3.1 Picard group

The Picard functor $\mathrm{\mathcal{P}ic}(\mathcal{A}/S)$ of $\mathcal{A}/S$ is a functor from the category of schemes over S to the category of groups defined by

\[\mathrm{\mathcal{P}ic}(\mathcal{A}/S)(T):=\mathrm{Pic}(\mathcal{A}_T)/\pi_T^*\mathrm{Pic}(T) \]

for any S-scheme T. We define the Picard group for $\mathcal{A}/S$ as

\[\mathrm{Pic}(\mathcal{A}/S):=\mathrm{\mathcal{P}ic}(\mathcal{A}/S)(S)=\mathrm{Pic}(\mathcal{A})/\pi^*\mathrm{Pic}(S). \]

A rigidified line bundle on $\mathcal{A}/S$ is a pair $(\mathcal{L},e^*\mathcal{L}\cong \mathcal{O}_S)$ with $\mathcal{L}\in \mathrm{Pic}(\mathcal{A})$ such that $e^*\mathcal{L}\in \mathrm{Pic}(S)$ is trivial and the fixed rigidification $e^*\mathcal{L}\cong \mathcal{O}_S$ is an isomorphism of line bundles. We usually leave out the rigidification when there is no ambiguity. An isomorphism of rigidified line bundles is an isomorphism of line bundles that is compatible with the rigidification. One checks immediately:

\[\mathrm{Pic}(\mathcal{A}/S)\cong \{\text{Rigidified line bundles on }\mathcal{A}/S\}/\sim_{\mathrm{isom}}. \]

Thus, we shall regard an element in $\mathrm{Pic}(\mathcal{A}/S)$ as a rigidified line bundle up to isomorphism.

Two $\mathbb{R}$ -line bundles $\mathcal{L}_1,\mathcal{L}_2\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ are said to be numerically equivalent, denoted as $\mathcal{L}_1\equiv\mathcal{L}_2$ , if $\deg(\mathcal{L}_1|_C)=\deg(\mathcal{L}_2|_C)$ for any curve C in a fiber of $\mathcal{A}\rightarrow S$ . If $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ is numerically equivalent to $\mathcal{O}_{\mathcal{A}}$ , we say $\mathcal{L}$ is numerically trivial. The Néron–Severi group of $\mathcal{A}/S$ is defined as

\[\mathrm{NS}(\mathcal{A}/S):=\mathrm{Pic}(\mathcal{A}/S)/{\equiv}. \]

A rigidified line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ is called symmetric if $[-1]^*\mathcal{L}\cong \mathcal{L}$ . Let $\mathrm{Pic}_0(\mathcal{A}/S)$ be the subgroup of $\mathrm{Pic}(\mathcal{A}/S)$ consisting of the rigidified symmetric line bundles on $\mathcal{A}$ . Any $\mathbb{Q}$ -line bundle $\mathcal{L}$ can be written as the sum of a symmetric line bundle ${1}/{2}(\mathcal{L}+[-1]^*\mathcal{L})$ with a numerically trivial line bundle ${1}/{2}(\mathcal{L}-[-1]^*\mathcal{L})$ . Thus, we may equivalently define $\mathrm{NS}(\mathcal{A}/S)_{\mathbb{Q}}$ as the quotient $\mathrm{Pic}_0(\mathcal{A}/S)_{\mathbb{Q}}/\equiv$ .

An $\mathbb{R}$ -line bundle on a projective variety is called ample if it is a positive linear combination of ample line bundles. $\mathbb{R}$ -ampleness is a numerical property; see [Reference LazarsfeldLaz04, Proposition 1.3.13]. An $\mathbb{R}$ -line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ is defined to be ample,Footnote ² respectively, nef, if $\mathcal{L}_s:=\mathcal{L}|_{\mathcal{A}_s}$ is ample, respectively, nef, for any $s\in S$ . Since ampleness and nefness are numerical, such definitions also make sense for a class in $\mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}$ . We use the notation $\mathcal{L}\geq0$ to mean $\mathcal{L}$ is nef.

According to Raynaud [Reference RaynaudRay70, Corollary VIII.7], a line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ is ample if $\mathcal{L}_s$ is ample for one point $s\in S$ . Since S is a normal variety, by a result of Grothendieck [Reference RaynaudRay70, Theorem XI.1.4], given any ample line bundle $\mathcal{L}_\eta\in \mathrm{Pic}(\mathcal{A}_\eta)$ , there is a symmetric line bundle $\mathcal{M}\in \mathrm{Pic}_0(\mathcal{A}/S)$ such that $\mathcal{M}_\eta\equiv 2\mathcal{L}_\eta$ . Then $\mathcal{M}$ is ample. In particular, the abelian scheme $\mathcal{A}/S$ is projective.

The following proposition summarizes some basic numerical properties of the rigidified line bundles on abelian schemes.

1. (1) The specialization map $\mathrm{sp}_s:\mathrm{NS}(\mathcal{A}/S)\rightarrow \mathrm{NS}(\mathcal{A}_s)$ is injective for any $s\in S$ . Moreover, $\mathrm{sp}_{\eta,\mathbb{Q}}:\mathrm{NS}(\mathcal{A}/S)_{\mathbb{Q}}\rightarrow\mathrm{NS}(\mathcal{A}_\eta)_{\mathbb{Q}}$ is an isomorphism.

2. (2) An $\mathbb{R}$ -line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ is ample, respectively, nef, if $\mathcal{L}_s$ is ample, respectively, nef for one point $s\in S$ .

3. (3) A numerically trivial $\mathbb{R}$ -line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ can be written as a linear combination of numerically trivial integral line bundles. In other words, there is a natural isomorphism

   \[ (\mathrm{Pic}(\mathcal{A}/S)/\equiv)\otimes_{\mathbb{Z}}\mathbb{R}=:\mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}\xrightarrow{\cong} \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}/\equiv. \]

4. (4) An ample $\mathbb{R}$ -line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ can be written as a positive linear combination of ample integral line bundles.

(2) Assume $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ such that $\mathcal{L}_s$ is ample for $s\in S$ . Then the Néron–Severi class $[\mathcal{L}_s]$ lies in the open convex ample cone $\mathrm{Amp}(\mathcal{A}_s)$ of $\mathcal{A}_s$ . In addition, $[\mathcal{L}_s]$ is in the $\mathbb{R}$ -vector subspace $\mathrm{sp}_{s}(\mathrm{NS}(\mathcal{A}/S))_{\mathbb{R}}$ . Thus,

\[ \mathrm{Amp}(\mathcal{A}_s)\cap\mathrm{sp}_{s}(\mathrm{NS}(\mathcal{A}/S))_{\mathbb{R}} \]

is a nonempty open convex subset of $\mathrm{sp}_{s,\mathbb{R}}(\mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}})$ containing $[\mathcal{L}_s]$ . By openness, there exists $[\mathcal{L}_1],[\mathcal{L}_2]\in \mathrm{NS}(\mathcal{A}/S)$ such that $[\mathcal{L}_s]=c_1[\mathcal{L}_{1,s}]+c_2[\mathcal{L}_{2,s}]$ for $c_1,c_2 \gt 0$ . By part (1), we have $[\mathcal{L}]=c_1[\mathcal{L}_1]+c_2[\mathcal{L}_2]$ . Note that $\mathcal{L}_1,\mathcal{L}_2$ are ample by Raynaud’s result. Thus, $\mathcal{L}$ is ample.

For nefness, one can use the same limit argument as in part (1).

(3) Assume $\mathcal{L}=\sum_i c_i \mathcal{L}_i$ is numerically trivial with $\mathcal{L}_i\in \mathrm{Pic}(\mathcal{A}/S)$ and $c_i\in \mathbb{R}$ . By part (2), this is equivalent to the condition that $\mathcal{L}_s$ is numerically trivial for a point $s\in S$ . The condition that $\sum_i c_i \mathcal{L}_{i,s}$ is numerically trivial, is given by finitely many integer-coefficient linear homogeneous equations in $c_i$ , determined by intersecting with a set of generators of the subgroup of $H_2(\mathcal{A}_s,\mathbb{Z})$ spanned by algebraic 1-cycles. It follows from linear algebra that the solution space is generated by vectors with integer coefficients. Hence, we can formally rewrite $\sum_i c_i\mathcal{L}_{i,s}=\sum_j c_j' \mathcal{L}_{j,s}'$ where $\mathcal{L}_{j}'$ is an integer linear combination of $\mathcal{L}_{i}$ and $\mathcal{L}_{j,s}'$ is numerically trivial. By part (2) again, $\mathcal{L}_{j}'$ is numerically trivial. Thus, $\mathcal{L}=\sum_j c_j'\mathcal{L}_j'$ is a linear combination of numerically trivial integral line bundles.

(4) The basic observation is that for any real number c and two line bundles $\mathcal{M}_1,\mathcal{M}_2$ , letting c’,c” be rationals with $c'<c<c''$ , there exists $0<t<1$ such that

\[ \mathcal{M}_1+c\mathcal{M}_2=t(\mathcal{M}_1+c'\mathcal{M}_2)+(1-t)(\mathcal{M}_1+c''\mathcal{M}_2). \]

We first use the proof of part (2) to express $\mathcal{L}\equiv c_1\mathcal{L}_1+c_2\mathcal{L}_2$ with ample $\mathcal{L}_1,\mathcal{L}_2\in \mathrm{Pic}(\mathcal{A}/S)$ and $c_1,c_2 \gt 0$ . Now by part (3), we can write $\mathcal{L}-c_1\mathcal{L}_1-c_2\mathcal{L}_2$ as a linear combination of numerically trivial integral line bundles. The observation allows us to merge those with $c_1\mathcal{L}_1$ and conclude.

