\unicode[STIX]{x1D714}$ is positive (respectively semipositive) in the ordinary sense (e.g. [Reference HuybrechtsHuy05, Definition 4.3.14]) if and only if $g_{\unicode[STIX]{x1D714}}$ is positive definitive (respectively positive semidefinite). We note that for a smooth function $\unicode[STIX]{x1D706}:Y\rightarrow \mathbb{R}$, the $(1,1)$-form $dd^{c}\unicode[STIX]{x1D706}$ is semipositive if and only if $\unicode[STIX]{x1D706}$ is plurisubharmonic (cf. [Reference GunningGun90a, Theorem K.8]).

For later reference, we remark that for any smooth function $f:Y\rightarrow \mathbb{R}$ the $(1,1)$-form $\unicode[STIX]{x1D714}=i(\unicode[STIX]{x2202}f\wedge \overline{\unicode[STIX]{x2202}}f)$ is of real type and

(23)$$\begin{eqnarray}g_{\unicode[STIX]{x1D714}}={\textstyle \frac{1}{2}}(\unicode[STIX]{x2202}f\otimes \overline{\unicode[STIX]{x2202}}f+\overline{\unicode[STIX]{x2202}}f\otimes \unicode[STIX]{x2202}f);\end{eqnarray}$$

for this is a local assertion that reduces by linearity to the fact that the $(1,1)$-form

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}=i\cdot (\unicode[STIX]{x1D6FC}\,dz_{i}\wedge d\overline{z}_{j}+\overline{\unicode[STIX]{x1D6FC}}\,dz_{j}\wedge d\overline{z}_{i}),\quad \unicode[STIX]{x1D6FC}\in \mathbb{C}, & & \displaystyle \nonumber\end{eqnarray}$$

on $\mathbb{C}^{n}$ is of real type and the fact that

$$\begin{eqnarray}\displaystyle i\,dz_{i}\wedge d\overline{z}_{j}(v,Iw)={\textstyle \frac{1}{2}}(dz_{i}(v)\,d\overline{z}_{j}(w)+d\overline{z}_{j}(v)\,dz_{i}(w)),\quad v,w\in T_{\mathbb{R},x}\mathbb{C}^{n},x\in \mathbb{C}^{n}. & & \displaystyle \nonumber\end{eqnarray}$$

To a $(1,1)$-form $\unicode[STIX]{x1D714}$ of real type is also associated a hermitian form (with respect to $I$)

(24)$$\begin{eqnarray}H_{\unicode[STIX]{x1D714}}:T_{\mathbb{R}}Y\times _{Y}T_{\mathbb{R}}Y\longrightarrow \mathbb{C}\times Y,\quad (v,w)\longmapsto g_{\unicode[STIX]{x1D714}}(v,w)-i\cdot \unicode[STIX]{x1D714}(v,w),\end{eqnarray}$$

and this can be also seen to be a one-to-one correspondence. Indeed, $\unicode[STIX]{x1D714}=-\text{Im}(H_{\unicode[STIX]{x1D714}})$.

Let $Z$ be a complex submanifold of a complex manifold $Y$ and $\unicode[STIX]{x1D714}$ a $(1,1)$-form of real type on $Y$. Restricting and taking exterior products, we obtain an alternating $\mathbb{R}$-multilinear map

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D714}|_{Z})^{\wedge \dim (Z)}:(T_{\mathbb{R},x}Z)^{2\dim (Z)}\longrightarrow \mathbb{R} & & \displaystyle \nonumber\end{eqnarray}$$

for each $x\in Z$. If the restriction of the $\mathbb{R}$-bilinear form $g_{\unicode[STIX]{x1D714},x}$ to $T_{\mathbb{R},x}Z$ is moreover positive definite, we have a non-zero Riemannian volume form [Reference HelgasonHel01, pp. 361–362]

$$\begin{eqnarray}\displaystyle \text{vol}(g_{\unicode[STIX]{x1D714},x}):(T_{\mathbb{R},x}Z)^{2\dim (Z)}\longrightarrow \mathbb{R}. & & \displaystyle \nonumber\end{eqnarray}$$

By [Reference VoisinVoi07, Lemma 3.8], $\text{vol}(g_{\unicode[STIX]{x1D714},x})$ agrees with $\dim (Z)!^{-1}(\unicode[STIX]{x1D714}|_{Z})^{\wedge \dim (Z)}$. If $\unicode[STIX]{x1D714}$ is continuous, this implies immediately that there is a euclidean neighborhood $U$ of $x$ in $Z$ such that $\int _{U}(\unicode[STIX]{x1D714}|_{Z})^{\wedge \dim (Z)}>0$.

To use this argument effectively, we need a criterion to check whether the restriction of $g_{\unicode[STIX]{x1D714},x}$ to $T_{\mathbb{R},x}Z$ is positive definite. For an arbitrary $\mathbb{R}$-bilinear form $g$ on a real vector space $V$, we define its kernel by

$$\begin{eqnarray}\displaystyle \text{ker}(g)=\{v\in V\mid \forall w\in V:g(v,w)=0\}. & & \displaystyle \nonumber\end{eqnarray}$$

In our applications, $\unicode[STIX]{x1D714}$ is always semipositive so that $g_{\unicode[STIX]{x1D714}}$ is positive semidefinite. For a positive semidefinite bilinear form $g$, we have

(25)$$\begin{eqnarray}\ker (g)=\{v\in V\mid g(v,v)=0\}\end{eqnarray}$$

and hence that $\ker (g|_{W})=\ker (g)\cap W$ for any $\mathbb{R}$-linear subspace $W\subset V$. Consequently, the restriction of $g_{\unicode[STIX]{x1D714},x}$ to $T_{\mathbb{R},x}Z$ is positive definite if and only if $\ker (g_{\unicode[STIX]{x1D714},x})\cap T_{\mathbb{R},x}Z=\{0\}$. Finally, let us note that for any positive semidefinite $\mathbb{R}$-bilinear forms $g_{1},g_{2}$ on $V$ their sum $g_{1}+g_{2}$ is also a positive semidefinite $\mathbb{R}$-bilinear form and (25) implies that

(26)$$\begin{eqnarray}\ker (g_{1}+g_{2})=\ker (g_{1})\cap \ker (g_{2}).\end{eqnarray}$$

Finally, let us remark that $\ker (\unicode[STIX]{x1D714}_{x})=\ker (g_{\unicode[STIX]{x1D714},x})$ for each $(1,1)$-form of real type on $Y$ and every point $x\in Y$, under the condition that we consider $\unicode[STIX]{x1D714}$ as a bilinear form on $T_{\mathbb{R}}Y$. We use this fact to simplify our notion in § 6.

5 Weil functions, hermitian metrics, and Chern forms

We provide here the necessary tools for § 7.4, in which bounds on certain intersection numbers are established. Our approach is to endow all line bundles under consideration with smooth hermitian metrics so that intersection numbers become integrals of the associated Chern forms. Throughout this section, we hence take $k=\mathbb{C}$ as our base field. A major issue is to interpolate between the Chern forms of different line bundles. For this purpose, we introduce certain explicit smooth $(1,1)$-forms of real type, namely the ‘toric’ $(1,1)$-forms $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$ in § 5.1 and the ‘abelian’ $(1,1)$-forms $\unicode[STIX]{x1D714}(N;\unicode[STIX]{x1D719}_{\text{ab}})$ in § 5.2.

5.1 ‘Toric’ $(1,1)$-forms

Our first aim is to endow the line bundles $M_{\overline{G}}$ from Construction 5 with a hermitian metric and to compute the associated Chern forms. Functoriality allows us to endow additionally the line bundles $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ from Construction 6 with a hermitian metric. A closer look at the associated Chern forms leads us to introduce the ‘toric’ $(1,1)$-forms $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$.

Our main instrument are Weil functions, on which the reader may find details in [Reference LangLan83, ch. 10] and [Reference LangLan88, ch. I]. Let $X$ be a complex algebraic variety and $D$ a Cartier divisor on $X$. In this situation, a function $\unicode[STIX]{x1D706}:(X\setminus \operatorname{supp}(D))(\mathbb{C})\rightarrow \mathbb{R}$ is called a Weil function for $D(\mathbb{C})$ if every point $x\in X(\mathbb{C})$ has an open neighborhood $U$ (in the euclidean topology) such that

(27)$$\begin{eqnarray}\unicode[STIX]{x1D706}=-\log |f|+\unicode[STIX]{x1D6FC}\quad \text{on}~U\setminus \operatorname{supp}(D)(\mathbb{C})\end{eqnarray}$$

with $f$ a meromorphic function on $U$ such that $\operatorname{div}(f)=D|_{U}$ (as formal sums of irreducible analytic varieties on $U$) and $\unicode[STIX]{x1D6FC}$ a continuous function on $U$. Furthermore, $\unicode[STIX]{x1D706}$ is called a smooth Weil function if $\unicode[STIX]{x1D6FC}$ can be even assumed smooth on $U$. Every (smooth) Weil function

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}:(X\setminus \operatorname{supp}(D))(\mathbb{C})\rightarrow \mathbb{R} & & \displaystyle \nonumber\end{eqnarray}$$

associated with $D$ yields a (smooth) hermitian metric $g$ on the associated line bundle $L(D)$. In fact, its sections ${\mathcal{O}}(D)$ form a ${\mathcal{O}}_{X}$-submodule of ${\mathcal{K}}_{X}$ and we can just set $g_{x}(f)=e^{-\unicode[STIX]{x1D706}(x)}|f(x)|$ for any meromorphic function $f$ on $U$ and any $x\in X(\mathbb{C})$ in its domain of definition. To a smooth Weil function $\unicode[STIX]{x1D706}$ for $D$ is associated a smooth closed $(1,1)$-form of real type, namely the Chern form $c_{1}(L(D),g)$ of the associated smooth metric on $L(D)$. On an open euclidean neighborhood $U$ such that (27) is true, we have $c_{1}(L,g)=dd^{c}\unicode[STIX]{x1D6FC}$. Additionally, $dd^{c}\unicode[STIX]{x1D6FC}=dd^{c}\unicode[STIX]{x1D706}$ outside $\operatorname{supp}(D)(\mathbb{C})$.

We now record a standard result on Weil functions. Let $D$ be a Cartier divisor on a complex projective variety $X$ and $\unicode[STIX]{x1D706}$ be a Weil function for $D$. Assume that $D$ is the difference $D_{1}-D_{2}$ of two effective Cartier divisors $D_{1},D_{2}$ with disjoint supports. From [Reference LangLan83, Propositions 10.2.1 and 10.3.2], we infer that $\sup \{\unicode[STIX]{x1D706},0\}$ (respectively $-\inf \{\unicode[STIX]{x1D706},0\}$) is a Weil function for $D_{1}$ (respectively $D_{2}$). The next lemma provides a smooth variant of this observation in the same situation.