3.2 Polarization and level structure

The identity component $\mathrm{\mathcal{P}ic}^0(\mathcal{A}/S)$ of the Picard functor $\mathrm{\mathcal{P}ic}(\mathcal{A}/S)$ is representable by an abelian scheme $\mathcal{A}^\vee/S$ , known as the dual abelian scheme to $\mathcal{A}/S$ . Given $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ , define $\Lambda(\mathcal{L}):\mathcal{A}\rightarrow \mathrm{\mathcal{P}ic}(\mathcal{A}/S)$ as

\[\mu^*\mathcal{L} -p_1^*\mathcal{L} -p_2^*\mathcal{L} \in \mathrm{\mathcal{P}ic}(\mathcal{A}/S)(\mathcal{A})=\mathrm{Pic}(\mathcal{A}\times_S\mathcal{A}/\mathcal{A}), \]

where $\mu,p_1,p_2:\mathcal{A}\times_S\mathcal{A}\rightarrow \mathcal{A}$ are the addition map, the first projection, and the second projection, respectively. One readily checks that the image is in the identity component; that is, $\Lambda(\mathcal{L})$ is a homomorphism from $\mathcal{A}$ to $\mathcal{A}^\vee$ .

A polarization of an abelian scheme is defined as a homomorphism $\lambda:\mathcal{A}\rightarrow \mathcal{A}^\vee$ such that for any geometric point $\bar{s}$ of S, $\lambda_{\bar{s}}=\Lambda(\mathcal{L}_{\bar{s}})$ for some ample line bundle $\mathcal{L}_{\bar{s}}$ on $\mathcal{A}_{\bar{s}}$ . Any ample line bundle $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ defines a polarization $\Lambda(\mathcal{L})$ . Conversely, $\lambda$ is not in general of the form $\Lambda(\mathcal{L})$ . However, $2\lambda=\Lambda(\mathcal{L})$ for some ample $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ by [Reference Mumford, Fogarty and KirwanMFK94, Proposition 6.10]. The polarization is called principal if it is an isomorphism. In general, the kernel of a polarization is a finite étale group subscheme over S, which over any geometric point, is isomorphic to

\[(\mathbb{Z}/d_1\mathbb{Z}\times \cdots\times \mathbb{Z}/d_g\mathbb{Z})^2 \]

for fixed positive integers $d_1|\cdots|d_g$ . The diagonal matrix $D:=\mathrm{diag}(d_1,\ldots,d_g)$ is called the type of the polarization. Observe that polarization types are stable under base change. Because any abelian scheme over the normal base S is projective, polarization always exists in our setting.

A level-n-structure on an abelian scheme in this article means a principal level-n-structure, i.e. an isomorphism $\alpha:(\mathbb{Z}/n\mathbb{Z})^{2g}\xrightarrow{\sim}\mathcal{A}[n]$ , where $n\in \mathbb{N}$ . Its existence thus requires all n-torsion points of $\mathcal{A}_\eta$ to be rational, and the converse clearly holds as well. In particular, if $\mathcal{A}/S$ has a level-n-structure, then so does any abelian subscheme $\mathcal{B}\subseteq\mathcal{A}$ , or the quotient abelian scheme $\mathcal{A}/\mathcal{B}$ . We have the following convenient lemma for later use.

3.3 Homomorphisms

Let $\mathcal{B}$ be another abelian scheme over S. A homomorphism of abelian schemes $\mathcal{A}\rightarrow \mathcal{B}$ restricts to a homomorphism of abelian varieties on the generic fiber $\mathcal{A}_\eta\rightarrow \mathcal{B}_\eta$ . Conversely, a homomorphism $\mathcal{A}_\eta\rightarrow \mathcal{B}_\eta$ spreads out uniquely to a homomorphism $\mathcal{A}\rightarrow\mathcal{B}$ by normality of S; cf. [Reference Faltings and ChaiFC90, Proposition I.2.7]. Thus, there is a natural isomorphism of homomorphism groups

\[\mathrm{Hom}(\mathcal{A},\mathcal{B})\cong \mathrm{Hom}(\mathcal{A}_\eta,\mathcal{B}_\eta). \]

In particular, $\mathrm{End}(\mathcal{A})\cong\mathrm{End}(\mathcal{A}_\eta)$ .

We sometimes make the following innocuous extra assumption in the later discussion

(3.1) \begin{equation} \mathrm{End}(\mathcal{A}_\eta)=\mathrm{End}(\mathcal{A}_{\bar\eta}).\end{equation}

This ensures that all abelian subvarieties of $\mathcal{A}_{\bar{\eta}}$ are defined over K(S). Since $\mathrm{End}(\mathcal{A}_{\bar\eta})$ is finitely generated, the assumption can be achieved by base changing $\mathcal{A}_\eta$ to a finite extension of K(S), which corresponds to taking a finite cover of S. Alternatively, one can apply a more precise result of Silverberg [Reference SilverbergSil92, Theorem 2.4] to say it suffices to assume that there exists a level-n-structure on $\mathcal{A}/S$ for some $n\geq3$ . By Lemma 3.2, we may achieve (3.1) by base changing to a finite étale cover of S.

Now suppose B is an abelian subvariety of $\mathcal{A}_\eta$ . Then by Poincaré’s complete reducibility [Reference MilneMil86, Proposition 12.1], there exists another abelian variety B’ and an isogeny $\mathcal{A}_\eta\rightarrow B\times B'$ . By composition with the projection and embedding, we get an endomorphism $\varphi_\eta:\mathcal{A}_\eta\rightarrow\mathcal{A}_\eta$ with $\varphi_\eta(\mathcal{A}_\eta)=B$ . Consider the corresponding homomorphism $\varphi\in \mathrm{End}(\mathcal{A})$ . Note that, for any $\varphi\in \mathrm{End}(\mathcal{A})$ , the image of $\varphi$ is an abelian subscheme of $\mathcal{A}$ ; see [Reference Achter, Casalaina-Martin and WiseACMW24] for a very general recent result. Thus, $\varphi(\mathcal{A})$ is an abelian subscheme of $\mathcal{A}$ with generic fiber B. As an immediate consequence, there is a bijection between

\[\{\text{abelian subschemes of }\mathcal{A}\}\leftrightarrow\{\text{abelian subvarieties of }\mathcal{A}_\eta\}.\]

By the complete reducibility of $\mathcal{A}_\eta$ , we have an isogeny decomposition

\[\mathcal{A}\sim \mathcal{A}_1^{r_1}\times_S\cdots\times_S \mathcal{A}_k^{r_k},\]

where $\mathcal{A}_i$ are simple abelian subschemes of $\mathcal{A}$ which are pairwise nonisogenous with $r_1,\ldots,r_k\in \mathbb{N}\setminus\{0\}$ . Here, $\mathcal{A}_i^{r_i}$ denotes the $r_i$ th fibered power over S. If we assume (3.1), then each $\mathcal{A}_i$ is also geometrically simple.

A significant portion to us is the fixed part (or constant part) of $\mathcal{A}_\eta$ . Denote the function field of S by F.

Since varieties are assumed to be geometrically connected, the extension $F/K$ is primary. The trace exists for the primary extension $F/K$ , and in characteristic 0, the homomorphism $\tau$ is an embedding; cf. [Reference ConradCon06, § 6]. Denote by $\mathcal{A}_0$ the abelian subscheme of $\mathcal{A}$ corresponding to $\tau(\mathrm{tr}_{F/K}(\mathcal{A}_\eta)_F)$ . Note that $\mathcal{A}_0\cong \mathrm{tr}_{F/K}(\mathcal{A}_\eta)\times_K S$ . The following definition is for later use.

If the constant (respectively, torsion) generic point is F-rational, then the corresponding constant (respectively, torsion) multi-section is called a constant (respectively, torsion) section.

For later purposes, we are mostly interested in surjective homomorphisms. Suppose $\varphi:\mathcal{A}\rightarrow\mathcal{B}$ is a surjective homomorphism. Then $\mathcal{B}$ is isogenous to $\mathcal{A}_1^{r_1'}\times_S\cdots\times_S\mathcal{A}_k^{r_k'}$ with $0\leq r_i'\leq r_i$ . In particular, we get the following lemma, which allows us to consider only finitely many targets eventually.

For the rest of the section, we fix $\mathcal{B}$ to be an abelian subscheme of $\mathcal{A}$ from one of the isogeny classes in Lemma 3.5. Let $\mathrm{Hom}(\mathcal{A},\mathcal{B})^\circ \subseteq \mathrm{Hom}(\mathcal{A},\mathcal{B})$ be the subset of surjective homomorphisms. Let $\mathrm{Hom}(\mathcal{A},\mathcal{B})^\circ_{\mathbb{Q}}\subseteq\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}$ be the subset of f such that N f is a surjective homomorphism for some integer $N\in \mathbb{Z}$ . To use continuity arguments later, we also need a more general notion of surjective $\mathbb{R}$ -homomorphisms in $\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}$ .