Lemma 13. In the situation described above, assume additionally that $\unicode[STIX]{x1D706}$ is a smooth Weil function. Then, $\log (1+e^{2\unicode[STIX]{x1D706}})/2$ (respectively $\log (1+e^{-2\unicode[STIX]{x1D706}})/2$) is a smooth Weil function for $D_{1}$ (respectively $D_{2}$).

Proof. By assumption, we know that for each $x\in X(\mathbb{C})$ there exists an open euclidean neighborhood $U$, a meromorphic function $f$ representing $D$ on $U$ and a smooth function $\unicode[STIX]{x1D6FC}$ satisfying (27). Since $D_{1}$ and $D_{2}$ have disjoint supports, we may shrink $U$ to guarantee that it is relatively compact and that its topological closure $\overline{U}$ does not intersect $\operatorname{supp}(D_{1})(\mathbb{C})$ or $\operatorname{supp}(D_{2})(\mathbb{C})$. Suppose $\overline{U}\cap D_{1}=\emptyset$ (respectively $\overline{U}\cap D_{2}=\emptyset$). Then, $|f|\geqslant \unicode[STIX]{x1D700}>0$ (respectively $|f|\leqslant \unicode[STIX]{x1D700}^{-1}$) for some sufficiently small $\unicode[STIX]{x1D700}>0$. Furthermore, 1 (respectively $f$) is a local equation for $D_{1}$. Note that $\unicode[STIX]{x1D6FD}=\log (1+|f|^{-2}e^{2\unicode[STIX]{x1D6FC}})/2$ (respectively $\unicode[STIX]{x1D6FD}=\log (|f|^{2}+e^{2\unicode[STIX]{x1D6FC}})/2$) is a smooth function on $U$.Footnote ⁷ In addition,

$$\begin{eqnarray}\displaystyle {\textstyle \frac{1}{2}}\log (1+e^{2\unicode[STIX]{x1D706}})=-\log |1|+\unicode[STIX]{x1D6FD}\quad (\text{respectively}~{\textstyle \frac{1}{2}}\log (1+e^{2\unicode[STIX]{x1D706}})=-\log |f|+\unicode[STIX]{x1D6FD}). & & \displaystyle \nonumber\end{eqnarray}$$

This demonstrates that $\frac{1}{2}\log (1+e^{2\unicode[STIX]{x1D706}})$ is a smooth Weil function for $D_{1}$. Similarly, $\frac{1}{2}\log (1+e^{-2\unicode[STIX]{x1D706}})$ can be shown to be a smooth Weil function for $D_{2}$.◻

Let us next recollect a fundamental result of Vojta [Reference VojtaVoj96]. Let $G$ be a semiabelian variety with split toric part $T=\mathbb{G}_{m}^{t}$ and abelian quotient $A$. Recall from Construction 5 its compactification $\overline{G}$ as well as the Cartier divisor $G(D_{t})$ on $\overline{G}$. With $\operatorname{pr}_{u}:(\mathbb{P}^{1})^{t}\rightarrow \mathbb{P}^{1}$ being the projection to the $u\text{th}$ component as in Construction 4, we set $D_{u,0}=G(\operatorname{pr}_{u}^{\ast }E_{0})$ and $D_{u,\infty }=G(\operatorname{pr}_{u}^{\ast }E_{\infty })$ so that $G(D_{t})=\sum _{u=1}^{t}(D_{u,0}+D_{u,\infty })$.

Lemma 14. For each divisor $D_{u,0}-D_{u,\infty }$, $1\leqslant u\leqslant t$, there exists a unique smooth Weil function

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{u}:\overline{G}(\mathbb{C})\setminus \operatorname{supp}(D_{u,0}-D_{u,\infty })(\mathbb{C})\longrightarrow \mathbb{R} & & \displaystyle \nonumber\end{eqnarray}$$

that satisfies

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{u}(x+y)=\unicode[STIX]{x1D706}_{u}(x)+\unicode[STIX]{x1D706}_{u}(y) & & \displaystyle \nonumber\end{eqnarray}$$

for all $x,y\in G(\mathbb{C})$. In addition, $e^{\unicode[STIX]{x1D706}_{u}}$ is locally the absolute value of a meromorphic function.

Outside $\operatorname{supp}(D_{u,0}-D_{u,\infty })(\mathbb{C})$, we have locally $\unicode[STIX]{x1D706}_{u}=\log |f|$ for some holomorphic function $f$. This implies $dd^{c}\unicode[STIX]{x1D706}_{u}=0$ on $G(\mathbb{C})$.

Proof. This is stated in [Reference VojtaVoj96, Proposition 2.6] except for the assertion about $e^{\unicode[STIX]{x1D706}_{u}}$. Inspecting (2.6.3) in the proof of the said proposition, one sees that it suffices to prove the same assertion for the Néron function $\unicode[STIX]{x1D706}_{(s)}$ (cf. [Reference LangLan83, Theorem 11.1.1]) attached to the divisor $(s)$ on $A$. As $s$ is a rational section of an algebraically trivial line bundle by construction, the divisor $(s)$ is algebraically equivalent to the zero divisor. The explicit formula for $\unicode[STIX]{x1D706}_{(s)}$ in terms of a normalized theta function [Reference LangLan83, Theorem 13.1.1] directly yields the assertion in this case; the hermitian form $H$ in the formula is zero because $(s){\sim}_{\text{alg}}0$ (cf. [Reference Birkenhake and LangeBL04, Proposition 2.2.2]).◻

Using the Weil functions $\unicode[STIX]{x1D706}_{u}$ we can define a subgroup

(28)$$\begin{eqnarray}\{x\in G(\mathbb{C})\mid \unicode[STIX]{x1D706}_{1}(x)=\unicode[STIX]{x1D706}_{2}(x)=\cdots =\unicode[STIX]{x1D706}_{t}(x)=0\}\subset G(\mathbb{C}).\end{eqnarray}$$

This coincides with the maximal compact subgroup $K_{G}$ of $G(\mathbb{C})$. Indeed, any homomorphism $K_{G}\rightarrow \mathbb{R}$ vanishes by compactness so that $\unicode[STIX]{x1D706}_{u}|_{K_{G}}=0$. By uniqueness, the restriction of $\unicode[STIX]{x1D706}_{u}$ to the maximal torus $T(\mathbb{C})$ equals $-\log |X_{u}|$ (in standard coordinates $X_{1},\ldots ,X_{t}$). Hence, the subgroup in (28) is topologically a fiber bundle with compact fiber $(S^{1})^{t}$, $S^{1}=\{z\in \mathbb{C}\mid |z|=1\}$, over the compact base $A(\mathbb{C})$. Therefore, it is compact itself and hence contained in $K_{G}$. As $G(\mathbb{C})$ is Hausdorff, its maximal compact subgroup $K_{G}$ is a closed subgroup. Using [Reference HelgasonHel01, Theorem II.2.3] and counting dimensions, we see that $K_{G}$ is a real Lie subgroup of (real) dimension $2\dim (A)+t$.

Recall that $M_{\overline{G}}$ is the line bundle associated to the $T$-invariant Weil divisor $G(D_{t})$. By Lemmas 13 and 14, the function

(29)$$\begin{eqnarray}\unicode[STIX]{x1D706}=\frac{1}{2}\mathop{\sum }_{u=1}^{t}(\log (1+e^{2\unicode[STIX]{x1D706}_{u}})+\log (1+e^{-2\unicode[STIX]{x1D706}_{u}}))\end{eqnarray}$$

is a smooth Weil function for $G(D_{t})$. For the associated smooth hermitian line bundle, which is denoted $\overline{M}_{\overline{G}}$ in the following, we have

$$\begin{eqnarray}\displaystyle c_{1}(\overline{M}_{\overline{G}})=\frac{1}{2}\mathop{\sum }_{u=1}^{t}(dd^{c}\log (1+e^{2\unicode[STIX]{x1D706}_{u}})+dd^{c}\log (1+e^{-2\unicode[STIX]{x1D706}_{u}}))\quad \text{on}~G(\mathbb{C}). & & \displaystyle \nonumber\end{eqnarray}$$

The Weil functions of Lemma 14 also satisfy some functoriality. To be precise, let $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ be a homomorphism of semiabelian varieties with toric component $\unicode[STIX]{x1D711}_{\text{tor}}:T=\mathbb{G}_{m}^{t}\rightarrow T^{\prime }=\mathbb{G}_{m}^{t^{\prime }}$. Let $X_{i}$ (respectively $Y_{j}$) be the standard algebraic coordinates on $\mathbb{G}_{m}^{t}$ (respectively $\mathbb{G}_{m}^{t^{\prime }}$) and write

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{tor}}^{\ast }(Y_{v})=X_{1}^{a_{1v}}\cdots X_{t}^{a_{tv}} & & \displaystyle \nonumber\end{eqnarray}$$

with integers $a_{uv}$. Lemma 14 supplies Weil functions $\unicode[STIX]{x1D706}_{v}^{\prime }$, $1\leqslant u\leqslant t^{\prime }$, on $G^{\prime }$ and there is an identity

(30)$$\begin{eqnarray}\unicode[STIX]{x1D711}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }=\unicode[STIX]{x1D706}_{v}^{\prime }\circ \unicode[STIX]{x1D711}=a_{1v}\unicode[STIX]{x1D706}_{1}+a_{2v}\unicode[STIX]{x1D706}_{2}+\cdots +a_{tv}\unicode[STIX]{x1D706}_{t}\quad \text{on}~G(\mathbb{C}).\end{eqnarray}$$

Indeed, the equality is valid on $T$ since the restriction of $\unicode[STIX]{x1D706}_{u}$ (respectively $\unicode[STIX]{x1D706}_{v}^{\prime }$) to the maximal torus $T(\mathbb{C})\approx (\mathbb{C}^{\times })^{t}$ (respectively $T^{\prime }(\mathbb{C})\approx (\mathbb{C}^{\times })^{t^{\prime }}$) is $(-\log |X_{u}|)$ (respectively $(-\log |Y_{v}|)$) as we noted above. It is also true on $K_{G}$ because $\unicode[STIX]{x1D711}(K_{G})\subset K_{G^{\prime }}$. As $K_{G}$ and $T(\mathbb{C})$ generate $G(\mathbb{C})$ as a group, (30) is true for all of $G(\mathbb{C})$. Note that $\unicode[STIX]{x1D711}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }$ is independent of the abelian component of $\unicode[STIX]{x1D711}$. Abusing notation, we therefore write $\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }$ instead of $\unicode[STIX]{x1D711}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }$. Even more, we can use (30) to formally define $\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }:G(\mathbb{C})\rightarrow \mathbb{R}$, $v=1,\ldots ,t^{\prime }$, for an arbitrary $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}_{\mathbb{R}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$.