Note first that the composition of homomorphisms extends linearly to a pairing

\begin{align*} \mathrm{Hom}(\mathcal{B},\mathcal{A})_{\mathbb{R}} \times \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}&\longrightarrow \mathrm{End}(\mathcal{B})_{\mathbb{R}}\nonumber\\ (\theta,f)&\longmapsto f\circ \theta.\end{align*}

Define the subset of surjective $\mathbb{R}$ -homomorphisms algebraically as

\[\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}^\circ:=\{f\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}\mid \exists \theta\in \mathrm{Hom}(\mathcal{B},\mathcal{A})_{\mathbb{R}}\text{ s.t. }f\circ \theta=\mathrm{id}_{\mathcal{B}}\}.\]

This is consistent with the previous definition of surjective $\mathbb{Q}$ -homomorphisms due to complete reducibility.

We would like to study the union $\cup\ker\varphi$ for all $\varphi\in \mathrm{Hom}(\mathcal{A},\mathcal{B})^\circ$ , which is equivalently the union of torsion translates of abelian subschemes of $\mathcal{A}$ of fixed-type isogenous to $\mathcal{A}/\mathcal{B}$ .

Equip the finite-dimensional vector space $\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}$ with the usual Euclidean topology. The following compactness result is one of the key ingredients to our main theorem, motivated by the analogous result of Habegger [Reference HabeggerHab09a, Lemma 2] on abelian varieties. We assume K is a number field below for simplicity.

Proof. For any $ s\in S(K^{\mathrm{alg}})$ , specialization gives an injection $\mathrm{End}(\mathcal{A})\hookrightarrow\mathrm{End}(\mathcal{A}_s)$ . Under the assumption (3.1), such specialization is indeed an isomorphism for ‘most’ of $s\in S(K^{\mathrm{alg}})$ by the main result of [Reference MasserMas96] (or by Hodge theory). We still denote such a point by $s\in S(K^{\mathrm{alg}})$ . By our assumption, $\mathcal{B}\subseteq\mathcal{A}$ . The isomorphism $\mathrm{End}(\mathcal{A})\cong\mathrm{End}(\mathcal{A}_{s})$ also implies that $\mathrm{End}(\mathcal{B})\cong\mathrm{End}(\mathcal{B}_{s})$ , $\mathrm{Hom}(\mathcal{A},\mathcal{B})\cong\mathrm{Hom}(\mathcal{A}_{s},\mathcal{B}_{s})$ and $\mathrm{Hom}(\mathcal{B},\mathcal{A})\cong\mathrm{Hom}(\mathcal{B}_s,\mathcal{A}_s)$ . By the purely algebraic definition of surjective $\mathbb{R}$ -homomorphisms, this means that we can identify

\[\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}^\circ=\mathrm{Hom}(\mathcal{A}_{s},\mathcal{B}_{s})_{\mathbb{R}}^\circ. \]

Now by [Reference HabeggerHab09a, Lemma 2] and its proof, there is a compact set $\mathcal{K}(\mathcal{A}_{s},\mathcal{B}_{s})\subseteq \mathrm{Hom}(\mathcal{A}_{ s},\mathcal{B}_{ s})_{\mathbb{R}}^\circ$ with the property that for any $f_s\in \mathrm{Hom}(\mathcal{A}_{s},\mathcal{B}_{ s})_{\mathbb{Q}}^\circ$ , there exists $\theta_s\in \mathrm{End}(\mathcal{B}_{ s})_{\mathbb{Q}}^\circ$ such that $\theta_s \circ f_s\in \mathcal{K}(\mathcal{A}_s,\mathcal{B}_s)$ . Here, the rationality of $\theta_s$ follows from the construction of $\theta$ in the proof of [Reference HabeggerHab09a, Lemma 1]. Let $\mathcal{K}(\mathcal{A},\mathcal{B})$ be the corresponding set in $\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}^\circ$ . If x is $\mathcal{B}$ -null, there exists a surjective homomorphism $\varphi:\mathcal{A}\rightarrow \mathcal{B}$ which kills x. Let $\theta\in \mathrm{End}(\mathcal{B})_{\mathbb{Q}}^\circ$ with $f:=\theta\circ \varphi\in \mathcal{K}(\mathcal{A},\mathcal{B})$ . Then for any nonzero $N\in \mathbb{N}$ such that $N\theta$ is a homomorphism, $Nf=N\theta \circ \varphi$ kills x.

3.4 Pullback by $\mathbb{R}$ -homomorphisms

For any $\varphi\in \mathrm{Hom}(\mathcal{A},\mathcal{B})$ , there is a pullback map $\varphi^*:\mathrm{Pic}(\mathcal{B}/S)_{\mathbb{R}}\rightarrow \mathrm{Pic}(\mathcal{A}/S)_{\mathbb{R}}$ which is a linear map preserving symmetry of line bundles. By abuse of notation, we also write $\varphi^*:\mathrm{Pic}_0(\mathcal{B}/S)_{\mathbb{R}}\rightarrow \mathrm{Pic}_0(\mathcal{A}/S)_{\mathbb{R}}$ for the map restricted to symmetric line bundles. First, we extend it to $\mathbb{Q}$ -homomorphisms. Let $f\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}$ . Suppose $Nf\in \mathrm{Hom}(\mathcal{A},\mathcal{B})$ for nonzero $N\in \mathbb{N}$ . Define the pullback

\[f^*:\mathrm{Pic}_0(\mathcal{B}/S)_{\mathbb{R}}\rightarrow \mathrm{Pic}_0(\mathcal{A}/S)_{\mathbb{R}}\]

as $(Nf)^*/N^2$ . This is well-defined since $[N]^*\mathcal{L}=N^2\mathcal{L}$ for a symmetric line bundle $\mathcal{L}$ on an abelian scheme. Therefore, we have

\[ \star_{\mathbb{Q}}: \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}\times \mathrm{Pic}_{0}(\mathcal{B}/S)_{\mathbb{R}} \rightarrow \mathrm{Pic}_{0}(\mathcal{A}/S)_{\mathbb{R}} \]

sending $(f,\mathcal{L})$ to $f^*\mathcal{L}$ .

1. (1) homogeneous, i.e. $(t f_1)^*\mathcal{L}=t^2 f_1^*\mathcal{L}$ ; and

2. (2) quadratic, i.e. $(f_1+t f_2)^*\mathcal{L}$ is a quadratic polynomial in t with coefficients from $\mathrm{Pic}_0(\mathcal{A}/S)$ .

Proof. It is clearly homogeneous of degree 2. We show by induction that

(3.2) \begin{equation} (f_1+n f_2)^*\mathcal{L}=(n^2-n)f_2^*\mathcal{L} + n(f_1+f_2)^*\mathcal{L} - (n-1)f_1^*\mathcal{L} \end{equation}

for any $n\in \mathbb{N}$ . It holds trivially for $n=0,1$ . Assume it holds for n. By the theorem of the cube [Reference Faltings and ChaiFC90, Theorem I.1.3], there is linear equivalence:

\[ (f_1+nf_2+f_2)^*\mathcal{L}= (f_1+nf_2)^*\mathcal{L} + (nf_2+f_2)^*\mathcal{L} + (f_1+f_2)^*\mathcal{L} - f_1^*\mathcal{L} - (nf_2)^*\mathcal{L} - f_2^*\mathcal{L}+ 0^*\mathcal{L}. \]

Note that $0^*\mathcal{L}=\pi^*e^*\mathcal{L}$ is trivial. Using the induction hypothesis and the symmetry of $\mathcal{L}$ , it is immediate to verify that

\[ (f_1+(n+1)f_2)^*\mathcal{L}=(n^2+n)f_2^*\mathcal{L}+(n+1)(f_1+f_2)^*\mathcal{L}-nf_1^*\mathcal{L}. \]

Meanwhile, we have

\[ (f_1-nf_2)^*\mathcal{L}=[(f_1+f_2)+(n+1)(-f_2)]^*\mathcal{L}=(n^2+n)f_2^*\mathcal{L}+(n+1)f_1^*\mathcal{L}-n(f_1+f_2)^*\mathcal{L}. \]

Thus, (3.2) holds for any $n\in \mathbb{Z}$ , or $n\in \mathbb{Q}$ by homogeneity.

As a consequence, we have the following proposition.

Proposition 3.9. There is a unique map

(3.3) \begin{equation} \star: \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}\times \mathrm{Pic}_{0}(\mathcal{B}/S)_{\mathbb{R}} \rightarrow \mathrm{Pic}_{0}(\mathcal{A}/S)_{\mathbb{R}}\end{equation}

sending $(f,\mathcal{L})$ to $f^*\mathcal{L}$ extending the pullback $\star_{\mathbb{Q}}$ , which is a real quadratic form in the first input and linear in the second input. Moreover, if $\mathcal{L}\in \mathrm{Pic}_0(\mathcal{B}/S)_{\mathbb{R}}$ is nef, then $f^*\mathcal{L}$ is nef for any $f\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}$ . In particular, $\star$ descends to

(3.4) \begin{equation} \star:\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}\times \mathrm{NS}(\mathcal{B}/S)_{\mathbb{R}}\rightarrow \mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}.\end{equation}

Proof. Let $\mathcal{L}\in \mathrm{Pic}_0(\mathcal{B}/S)_{\mathbb{R}}$ . Let $f_1,\ldots,f_r\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}$ and $n_1,\ldots,n_r\in \mathbb{Q}$ . We use induction on r to show that a quadratic form satisfies the following generalization of (3.2):

(3.5) \begin{equation} \bigg(\sum_{i=1}^r n_i f_i\bigg)^{\!\!*}\mathcal{L} =\frac{1}{2}\sum_{i,j=1}^r n_i n_j [(f_i+f_j)^*\mathcal{L}-f_i^*\mathcal{L}-f_j^*\mathcal{L}].\end{equation}