We now apply the results of the previous paragraph to endow the line bundles $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$, $\unicode[STIX]{x1D711}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$, with hermitian metrics. For this, we use the homomorphism $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ from Construction 7 with $\overline{\unicode[STIX]{x1D711}}^{\ast }M_{\overline{G}^{\prime }}\approx M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$. We may endow $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ with a hermitian metric such that $\overline{\unicode[STIX]{x1D711}}^{\ast }\overline{M}_{\overline{G}^{\prime }}\approx \overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$. Since the isomorphism between $\overline{\unicode[STIX]{x1D711}}^{\ast }M_{\overline{G}^{\prime }}$ and $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ is unique up to multiplication with a non-zero constant, this singles out a hermitian metric on $\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ up to a non-zero constant scaling factor. Regardless of this indeterminate scaling factor, we have an identity of Chern forms $c_{1}(\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}})=\overline{\unicode[STIX]{x1D711}}^{\ast }c_{1}(\overline{M}_{\overline{G}^{\prime }})$. Thus,

(31)$$\begin{eqnarray}c_{1}(\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}})=\frac{1}{2}\mathop{\sum }_{v=1}^{t^{\prime }}(dd^{c}\log (1+e^{2\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})+dd^{c}\log (1+e^{-2\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }}))\quad \text{on}~G(\mathbb{C}).\end{eqnarray}$$

Since the indeterminacy in the metric is negligible for our purposes, we suppress it in writing $\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$ for any hermitian line bundle as constructed above. Again, the right-hand side of (31) depends only on $\unicode[STIX]{x1D711}_{\text{tor}}$ and is moreover well-defined for any $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}_{\mathbb{R}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$. In other words, we can associate with each $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}_{\mathbb{R}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ a $(1,1)$-form

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})=\frac{1}{2}\mathop{\sum }_{v=1}^{t^{\prime }}(dd^{c}\log (1+e^{2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})+dd^{c}\log (1+e^{-2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})) & & \displaystyle \nonumber\end{eqnarray}$$

on $G(\mathbb{C})$. (Note that we do not claim that $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$ extends to any compactification of $G(\mathbb{C})$. In the proof of Lemma 17 we give such an extension in the case where $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$, but we neither can prove the existence of an extension in general nor do we need it.) In the remainder of this section, we establish basic properties of this $(1,1)$-form.

Lemma 15. Each $dd^{c}\log (1+e^{\pm 2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})$, $1\leqslant v\leqslant t^{\prime }$, is a semipositive $(1,1)$-form of real type on $G(\mathbb{C})$. Consequently, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$ is a semipositive $(1,1)$-form of real type.

Proof. It suffices to prove that $\log (1+e^{\pm 2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})$ is a plurisubharmonic function. This follows directly from $dd^{c}\unicode[STIX]{x1D706}_{u}=0$ (i.e., both $\unicode[STIX]{x1D706}_{u}$ and $-\unicode[STIX]{x1D706}_{u}$ are plurisubharmonic on $G(\mathbb{C})$) and the fact that $\log (1+e^{x})$ is a convex monotonously increasing function (cf. [Reference GunningGun90a, Theorem K.5(d)]).◻

Furthermore, the map $\unicode[STIX]{x1D719}_{\text{tor}}\mapsto \unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$ is continuous with respect to the euclidean topology on $\operatorname{Hom}_{\mathbb{R}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})\approx \mathbb{R}^{t\times t^{\prime }}$ and the usual topology on smooth $(1,1)$-forms (cf. [Reference DemaillyDem12, § I.2] and [Reference RudinRud91, § 1.46]). By Lemma 14, there exists locally on $G(\mathbb{C})$ a non-zero holomorphic function $\unicode[STIX]{x1D705}_{u}$ such that $e^{\unicode[STIX]{x1D706}_{u}^{\prime }}=|\unicode[STIX]{x1D705}_{u}|$. From $\unicode[STIX]{x1D706}_{u}^{\prime }=\log (|\unicode[STIX]{x1D705}_{u}|^{2})/2$, we deduce

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}\unicode[STIX]{x1D706}_{u}^{\prime }=\frac{\overline{\unicode[STIX]{x1D705}_{u}}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2|\unicode[STIX]{x1D705}_{u}|^{2}}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\quad \text{and}\quad \overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D706}_{u}^{\prime }=\frac{\unicode[STIX]{x1D705}_{u}\overline{\unicode[STIX]{x2202}}\overline{\unicode[STIX]{x1D705}_{u}}}{2|\unicode[STIX]{x1D705}_{u}|^{2}}=\overline{\biggl(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}. & & \displaystyle \nonumber\end{eqnarray}$$

Using Lemma 16 below, we hence obtain

(32)$$\begin{eqnarray}dd^{c}\log (1+e^{2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})+dd^{c}\log (1+e^{-2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})=\frac{2i(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }\wedge \overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime })}{\unicode[STIX]{x1D70B}(1+e^{2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})(1+e^{-2\unicode[STIX]{x1D719}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})}.\end{eqnarray}$$

Lemma 16. Let $\unicode[STIX]{x1D705}_{1},\ldots ,\unicode[STIX]{x1D705}_{m}$ be zero-free holomorphic functions on an open subset $U\subset \mathbb{C}^{n}$. Then,

$$\begin{eqnarray}\displaystyle dd^{c}\log (1+|\unicode[STIX]{x1D705}_{1}|^{a_{1}}\cdots |\unicode[STIX]{x1D705}_{m}|^{a_{m}})=\frac{i}{4\unicode[STIX]{x1D70B}}\frac{|\unicode[STIX]{x1D705}_{1}|^{a_{1}}\cdots |\unicode[STIX]{x1D705}_{m}|^{a_{m}}}{(1+|\unicode[STIX]{x1D705}_{1}|^{a_{1}}\cdots |\unicode[STIX]{x1D705}_{m}|^{a_{m}})^{2}}\biggl(\mathop{\sum }_{u=1}^{m}\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{\unicode[STIX]{x1D705}_{u}}\wedge \overline{\mathop{\sum }_{u=1}^{m}\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{\unicode[STIX]{x1D705}_{u}}}\biggr) & & \displaystyle \nonumber\end{eqnarray}$$

for any real numbers $a_{1},\ldots ,a_{m}$.

Proof. Without loss of generality, we assume that all $a_{1},\ldots ,a_{m}$ are non-zero. To simplify our notation, we set $f(z)=|\unicode[STIX]{x1D705}_{1}|^{a_{1}}\cdots |\unicode[STIX]{x1D705}_{m}|^{a_{m}}$. First, we note that $\unicode[STIX]{x2202}|\unicode[STIX]{x1D705}|^{q}=\unicode[STIX]{x2202}(\unicode[STIX]{x1D705}\overline{\unicode[STIX]{x1D705}})^{q/2}=(q/2)|\unicode[STIX]{x1D705}|^{q-2}\overline{\unicode[STIX]{x1D705}}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}=(q/2)|\unicode[STIX]{x1D705}|^{q}(\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x1D705})$ (respectively $\overline{\unicode[STIX]{x2202}}|\unicode[STIX]{x1D705}|^{q}=(q/2)|\unicode[STIX]{x1D705}|^{q}\overline{(\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x1D705})}$) implies that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}f(z)=f(z)\cdot \mathop{\sum }_{u=1}^{m}\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)\quad \biggl(\text{respectively}~\overline{\unicode[STIX]{x2202}}f(z)=f(z)\cdot \mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

Since both $\unicode[STIX]{x1D705}_{u}$ and $\unicode[STIX]{x1D705}_{u}^{-1}$ are holomorphic, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D705}_{u}^{-1}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}=\overline{\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x1D705}_{u}^{-1}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u})}=\overline{\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x1D705}_{u}^{-1})\wedge \unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}+\unicode[STIX]{x1D705}_{u}^{-1}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}(\unicode[STIX]{x1D705}_{u})}=0 & & \displaystyle \nonumber\end{eqnarray}$$

and hence

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}f(z) & = & \displaystyle \unicode[STIX]{x2202}f(z)\wedge \biggl(\mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr)+f(z)\cdot \unicode[STIX]{x2202}\biggl(\mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr)\nonumber\\ \displaystyle & = & \displaystyle f(z)\biggl(\mathop{\sum }_{u=1}^{m}\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)\wedge \biggl(\mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr).\nonumber\end{eqnarray}$$

We compute

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\log (1+f(z)) & = & \displaystyle \unicode[STIX]{x2202}\biggl(\frac{1}{1+f(z)}\overline{\unicode[STIX]{x2202}}f(z)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{-1}{(1+f(z))^{2}}\unicode[STIX]{x2202}f(z)\wedge \overline{\unicode[STIX]{x2202}}f(z)+\frac{1}{1+f(z)}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}f(z)\nonumber\\ \displaystyle & = & \displaystyle \biggl(\frac{-f(z)^{2}}{(1+f(z))^{2}}+\frac{f(z)}{1+f(z)}\biggr)\biggl(\mathop{\sum }_{u=1}^{m}\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\wedge \mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{2\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{f(z)}{4(1+f(z))^{2}}\biggl(\mathop{\sum }_{u=1}^{m}\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{\unicode[STIX]{x1D705}_{u}}\wedge \mathop{\sum }_{u=1}^{m}\overline{\biggl(\frac{a_{u}\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{u}}{\unicode[STIX]{x1D705}_{u}}\biggr)}\biggr).\nonumber\end{eqnarray}$$

The assertion follows directly as $dd^{c}=(i/2\unicode[STIX]{x1D70B})(\unicode[STIX]{x2202}+\overline{\unicode[STIX]{x2202}})(\overline{\unicode[STIX]{x2202}}-\unicode[STIX]{x2202})=(i/\unicode[STIX]{x1D70B})\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$.◻

The next lemma establishes an essential homogeneity property for $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})$.