Indeed, the cases with $r=1,2$ follow from homogeneity and (3.2). Assuming (3.5) holds for r, to prove it for $(r+1)$ , it is easy to check that by (3.2) and homogeneity, we only need to show (3.5) for the situation where $n_1=\cdots=n_{r+1}=1$ . By taking $n=-1$ in (3.2), we have the identity

\[(f_1+f_2)^*\mathcal{L}-f_1^*\mathcal{L}-f_2^*\mathcal{L}=-(f_1-f_2)^*\mathcal{L}+f_1^*\mathcal{L}+f_2^*\mathcal{L}. \]

Thus, the induction hypothesis gives

\begin{align*} \bigg(\sum_{i=1}^{r+1} f_i\bigg)^{\!\!*}\mathcal{L}&=\bigg(\sum_{i=1}^{r} \bigg(f_i+\frac{1}{r}f_{r+1}\bigg)\bigg)^{\!\!*}\mathcal{L}\nonumber\\ &=\frac{1}{2}\sum_{i,j=1}^r \bigg[-(f_i-f_j)^{\!\!*}\mathcal{L}+\bigg(f_i+\frac{1}{r}f_{r+1}\bigg)^{\!\!*}\mathcal{L} +\bigg(f_j+\frac{1}{r}f_{r+1}\bigg)^{\!\!*}\mathcal{L}\bigg].\end{align*}

Finally, the desired equation follows by applying (3.2) several times to expand the terms above. This completes the induction.

Now, $\star$ can be uniquely defined by taking $f_1,\ldots,f_r$ as a $\mathbb{Q}$ -basis of the finite-dimensional space $\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}$ and allowing $n_i\in \mathbb{R}$ . One checks that (3.5) implies $\star$ is a real quadratic form in the first input. The ‘moreover’ part follows from continuity, since $f^*\mathcal{L}$ is nef for any $f\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{Q}}$ .

4.1 Universal abelian variety and its uniformization

Let $g,n\geq 1$ be integers. Let $D:=\mathrm{diag}(d_1,\ldots,d_g)$ denote a polarization type with positive integers $d_1|\cdots|d_g$ . Denote by $\mathbb{A}_{g,D,n}$ the moduli space of complex abelian varieties of dimension g of polarization type D with level-n-structures, which admits a uniformization map from the Siegel upper half-space

\[\mathfrak{H}_g:=\{Z\in M_{g}(\mathbb{C}) \mid Z=Z^t, \mathrm{Im}(Z) \gt 0\}\]

denoted by $u_{g,D,n}:\mathfrak{H}_g\rightarrow \mathbb{A}_{g,D,n}$ . When $n\geq 3$ , the moduli space $\mathbb{A}_{g,D,n}$ is a fine moduli space, that is, there exists a universal abelian variety $\pi_{g,D,n}:\mathfrak{A}_{g,D,n}\rightarrow \mathbb{A}_{g,D,n}$ ; see [Reference Mumford, Fogarty and KirwanMFK94, Theorem 7.9] and its remark. We note that $\mathfrak{A}_{g,D,n}$ and $\mathbb{A}_{g,D,n}$ are defined over $\mathbb{Q}^{\mathrm{alg}}$ .

The fiber product in the category of complex analytic varieties gives the following diagram.

The fiber of $\mathcal{A}_{\mathfrak{H}_g}$ over $Z\in \mathfrak{H}_g$ is the complex abelian variety $\mathbb{C}^g/(D\mathbb{Z}^g+Z\mathbb{Z}^g)$ with the obvious polarization and level structure. There is a natural uniformization of $\mathcal{A}_{\mathfrak{H}_g}$ given by $\mathbb{C}^g\times\mathfrak{H}_g \rightarrow \mathcal{A}_{\mathfrak{H}_g}$ , such that, over $Z\in \mathfrak{H}_g$ , the map reduces to $\exp:\mathbb{C}^g\rightarrow \mathbb{C}^g/(D\mathbb{Z}^g+Z\mathbb{Z}^g)$ . The composed map

\[u_{g,D,n,a}:\mathbb{C}^g\times \mathfrak{H}_g\rightarrow \mathcal{A}_{\mathfrak{H}_g}\rightarrow \mathfrak{A}_{g,D,n}\]

is a uniformization of the universal abelian variety.

The polarization and level structure are not central to our discussion, as our primary focus is on dimension theory, which is stable under finite coverings. While some flexibility in polarization is needed to handle abelian subschemes or quotients, the specific choice of polarization is not crucial. Therefore, for the rest of this paper, we fix $n=n_0\geq 3$ and omit the indices of polarization and level structure from the notation.

The uniformization maps $u_g$ and $u_{g,a}$ are complex analytic and transcendental. However, the targets have natural algebraic structures, as do the sources, which we now describe. For $\mathfrak{H}_g$ , the embedding

\[\mathfrak{H}_g\hookrightarrow \{Z\in M_g(\mathbb{C}) \mid Z=Z^t\}\cong \mathbb{C}^{g(g+1)/2}\]

realizes $\mathfrak{H}_g$ as an unbounded semi-algebraic subset of the algebraic variety $\mathbb{C}^{g(g+1)/2}$ . A closed analytic subset $Y\subseteq \mathfrak{H}_g$ is said to be irreducible algebraic if Y is a component of the intersection of an algebraic subvariety of $\mathbb{C}^{g(g+1)/2}$ with $\mathfrak{H}_g$ . Similarly, we define the algebraic structure on $\mathbb{C}^g\times \mathfrak{H}_g$ by inheriting algebraic structure from $\mathbb{C}^{g}\times \mathbb{C}^{g(g+1)/2}$ . One goal of bi-algebraic geometry is to understand the interplay between the two algebraic structures of the source and the target for such uniformization maps. The closed analytic subsets that are algebraic both on the source and the target, are of particular importance.

4.2 Weakly special subvarieties

Let $\pi:\mathcal{A}\rightarrow S$ be an abelian scheme over a normal complex algebraic variety S. We also assume $\mathcal{A}$ has a level- $n_0$ -structure and, hence, (3.1). Since $\mathbb{C}(S)/\mathbb{C}$ is a primary extension, the trace $(\mathrm{tr}_{\mathbb{C}(S)/\mathbb{C}}(\mathcal{A}_\eta),\tau)$ exists and constant multi-sections as defined in Definition 3.4 make sense.

1. (1) We say $\mathcal{Y}$ is generically weakly special (respectively, generically special) if $\mathcal{Y}=\mathcal{B}+\sigma$ where $\mathcal{B}\subseteq\mathcal{A}$ is an abelian subscheme and $\sigma$ is a constant (respectively, torsion) multi-section.

2. (2) We say $\mathcal{Y}$ is weakly special (respectively, special), if $\mathcal{Y}$ is generically weakly special (respectively, generically special) as a subvariety of the abelian scheme $\mathcal{A}_{\pi(\mathcal{Y})}$ .

Our definition of weakly special subvarieties differs slightly from the definition of weakly special subvarieties in the context of mixed Shimura varieties by Pink, but is in conformity with Klingler’s convention [Reference KlinglerKli17] in the context of variation of mixed Hodge structures; see Proposition 4.7. Note that generically weakly special subvariety is called ‘generically special subvariety of sg type’ in Gao [Reference GaoGao20a, Definition 1.5].

1. (1) If $\mathcal{Y}'$ is weakly special in $\mathcal{A}'$ , then $\varphi(\mathcal{Y}')$ is weakly special in $\mathcal{A}$ .

2. (2) If $\mathcal{Y}$ is weakly special in $\mathcal{A}$ , then any irreducible component of $\varphi^{-1}(\mathcal{Y})$ is weakly special in $\mathcal{A}'$ .

Before proceeding, let us introduce the following terminology to avoid mentioning the specific choice of polarization and level structure.

The following lemma describes weakly special subvarieties as those that can be contracted by quotients followed by moduli sieves.

1. (1) Let $\mathcal{B}\subseteq \mathcal{A}$ be an abelian subscheme with $\dim(\mathcal{B}/S)=g'$ . Let $\mathcal{A}/\mathcal{B}\rightarrow \mathfrak{A}_{g-g'}$ be a moduli sieve of the quotient abelian scheme. Then an irreducible component $\mathcal{Y}$ of any fiber of the composition $\mathcal{A}\rightarrow \mathcal{A}/\mathcal{B}\rightarrow \mathfrak{A}_{g-g'}$ is weakly special of relative dimension g’.

2. (2) Conversely, if $\mathcal{Y}\subseteq\mathcal{A}$ is a weakly special subvariety, then there exist abelian subschemes $\mathcal{B}_1\subseteq \mathcal{A}_1\subseteq\mathcal{A}|_{\pi(\mathcal{Y})}$ together with a positive integer N, such that $[N]\mathcal{Y}\subseteq\mathcal{A}_1$ and $[N]\mathcal{Y}$ is an irreducible component of a fiber of the composition $\mathcal{A}_1\rightarrow \mathcal{A}_1/\mathcal{B}_1\rightarrow \mathfrak{A}_{g_1-g_1'}$ , where $g_1,g_1'$ are the relative dimensions of $\mathcal{A}_1,\mathcal{B}_1$ , respectively, and $\mathcal{A}_1/\mathcal{B}_1\rightarrow \mathfrak{A}_{g_1-g_1'}$ is a moduli sieve.

Note that for any $t\in \iota(S)$ , an irreducible component T of $\iota^{-1}(t)\subseteq S$ gives rise to an isotrivial abelian scheme $\mathcal{A}/\mathcal{B}\times_S T\rightarrow T$ . Suppose $x\in \mathfrak{A}_{g-g'}$ is a point over t. Then $\iota_a^{-1}(x)|_T$ is a constant section of the above isotrivial abelian scheme. By Poincaré’s complete reducibility § 3.3, there exists a homomorphism $\varphi:\mathcal{A}/\mathcal{B}\rightarrow \mathcal{A}$ such that $q\varphi$ is an isogeny. Then $q^{-1}(\iota_a^{-1}(x)|_T)$ is the translate of $\mathcal{B}\times_S T$ by the constant section $\varphi(\iota_a^{-1}(x)|_T)$ . Therefore, any irreducible component of $q^{-1}(\iota_a^{-1}(x)|_T)$ , which is also an irreducible component of $(\iota_a\circ q)^{-1}(x)$ , is weakly special. We stress that it is, in general, not generically weakly special.

Conversely, if $\mathcal{Y}\subseteq\mathcal{A}$ is weakly special, by definition, there exists a positive integer N such that $[N]\mathcal{Y}$ is a translate of an abelian subscheme $\mathcal{B}_1\subseteq \mathcal{A}|_{\pi(\mathcal{Y})}$ by a constant section $\sigma_c$ . Let $\mathcal{A}_1$ be the abelian scheme which is the sum of $\mathcal{B}_1$ and the constant abelian subscheme in $\mathcal{A}|_{\pi(\mathcal{Y})}$ generated by $\sigma_c$ . Then it is clear that $\mathcal{A}_1,\mathcal{B}_1,N$ satisfy the requirement.

4.3 Characterization and the Ax–Schanuel theorem

First, we note that an arbitrary intersection of algebraic subvarieties is algebraic. It follows formally that any intersection of bi-algebraic subvarieties is bi-algebraic.

1. (1) The Zariski closure $\mathcal{U}^{\mathrm{Zar}}\subseteq \mathbb{C}^g\times \mathfrak{H}_g$ (respectively, $\mathcal{Y}^{\mathrm{Zar}}\subseteq \mathfrak{A}_g$ ) is defined to be the smallest algebraic subvariety containing $\mathcal{U}$ (respectively, $\mathcal{Y}$ ).

2. (2) The bi-algebraic closure $\mathcal{U}^{\mathrm{biZar}}\subseteq \mathbb{C}^g\times \mathfrak{H}_g$ (respectively, $\mathcal{Y}^{\mathrm{biZar}}\subseteq \mathfrak{A}_g$ ) is defined to be the smallest bi-algebraic subvariety containing $\mathcal{U}$ (respectively, $\mathcal{Y}$ ).

The relevant bi-algebraic subvarieties for $u_g$ (respectively, $u_{g,a}$ ) are characterized in [Reference Ullmo and YafaevUY11] (respectively, [Reference GaoGao17a, Reference GaoGao17b]). We have the following characterization of the bi-algebraic subvarieties by Gao in terms of the weakly special subvarieties.

Proposition 4.7. The bi-algebraic subvarieties of $\mathfrak{A}_g$ are exactly those weakly special subvarieties of $\mathfrak{A}_g/\mathbb{A}_g$ which dominate bi-algebraic subvarieties of $\mathbb{A}_g$ . Moreover, if $S\subseteq\mathbb{A}_g$ and $\mathcal{A}=\mathfrak{A}_{g,S}:=\mathfrak{A}_g|_S$ , then the weakly special subvarieties of $\mathcal{A}/S$ come from restriction of bi-algebraic subvarieties, i.e.

\[ \{\mathcal{Y}\subseteq\mathcal{A}\mid \mathcal{Y} \text{ is weakly special}\}=\{\mathcal{Y}\subseteq\mathcal{A}\mid \mathcal{Y} \text{ is an irreducible component of }\mathcal{Y}^{\mathrm{biZar}}\cap\mathcal{A}\}. \]

One of the most powerful tools in studying bi-algebraic geometry is a functional transcendence result called the Ax–Schanuel theorem. Here, we recall a weak version [Reference GaoGao20b, Theorem 3.5] of the Ax–Schanuel theorem in the context of the universal abelian variety, which is sufficient for our purpose.

Theorem 4.8. Let $u_{g,a}:\mathbb{C}^g\times \mathfrak{H}_g\rightarrow \mathfrak{A}_g$ be the uniformization map, and let $\mathcal{U}\subseteq \mathbb{C}^g\times\mathfrak{H}_g$ be a closed analytic irreducible subset. Then

\[ \dim \mathcal{U}^{\mathrm{Zar}} -\dim \mathcal{U} \geq \dim u_{g,a}(\mathcal{U})^{\mathrm{biZar}}-\dim u_{g,a}(\mathcal{U})^\mathrm{Zar}. \]

Since $\pi_g(u_{g,a}(\mathcal{U})^{\mathrm{biZar}})$ contains $u_g(p(\mathcal{U}^{\mathrm{Zar}}))$ , where $p:\mathbb{C}^g\times\mathfrak{H}_g\rightarrow \mathfrak{H}_g$ is the projection, we immediately have the following consequence.

Corollary 4.9. Let $u_{g,a}:\mathbb{C}^g\times \mathfrak{H}_g\rightarrow \mathfrak{A}_g$ be the uniformization map, and let $\mathcal{U}\subseteq \mathbb{C}^g\times\mathfrak{H}_g$ be a closed analytic irreducible subset. Then

\[ \mathrm{rel. dim} \mathcal{U}^{\mathrm{Zar}} -\dim \mathcal{U} \geq \mathrm{rel. dim} u_{g,a}(\mathcal{U})^{\mathrm{biZar}}-\dim u_{g,a}(\mathcal{U})^\mathrm{Zar}. \]

4.4 Finiteness à la Bogomolov–Ullmo

The weakly defect of a locally closed subvariety $\mathcal{Y}\subseteq \mathfrak{A}_g$ is defined as

\[\delta_{ws}(\mathcal{Y}):=\dim \mathcal{Y}^{\mathrm{biZar}} -\dim \mathcal{Y}. \]

Let $\mathcal{X}\subseteq\mathfrak{A}_g$ be a locally closed subvariety. A subvariety $\mathcal{Y}\subseteq\mathcal{X}$ is called weakly optimal in $\mathcal{X}$ if, for any subvariety $\mathcal{Y}'\subseteq\mathcal{X}$ strictly containing $\mathcal{Y}$ , one has $\delta_{ws}(\mathcal{Y}')\gt \delta_{ws}(\mathcal{Y})$ .

Later we need the following finiteness à la Bogomolov–UllmoFootnote ³ result of Gao for weakly optimal subvarieties, which is itself an application of Ax–Schanuel, reformulated into our language as follows.

For any weakly optimal subvariety $\mathcal{Z}$ of $\mathcal{X}$ , there is a triple $(\mathcal{A}_1,\mathcal{B}_1,N)\in \Sigma(\mathcal{X})$ such that $[N]\mathcal{Z}\subseteq\mathcal{A}_1$ and $([N]\mathcal{Z})^{\mathrm{biZar}}$ is an irreducible component of a fiber of the composition

\[ \mathcal{A}_1\rightarrow \mathcal{A}_1/\mathcal{B}_1\rightarrow \mathfrak{A}_{g_1-g_1'}, \]

where the first map is the quotient map and the second is a moduli sieve with $g_1,g_1'$ being the relative dimensions of $\mathcal{A}_1,\mathcal{B}_1$ .Footnote ⁴

Proof. The result [Reference GaoGao20b, Theorem 1.4] (or [Reference GaoGao21, Theorem 3.2.4]) claims that there are finitely many pairs

\[ ((Q,\mathcal{Y}^+),H)\in \Sigma'=\Sigma'(\mathcal{X}), \]

where $(Q,\mathcal{Y}^+)$ is a connected mixed Shimura subdatum and H is a normal subgroup of Q with semi-simple reductive part, such that the bi-algebraic closure of any weakly optimal subvariety $\mathcal{Z}$ is equal to $u_{g,a}(H(\mathbb{R})^+\tilde y)$ for one of the triples with some $\tilde y\in \mathcal{Y}^+$ . By [Reference GaoGao20a, Proposition 5.6], a connected mixed Shimura subdatum $(Q,\mathcal{Y}^+)$ corresponds to a torsion translate of an abelian subscheme $\mathcal{A}_1$ over a special subvariety of $\mathbb{A}_g$ . Let $N\in \mathbb{N}$ be an integer that kills the torsion. By § 5.4 of [Reference GaoGao20a], H gives rise to an abelian subscheme $\mathcal{B}_1\subseteq\mathcal{A}_1$ such that the quotient of $(Q,\mathcal{Y}^+)$ by H corresponds to a moduli sieve of the quotient $\mathcal{A}_1/\mathcal{B}_1$ . The fact that $\mathcal{Z}^{\mathrm{biZar}}=u_{g,a}(H(\mathbb{R})^+\tilde y)$ means that $([N]\mathcal{Z})^{\mathrm{biZar}}$ is an irreducible component of a fiber of $\mathcal{A}_1\rightarrow \mathcal{A}_1/\mathcal{B}_1\rightarrow \mathfrak{A}_{g_1-g_1'}$ . Thus, we are done.