Lemma 17. Let $G$ be a semiabelian variety with abelian quotient $\unicode[STIX]{x1D70B}:G\rightarrow A$ and toric part $\mathbb{G}_{m}^{t}$. In addition, let $t^{\prime }$ be a non-negative integer and $\unicode[STIX]{x1D71B}$ a smooth closed $(1,1)$-form on $A(\mathbb{C})$. For every $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}_{\mathbb{Q}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$, every algebraic subvariety $X\subset G$, and every non-negative integer $s_{1}$, $s_{2}$ satisfying $s_{1}+s_{2}=\dim (X)$, the integral

$$\begin{eqnarray}\displaystyle \int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}} & & \displaystyle \nonumber\end{eqnarray}$$

is finite and

(33)$$\begin{eqnarray}\int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(n\cdot \unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}=n^{s_{1}}\cdot \int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}\end{eqnarray}$$

for each non-negative integer $n$.

Proof. Let us first prove the lemma assuming that $\unicode[STIX]{x1D719}_{\text{tor}}=\unicode[STIX]{x1D711}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$. In this situation, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D711}_{\text{tor}})$ extends to a smooth closed $(1,1)$-form on the (complex) analytic space $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})$. Indeed, $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D711}_{\text{tor}})$ is precisely defined to agree with the restriction of the smooth differential form $\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})=c_{1}(\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}})$ on $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})\supset G(\mathbb{C})$. Writing $\overline{X}$ for the Zariski closure of $X$ in $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}$, we have hence

$$\begin{eqnarray}\displaystyle \int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}=\int _{\overline{X}(\mathbb{C})}\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}} & & \displaystyle \nonumber\end{eqnarray}$$

because $(\overline{X}\setminus X)(\mathbb{C})$ is of positive codimension in $X(\mathbb{C})$. As we are integrating a smooth differential form over a compact analytic space, the integral on the right-hand side is evidently finite.

To show the second part of the assertion, still assuming that $\unicode[STIX]{x1D719}_{\text{tor}}=\unicode[STIX]{x1D711}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$, we start by noting that also $c_{1}(\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}})$ is an extension of $\unicode[STIX]{x1D714}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})$ on $G_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})\supset G(\mathbb{C})$. In addition, Construction 6 supplies us with a map $\unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow G_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}}$, which is the identity on $G$. Therefore, the smooth closed $(1,1)$-form $\widetilde{\unicode[STIX]{x1D714}}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})=\unicode[STIX]{x1D717}_{\unicode[STIX]{x1D711}_{\text{tor}},n}^{\ast }c_{1}(\overline{M}_{\overline{\unicode[STIX]{x1D6E4}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})}})$ extends $\unicode[STIX]{x1D714}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})$ to $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})\supset G(\mathbb{C})$. (Note that both extensions $\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})$ and $\widetilde{\unicode[STIX]{x1D714}}(n\cdot \unicode[STIX]{x1D711}_{\text{tor}})$ are actually unique.)

Denote by $\overline{\unicode[STIX]{x1D70B}}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow A$ the abelian quotient. Since the boundary $(\overline{X}\setminus X)(\mathbb{C})$ has measure zero, (33) would follow from the equality

(34)$$\begin{eqnarray}\int _{\overline{X}(\mathbb{C})}\widetilde{\unicode[STIX]{x1D714}}(m\cdot \unicode[STIX]{x1D711}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}=m^{s_{1}}\cdot \int _{\overline{X}(\mathbb{C})}\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}.\end{eqnarray}$$

For this, we claim that any function

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{v}^{\pm }:G(\mathbb{C})\rightarrow \mathbb{R}^{{>}0},\quad x\mapsto (1+e^{\pm 2m\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})/(1+e^{\pm 2\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }})^{m},\quad 1\leqslant v\leqslant t^{\prime }, & & \displaystyle \nonumber\end{eqnarray}$$

extends smoothly to $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})$. It suffices to prove that each $x\in (G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\setminus G)(\mathbb{C})$ has a (euclidean) neighborhood on which $\unicode[STIX]{x1D70C}_{v}^{\pm }$ extends smoothly. For this, we let $\unicode[STIX]{x1D711}:G\rightarrow G^{\prime }$ be again the homomorphism from Construction 7 so that $M_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\approx \overline{\unicode[STIX]{x1D711}}^{\ast }M_{\overline{G}^{\prime }}$. As before, Lemma 14 affords a Weil function $\unicode[STIX]{x1D706}_{v}^{\prime }$, $1\leqslant v\leqslant t^{\prime }$, for each divisor $D_{v,0}^{\prime }-D_{v,\infty }^{\prime }$ on $\overline{G}^{\prime }$. Its pullback $\overline{\unicode[STIX]{x1D711}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }$ along $\overline{\unicode[STIX]{x1D711}}:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}\rightarrow \overline{G}^{\prime }$ restricts to the function $\unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }:G(\mathbb{C})\rightarrow \mathbb{R}$ formally defined by (30). There exists a euclidean neighborhood $U$ of $\overline{\unicode[STIX]{x1D711}}(x)$ and a meromorphic function $f$ on $U$ with $\operatorname{div}(f)=(D_{v,0}^{\prime }-D_{v,\infty }^{\prime })|_{U}$ such that $\unicode[STIX]{x1D706}_{v}^{\prime }+\log |f|$ extends to a smooth function $\unicode[STIX]{x1D6FC}$ on $U$. On $G(\mathbb{C})\cap \overline{\unicode[STIX]{x1D711}}^{-1}(U)\subset G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})$, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{tor}}^{\ast }\unicode[STIX]{x1D706}_{v}^{\prime }=\unicode[STIX]{x1D706}_{v}^{\prime }\circ \overline{\unicode[STIX]{x1D711}}=-\log |f\circ \overline{\unicode[STIX]{x1D711}}|+(\unicode[STIX]{x1D6FC}\circ \overline{\unicode[STIX]{x1D711}}) & & \displaystyle \nonumber\end{eqnarray}$$