5.1 Betti map and Betti foliation

Let $ \Delta\subseteq S$ be a simply connected analytic open neighborhood of $s\in S$ . The local system $\mathbb{V}_{\mathbb{Z}}:=(R^1\pi_*\mathbb{Z})^\vee$ is trivial on $\Delta$ and there are trivializations $\mathbb{V}_{\mathbb{Z},\Delta}\cong \mathbb{Z}^{2g}\times \Delta$ , and $\mathcal{V}_{\Delta}\cong \mathbb{C}^{2g}\times \Delta$ as holomorphic vector bundles. Denote the induced Hodge filtration on $\mathbb{C}^{2g}\times \Delta$ by $\mathcal{F}^{\prime \bullet}$ . The composition map

\[ \mathbb{R}^{2g}\times \Delta\rightarrow \mathbb{C}^{2g}\times \Delta\rightarrow \mathbb{C}^{2g}\times \Delta/\mathcal{F}^{\prime 0} \]

is a real analytic diffeomorphism which restricts to a group isomorphism over any $s\in \Delta$ . Taking the quotient by $\mathbb{Z}^{2g}\times \Delta$ , it induces a real analytic diffeomorphism

(5.1) \begin{equation} \mathbb{T}^{2g}\times \Delta\cong \mathcal{A}_\Delta, \end{equation}

which restricts to a real Lie group isomorphism of tori $\mathbb{T}^{2g}\cong\mathcal{A}_s$ for any $s\in \Delta$ . The composition of its inverse with the projection

\[ \beta_\Delta:\mathcal{A}_\Delta\cong \mathbb{T}^{2g}\times \Delta \rightarrow \mathbb{T}^{2g} \]

is called a Betti map associated to the data $(\mathcal{A}/S,\Delta)$ . The Betti map $\beta_\Delta$ is uniquely determined up to an action of $\mathrm{GL}_{2g}(\mathbb{Z})$ on the target. To further remove the ambiguity, we can use the isomorphism $\beta_\Delta|_{\mathcal{A}_{s}}:\mathcal{A}_s\rightarrow \mathbb{T}^{2g}$ to define the composition

\[ \beta_{\Delta,s}:\mathcal{A}_\Delta\xrightarrow{\beta_\Delta} \mathbb{T}^{2g}\xrightarrow {(\beta_{\Delta}|_{\mathcal{A}_s})^{-1}} \mathcal{A}_s \]

which is called the Betti map associated to the data $(\mathcal{A}/S,\Delta,s)$ .

The Betti map is real analytic but, in general, not holomorphic. On the other hand, any fiber of $\beta_\Delta$ , being the image of

\[ \{c\}\times \Delta\subseteq \mathbb{C}^{2g}\times \Delta\rightarrow (\mathbb{Z}^{2g}\times \Delta)\backslash(\mathbb{C}^{2g}\times \Delta)/\mathcal{F}'^0 \]

for some $c\in \mathbb{C}^{2g}$ , is complex analytic, and gives rise to a local complex analytic foliation on $\mathcal{A}_\Delta$ . These local foliations patch together to a global foliation $\mathcal{F}_{\text{Betti}}$ on $\mathcal{A}$ , which is called the Betti foliation. A Betti leaf is a path-connected piece of the Betti foliation, and a path-connected component in the Betti foliation is called a maximal Betti leaf. We denote the maximal Betti leaf passing through x by $\mathcal{F}_{\text{Betti},x}$ . For instance, a constant multi-section is a finite union of maximal Betti leaves. In general, a Betti leaf can be dense in $\mathcal{A}$ under the analytic topology. The Betti foliation induces a splitting of the holomorphic tangent space

(5.2) \begin{equation} T_x'\mathcal{A}= T_x'\mathcal{F}_{\text{Betti}} \oplus T_x'\mathcal{A}_s \end{equation}

at a point $x\in \mathcal{A}_s$ with $s\in S$ . Pulling back via the identity section, the splitting gives

(5.3) \begin{equation} T'\mathcal{A}|_S:=e^*T'\mathcal{A}=T' S\oplus \mathrm{Lie}(\mathcal{A}/S)\end{equation}

where $\mathrm{Lie}(\mathcal{A}/S)$ represents the Lie algebra of $\mathcal{A}/S$ .

5.2 An interlude to complex geometry

Before defining Betti forms, let us quickly review a few concepts from complex geometry. Let Y be a complex manifold and let $\omega$ be a real $C^\in fty$ (1, 1)-form on Y. One way of thinking about the real $C^\in fty$ (1,1)-form $\omega$ is as an alternating $\mathbb{R}$ -bilinear pairing $\omega_y:T_{y}Y \otimes_{\mathbb{R}} T_{y}Y\rightarrow \mathbb{R}$ on the real tangent bundle T Y of Y, varying smoothly in $y\in Y$ , which satisfies $\omega_{y}(Jv,Jv')=\omega_y(v,v')$ for any $v,v'\in T_{y} Y$ with $J:T Y\rightarrow T Y$ the almost complex structure. If we view $\omega$ as a $C^\in fty$ map of vector bundles $TY\rightarrow T^*Y$ , then its kernel $\ker\omega$ is given by

\[(\ker\omega)_y:=\ker\omega_y:=\{v\in T_y Y: \omega_y(v,v')=0\text{ for any }v'\in T_yY\}.\]

Clearly, $\ker \omega$ is stable under J. The kernel $\ker\omega$ is a vector bundle if and only if the rank of $\ker\omega$ is locally constant. Assume, moreover, that $\omega$ is semipositive, i.e. $\omega(v,Jv)\geq0$ for any $v\in TY$ . Then

\[\ker\omega_y=\{v\in T_{y} Y: \omega(v,Jv)=0\}. \]

Indeed, for the less obvious direction with $\omega(v,Jv)=0$ , note that for $v'\in T_{y} Y$ , we have

\[\omega_y(v+kv',J(v+kv'))=k^2\omega_y(v',Jv')+2k\omega_y(v,Jv')\geq 0 \quad (\forall k\in \mathbb{R}) \]

from which we derive $\omega_y(v,Jv')=0$ . If $Z\subseteq Y$ is a complex submanifold and $z\in Z$ , then the volume form $\omega_z^{\wedge \dim Z}|_Z\neq 0$ if and only if $\omega_z$ is positive on $T_z Z$ , if and only if $ \ker\omega_z \cap T_z Z=\{0\}$ .

The next thing we recall is the first Chern class of line bundles. It is well-known that there is a first Chern class map for holomorphic line bundles

\[c_1: \mathrm{Pic}(Y)= H^1(Y,\mathcal{O}_Y^*)\rightarrow H^2(Y,\mathbb{Z}).\]

There is an analogous map for $C^\in fty$ complex line bundles. The exact sequence

\[0\rightarrow \mathbb{Z}\rightarrow \mathcal{R}_Y\xrightarrow{\exp} \mathcal{R}_Y^*\rightarrow 0, \]

where $\mathcal{R}_Y$ (respectively, $\mathcal{R}_Y^*$ ) is the sheaf of $C^\in fty$ complex functions (respectively, nonzero $C^\in fty$ complex functions), gives rise to an injective homomorphism

(5.4) \begin{equation}c_1': H^1(Y,\mathcal{R}_Y^*)\hookrightarrow H^2(Y,\mathbb{Z})\end{equation}

such that $c_1$ factors through $c_1'$ ; cf. [Reference Griffiths and HarrisGH94, p. 140]. Here $H^1(Y,\mathcal{R}_Y^*)$ is canonically identified as the group of $C^\in fty$ complex line bundles L up to isomorphism using transition functions. Via the de Rham theory, we also regard the first Chern class $c_1'(L)$ as an integral class of real 2-forms in $H^2(Y,\mathbb{R})\cong H^2_{\mathrm{dR}}(Y,\mathbb{R})$ ; a representative in the class can be given by ${\sqrt{-1}}/{2\pi}\Theta(D)$ , called the Chern form of (L,D), where $\Theta(D)$ is the curvature form of a hermitian connection D with respect to a hermitian metric on L; see [Reference DemaillyDem12, V.9].

5.3 Betti form

Now let $\mathcal{L}\in \mathrm{Pic}(\mathcal{A}/S)$ . For $s\in S$ , there is a unique translation-invariant closed (1,1)-form $\omega_s$ on $\mathcal{A}_s$ representing the Chern class $c_1(\mathcal{L}_s)$ on $\mathcal{A}_s$ . The Betti form associated to $\mathcal{L}$ is the (1,1)-form $\omega=\omega(\mathcal{L})$ on $\mathcal{A}$ such that:

1. (1) for any $s\in S$ , the restriction $\omega|_{T\mathcal{A}_s}=\omega_s$ ; and

2. (2) the form $\omega$ is rigidified along the Betti foliation, namely, $\omega|_{T\mathcal{F}_{\text{Betti}}}=0$ .

This uniquely determines $\omega$ since it specifies the pairing $\omega_x:T_x\mathcal{A}\times T_x\mathcal{A}\rightarrow \mathbb{R}$ at any point $x\in \mathcal{A}$ by the splitting (5.2).

To see the continuous structure of Betti forms, we give a local construction. Let $\beta_{\Delta,s_0}:\mathcal{A}_\Delta \rightarrow \mathcal{A}_{s_0}$ be a Betti map and let $\omega_\Delta:=\beta_{\Delta,s_0}^*\omega_{s_0}$ . By definition, $\omega_\Delta$ satisfies part (2) as fibers of $\beta_{\Delta,s_0}$ are the Betti leaves. To see part (1), we regard $\mathcal{L}_\Delta$ as a $C^\in fty$ complex line bundle on $\mathcal{A}_\Delta$ , and consider the corresponding line bundle $\mathcal{L}_{\Delta}'$ on $\mathbb{T}^{2g}\times \Delta$ via the $C^\in fty$ diffeomorphism (5.1). Given any hermitian metric on $\mathcal{L}'_\Delta$ and a hermitian connection D, the association of Chern forms on $\mathcal{L}'_\Delta$ restricted to $\mathbb{T}^{2g}\times \{s\}$ induces a continuous map

\[\Delta\rightarrow H^2_{\mathrm{dR}}(\mathbb{T}^{2g},\mathbb{R})\cong H^2(\mathbb{T}^{2g},\mathbb{R}). \]

It factors through the lattice $H^2(\mathbb{T}^{2g},\mathbb{Z})$ and hence must be constant. This implies that for any $s\in \Delta$ , the invariant form $\omega_s$ when viewed on $\mathbb{T}^{2g}$ is independent of s. In particular, $\omega_{\Delta}|_{T\mathcal{A}_s}=\omega_s$ for any $s\in \Delta$ . Patching together by uniqueness, we obtain the desired Betti form.

By definition, the Betti form of $\mathcal{L}$ only depends on its class $[\mathcal{L}]\in \mathrm{NS}(\mathcal{A}/S)$ . Linearity allows us to extend the definition to $\mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}$ . It is not hard to establish the following properties of Betti forms.

Proposition 5.1. The association of Betti forms

\[ \omega:\mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}\rightarrow \Gamma(\Lambda^2 T^*\mathcal{A}) \]

is an injective linear map. Here, $\Gamma(\Lambda^2 T^*\mathcal{A})$ denotes the set of global sections of alternating 2-forms on $\mathcal{A}$ . For $[\mathcal{L}]\in \mathrm{NS}(\mathcal{A}/S)_{\mathbb{R}}$ , we have:

1. (1) $\omega(\mathcal{L})$ is a closed, real $C^\in fty$ (1,1)-form;

2. (2) $\omega(\mathcal{L})$ is semipositive if $\mathcal{L}$ is nef;

3. (3) $[N]^*\omega(\mathcal{L})=N^2\omega(\mathcal{L})$ ;

4. (4) $\ker\omega(\mathcal{L})$ is a $C^\in fty$ complex vector bundle on $\mathcal{A}$ .

Next, we discuss pullback of Betti forms under $\mathbb{R}$ -homomorphisms. One concrete way to understand $\mathbb{R}$ -homomorphisms in the complex analytic setting is via Hodge theory. Let $\mathbb{V}_{\mathbb{Z}}=\mathbb{V}_{\mathbb{Z},\mathcal{A}}:=(R^1\pi_{*}\mathbb{Z})^\vee$ be the $\mathbb{Z}\mathrm{VHS}$ induced by $\pi:\mathcal{A}\rightarrow S$ . Let $\mathcal{V}$ be the associated holomorphic vector bundle to $\mathbb{V}_{\mathbb{C}}$ with Hodge filtration $F^\bullet \mathcal{V}$ . Then we can identify

\[\mathrm{Lie}(\mathcal{A}/S)=\mathcal{V}/F^0\mathcal{V}\text{ and }\mathcal{A}=\mathbb{V}_{\mathbb{Z}}\backslash\mathcal{V}/F^0\mathcal{V}.\]

The quotient map $\exp:\mathrm{Lie}(\mathcal{A}/S)\rightarrow \mathcal{A}$ is called the exponential map.

Now an $\mathbb{R}$ -homomorphism $f:\mathcal{A}\dashrightarrow \mathcal{B}$ can be regarded as a map of $\mathbb{R}\mathrm{VHS}$

\[f_{\mathbb{R}}:\mathbb{V}_{\mathbb{R},\mathcal{A}}\rightarrow \mathbb{V}_{\mathbb{R},\mathcal{B}}. \]

Note that f is integral if and only if $f_{\mathbb{R}}$ maps $\mathbb{V}_{\mathbb{Z},\mathcal{A}}$ to $V_{\mathbb{Z},\mathcal{B}}$ . Taking into account the Hodge filtration, we have the following commutative diagram.

This defines a functorial embedding

(5.5) \begin{equation}d:\mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}\hookrightarrow \mathrm{Hom}(\mathrm{Lie}(\mathcal{A}/S), \mathrm{Lie}(\mathcal{B}/S))\end{equation}

into the group of global sections of the Hom-bundle. If f is surjective, then df is surjective since a left inverse exists by functoriality.

Alternatively, we can regard df as the essential component of the differential map of f on tangent spaces over the identity sections. Namely, under the decomposition (5.3), the differential of f is identity on T’S and df on the Lie algebras.

Define the pullback of a Betti form $\omega(\mathcal{L})$ for $[\mathcal{L}]\in \mathrm{NS}(\mathcal{B}/S)_{\mathbb{R}}$ by $f\in \mathrm{Hom}(\mathcal{A},\mathcal{B})_{\mathbb{R}}$ as

\[f^*\omega(\mathcal{L}):=\omega(f^*\mathcal{L})\]

with $f^*\mathcal{L}$ defined in (3.4). This is consistent with the usual pullback of differential forms when f is integral. By continuity, we have $(df)^*\exp^*\omega(\mathcal{L})=\exp^*f^*\omega(\mathcal{L})$ .

Note first $\ker\omega(\mathcal{L})|_S=T'S$ on the identity section of $\mathcal{B}$ by ampleness. Then $\ker f^*\omega(\mathcal{L})|_S$ consists of the tangent vectors which are mapped to T’S by the differential of f. But the differential of f is identity on T’S and df on the Lie algebras under (5.3). Hence,

\[\ker f^*\omega(\mathcal{L})|_S=T'S\oplus \mathcal{W}\]

is identical to the tangent space of $\mathcal{W}$ or $\mathcal{H}$ on the zero section, and we are done.

6.1 Degeneracy loci

We defined earlier in Definition 4.2 the notion of a weakly special (respectively, special) subvariety of $\mathcal{A}/S$ . Components of intersections of (weakly) special subvarieties are (weakly) special.

1. (1) The weakly special closure $\langle \mathcal{Y} \rangle$ (respectively, special closure) of $\mathcal{Y}$ is defined as the smallest weakly special (respectively, special) subvariety containing $\mathcal{Y}$ .

2. (2) The vertical defect $\delta_v(\mathcal{Y})$ is defined as the difference of the relative dimension of its weakly special closure $\langle\mathcal{Y}\rangle$ with the dimension of $\mathcal{Y}$ . That is,

   \[ \delta_v(\mathcal{Y}):=\mathrm{rel. dim} \langle\mathcal{Y}\rangle -\dim\mathcal{Y}. \]

We have the following lemma.

Lemma 6.2. Let $\iota_a:\mathcal{A}\rightarrow \mathfrak{A}_g$ be a moduli sieve. For a subvariety $\mathcal{Y}\subseteq \mathcal{A}$ , we have

\[ \mathrm{rel. dim} \langle \mathcal{Y}\rangle =\mathrm{rel. dim}\langle \iota_a(\mathcal{Y})\rangle. \]

Proof. The pullback of $\langle\iota_a(\mathcal{Y})\rangle$ is weakly special and contains $\mathcal{Y}$ . Thus,

\[ \mathrm{rel. dim} \langle \mathcal{Y}\rangle \leq \mathrm{rel. dim}\langle \iota_a(\mathcal{Y})\rangle. \]

The reverse inequality follows from Lemma 4.3.

The following definition was introduced by Gao [Reference GaoGao20a] to study Betti maps.

A subvariety $\mathcal{Z}\subseteq\mathcal{X}$ is said to be vertically optimal in $\mathcal{X}$ if, for any subvariety $\mathcal{Y}\subseteq\mathcal{X}$ strictly containing $\mathcal{Z}$ , one has $\delta_v(\mathcal{Y})\gt\delta_v(\mathcal{Z})$ . Note that the degeneracy locus $\mathcal{X}^{\mathrm{deg}}(t)$ can be viewed as the union of positive-dimensional vertically optimal subvarieties with vertical defect $<t$ .

As a possibly infinite union of subvarieties, the degeneracy locus is indeed algebraic by [Reference GaoGao20a, Theorem 1.8]. Gao’s proof works for subvarieties of $\mathfrak{A}_g$ . For the reader’s convenience, we give a proof below following a similar idea. First, we make a convenient definition.

1. (1) $[N]\mathcal{Y} \subseteq \mathcal{A}_1$ ; and

2. (2) $\langle [N]\mathcal{Y}\rangle=\mathcal{B}_1|_{\pi(\mathcal{Y})}+\sigma_1$ for a constant multi-section $\sigma_1\subseteq \mathcal{A}_1|_{\pi(\mathcal{Y})}$ .

In this case, we shall say $\mathcal{Y}$ belongs to the type $(\mathcal{A}_1,\mathcal{B}_1,N)$ .

The idea is to first generalize the finiteness result in Theorem 4.10 to the finiteness of types of vertically optimal subvarieties.