and thus

(35)$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{v}^{\pm }=\frac{1+|f\circ \overline{\unicode[STIX]{x1D711}}|^{\mp 2m}e^{\pm 2m(\unicode[STIX]{x1D6FC}\circ \overline{\unicode[STIX]{x1D711}})}}{(1+|f\circ \overline{\unicode[STIX]{x1D711}}|^{\mp 2}e^{\pm 2(\unicode[STIX]{x1D6FC}\circ \overline{\unicode[STIX]{x1D711}})})^{m}}=\frac{|f\circ \overline{\unicode[STIX]{x1D711}}|^{\pm 2m}+e^{\pm 2m(\unicode[STIX]{x1D6FC}\circ \overline{\unicode[STIX]{x1D711}})}}{(|f\circ \overline{\unicode[STIX]{x1D711}}|^{\pm 2}+e^{\pm 2(\unicode[STIX]{x1D6FC}\circ \overline{\unicode[STIX]{x1D711}})})^{m}}.\end{eqnarray}$$

Since $\operatorname{supp}(D_{v,0}^{\prime })\cap \operatorname{supp}(D_{v,\infty }^{\prime })=\emptyset$, we have $x\notin \operatorname{supp}(D_{v,0}^{\prime })$ or $x\notin \operatorname{supp}(D_{v,\infty }^{\prime })$. Shrinking $U$ if necessary, we may hence assume that $f$ or $f^{-1}$ is holomorphic on $U$. In either case, (35) yields a smooth extension of $\unicode[STIX]{x1D70C}_{v}^{\pm }$ on $\overline{\unicode[STIX]{x1D711}}^{-1}(U)$. By uniqueness, these extensions glue together to a smooth function $\widetilde{\unicode[STIX]{x1D70C}}_{v}^{\pm }:G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})\rightarrow \mathbb{R}^{{>}0}$. (In fact, $\widetilde{\unicode[STIX]{x1D70C}}_{v}^{\pm }(x)=1$ for all $x\in \operatorname{supp}(D_{v,0}^{\prime })(\mathbb{C})\cup \operatorname{supp}(D_{v,\infty }^{\prime })(\mathbb{C})$ because $(1+e^{mx})/(1+e^{x})^{m}\rightarrow 1$ if $x\rightarrow \pm \infty$.) In addition,

(36)$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D714}}(m\cdot \unicode[STIX]{x1D711}_{\text{tor}})-m\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})=\frac{1}{2}\mathop{\sum }_{v=1}^{t^{\prime }}(dd^{c}\log (\widetilde{\unicode[STIX]{x1D70C}}_{v}^{+})+dd^{c}\log (\widetilde{\unicode[STIX]{x1D70C}}_{v}^{-}));\end{eqnarray}$$

indeed, this equality is obvious on $G(\mathbb{C})$ and any $(1,1)$-form on $G(\mathbb{C})$ has at most one smooth extension to the compactification $G_{\overline{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}_{\text{tor}})}}(\mathbb{C})$. We deduce from (36) that $\widetilde{\unicode[STIX]{x1D714}}(m\cdot \unicode[STIX]{x1D711}_{\text{tor}})-m\widetilde{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D711}_{\text{tor}})$ is exact and hence Stokes’ theorem [Reference Griffiths and HarrisGH94, p. 33] in combination with a partition of unity implies (34).

Now, let us consider a general $\unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}_{\mathbb{Q}}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$ and a positive integer $n$ that is a denominator for $\unicode[STIX]{x1D719}_{\text{tor}}$ (i.e., $n\cdot \unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$). Setting $Y=[n]^{-1}(X)$, the restriction $[n]|_{Y}:Y\rightarrow X$ is finite étale of degree $n^{2\dim (A)+t}$. By functoriality (30), we have $[n]^{\ast }\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})=\unicode[STIX]{x1D714}(n\cdot \unicode[STIX]{x1D719}_{\text{tor}})$ and $[n]^{\ast }\unicode[STIX]{x1D714}(m\cdot \unicode[STIX]{x1D719}_{\text{tor}})=\unicode[STIX]{x1D714}(m\cdot n\cdot \unicode[STIX]{x1D719}_{\text{tor}})$. We infer that

$$\begin{eqnarray}\displaystyle n^{2\dim (A)+t}\int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(m\cdot \unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}=\int _{Y(\mathbb{C})}\unicode[STIX]{x1D714}(m\cdot n\cdot \unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }[n]^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}} & & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle n^{2\dim (A)+t}\int _{X(\mathbb{C})}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}=\int _{Y(\mathbb{C})}\unicode[STIX]{x1D714}(n\cdot \unicode[STIX]{x1D719}_{\text{tor}})^{\wedge s_{1}}\wedge (\overline{\unicode[STIX]{x1D70B}}^{\ast }[n]^{\ast }\unicode[STIX]{x1D71B})^{\wedge s_{2}}. & & \displaystyle \nonumber\end{eqnarray}$$

Since $n\cdot \unicode[STIX]{x1D719}_{\text{tor}}\in \operatorname{Hom}(\mathbb{G}_{m}^{t},\mathbb{G}_{m}^{t^{\prime }})$, this reduces the assertion of the lemma to the already proven special case.◻