Proof. ${\textit{Case }\mathcal{A}=\mathfrak{A}_g}.$ We claim that vertically optimal implies weakly optimal. Assuming the claim, the result follows directly from Theorem 4.10. To prove the claim, if $\mathcal{Z}\subseteq\mathcal{X}$ is vertically optimal, and let $\mathcal{Y}\subseteq\mathcal{X}$ contain $\mathcal{Z}$ , by definition and Proposition 4.7, we have

\[ \mathrm{rel. dim}\mathcal{Y}^{\mathrm{biZar}}-\dim \mathcal{Y}\gt\mathrm{rel. dim}\mathcal{Z}^{\mathrm{biZar}}-\dim \mathcal{Z}. \]

Since $\pi(\mathcal{Z}^{\mathrm{biZar}})=\pi(\mathcal{Z})^{\mathrm{biZar}}$ and similarly for $\mathcal{Y}'$ , we derive from $\pi(\mathcal{Y})\supseteq\pi(\mathcal{Z})$ that

\[ \dim \pi(\mathcal{Y}^{\mathrm{biZar}})\geq\dim \pi(\mathcal{Z}^{\mathrm{biZar}}). \]

Combining the two inequalities, we see that $\delta_{\mathrm{ws}}(\mathcal{Y})\gt\delta_{\mathrm{ws}}(\mathcal{Z})$ .

${\textit{General Case}}$ . We claim that if $\mathcal{X}=\mathcal{U}\sqcup\mathcal{X}_1$ with $\mathcal{U}$ Zariski open, and the result holds for $\mathcal{U},\mathcal{X}_1$ , then we can take $\Sigma(\mathcal{X}):=\Sigma(\mathcal{U})\cup\Sigma(\mathcal{X}_1)$ . To show this, if $\mathcal{Z}$ is a vertically optimal subvariety of $\mathcal{X}$ , then $\mathcal{Z}$ must be Zariski closed. If $\mathcal{Z}\subseteq\mathcal{X}_1$ , then $\mathcal{Z}$ is vertically optimal in $\mathcal{X}_1$ by definition. Otherwise, $\mathcal{Z}\cap \mathcal{U}$ is vertically optimal in $\mathcal{U}$ . Thus, the union $\Sigma(\mathcal{U})\cup\Sigma(\mathcal{X}_1)$ takes into account all possible vertically optimal subvarieties of $\mathcal{X}$ .

Now take a moduli sieve $\iota:\mathcal{A}\rightarrow \mathfrak{A}_{g}$ . By Chevalley’s upper semi-continuity theorem [Reference GrothendieckGro66, Theorem 13.1.3], there exists a nonempty open subset of $\mathcal{U}\subseteq \mathcal{X}$ such that for any scheme-theoretic point $x\in \mathcal{U}$ , the fiber of $\iota|_{\mathcal{X}}$ over $\iota(x)$ has pure dimension $\dim \mathcal{X}-\dim\iota(\mathcal{X})$ . By the claim, it suffices to work on $\mathcal{U}$ using Noetherian induction. Without loss of generality, we assume $\iota|_{\mathcal{X}}$ has pure dimension r over any point of $\iota(\mathcal{X})$ . In particular, if $\mathcal{Y}'$ is a subvariety of $\iota(\mathcal{X})$ and $\mathcal{Y}$ is an irreducible component of $\iota|_{\mathcal{X}}^{-1}(\mathcal{Y}')$ , then using Lemma 6.2,

\[ \delta_v(\mathcal{Y})=\mathrm{rel. dim}\langle \mathcal{Y}\rangle -\dim\mathcal{Y} =\mathrm{rel. dim}\langle \mathcal{Y}'\rangle-\dim\mathcal{Y}'-r=\delta_v(\mathcal{Y}')-r. \]

Next, we observe that if $\mathcal{Z}$ is vertically optimal in $\mathcal{X}$ , then $\mathcal{Z}$ must be a component of $\iota|_{\mathcal{X}}^{-1}(\iota(\mathcal{Z}))$ . Indeed, let $\mathcal{Y}$ be a component of $\iota|_{\mathcal{X}}^{-1}(\iota(\mathcal{Z}))$ containing $\mathcal{Z}$ . By Lemma 6.2,

\[ \mathrm{rel. dim}\langle \mathcal{Z} \rangle =\mathrm{rel. dim}\langle \iota(\mathcal{Z}) \rangle =\mathrm{rel. dim}\langle \mathcal{Y}\rangle.\]

Since $\delta_v(\mathcal{Y})\geq \delta_v(\mathcal{Z})$ , we must have $\dim \mathcal{Y}=\dim\mathcal{Z}$ and $\mathcal{Y}=\mathcal{Z}$ .

As a consequence, the vertically optimal subvarieties of $\mathcal{X}$ correspond to those of $\iota(\mathcal{X})$ in the following way.

1. (1) If $\mathcal{Z}$ is vertically optimal in $\mathcal{X}$ , then $\iota(\mathcal{Z})$ is vertically optimal in $\iota(\mathcal{X})$ .

2. (2) If $\mathcal{Z}'$ is vertically optimal in $\iota(\mathcal{X})$ , then any irreducible component of $\iota|_{\mathcal{X}}^{-1}(\mathcal{Z}')$ is vertically optimal in $\iota(\mathcal{X})$ .

Finally, we construct $\Sigma(\mathcal{X})$ by pulling back the triples in the set $\Sigma(\iota(\mathcal{X}))$ , which is defined in the case $\mathcal{A}=\mathfrak{A}_g$ . More precisely, let $\Sigma(\mathcal{X})$ be the set of $(\mathcal{A}_1,\mathcal{B}_1,N)$ with the property that $(\iota(\mathcal{A}_1),\iota(\mathcal{B}_1),N)\in \Sigma(\iota(\mathcal{X}))$ and $\mathcal{A}_1$ is a component of $\iota^{-1}(\iota(\mathcal{A}_1))$ . Then it is immediate to check that $\Sigma(\mathcal{X})$ has the required property. This completes the Noetherian induction.

With the above result, we can conclude the proof as in [Reference GaoGao20a, § 7.2].

Proof. Take a type $(\mathcal{A}_1,\mathcal{B}_1,N)\in \Sigma(\mathcal{X})$ as defined in Theorem 6.5. Consider the composition

\[\varphi:[N]^{-1}\mathcal{A}_1\xrightarrow{[N]} \mathcal{A}_1\rightarrow \mathcal{A}_1/\mathcal{B}_1 \rightarrow \mathfrak{A}_{g_1-g'_1},\]

where $[N]^{-1}\mathcal{A}_1$ is the preimage of $\mathcal{A}_1$ under $[N]:\mathcal{A}\rightarrow \mathcal{A}$ , $g_1,g_1'$ are the respective relative dimensions of $\mathcal{A}_1,\mathcal{B}_1$ , and the last morphism is a moduli sieve. Irreducible components of fibers of $\varphi$ are weakly special of relative dimension $g_1'$ by Lemma 4.5. Denote the restriction of $\varphi$ onto $\mathcal{X}\cap[N]^{-1}\mathcal{A}_1$ as $\varphi|_{\mathcal{X}}$ . Let

\[Z=Z(\mathcal{A}_1,\mathcal{B}_1,N):=\{x\in \mathcal{X}\cap[N]^{-1}\mathcal{A}_1 \mid \dim_x \varphi|_{\mathcal{X}}^{-1}(\varphi(x))\gt\max \{0, g_1'-t\}\},\]

which is Zariski closed by Chevalley’s upper semi-continuity theorem [Reference GrothendieckGro66, Theorem 13.1.3]. Then $Z\subseteq \mathcal{X}^{\mathrm{deg}}(t)$ since the vertical defect of any (positive-dimensional) irreducible component of fiber $\varphi|_{\mathcal{X}}^{-1}(\varphi(x))$ is less than

\[g_1'-\max\{0,g_1'-t\}\leq \min\{g_1',t\}\leq t.\]

Now the union of such Z for the finitely many triples $(\mathcal{A}_1,\mathcal{B}_1,N)\in \Sigma(\mathcal{X})$ is also Zariski closed, and is indeed $\mathcal{X}^{\mathrm{deg}}(t)$ since all vertically optimal subvarieties have been taken into account.

Along the path, we obtain the following natural generalization of Gao’s criterion [Reference GaoGao20a, Theorem 8.1] for the case $t\leq 0$ and [Reference GaoGao24, Theorem 2.4] for the case $\mathcal{A}=\mathfrak{A}_g$ .

1. (1) $\mathcal{X}=\mathcal{X}^{\deg}(t)$ ;

2. (2) there exists an abelian subscheme $\mathcal{B}$ of relative dimension g’ such that the composition

   \[ \varphi: \mathcal{A}\rightarrow \mathcal{A}/\mathcal{B} \rightarrow \mathfrak{A}_{g-g'}, \]
   where $\mathcal{A}\rightarrow\mathcal{A}/\mathcal{B}$ is the quotient map and $\mathcal{A}/\mathcal{B}\rightarrow \mathfrak{A}_{g-g'}$ is a moduli sieve, satisfies
   \[ \dim\varphi(\mathcal{X})<\dim \mathcal{X} -\max \{0,g'-t\}. \]

Proof. For sufficiency, note that irreducible components of the fibers of $\varphi$ are weakly special of relative dimension g’ by Lemma 4.5. By assumption, any irreducible component $\mathcal{Y}$ of a fiber of $\varphi|_{\mathcal{X}}$ has dimension $\dim \mathcal{Y}\gt\max\{0,g'-t\}$ . Thus, $\mathcal{Y}$ is positive-dimensional and

\[ \delta_v(\mathcal{Y})=\dim\langle \mathcal{Y}\rangle -\dim \mathcal{Y}<g'-\max\{0,g'-t\}=\min\{g',t\}\leq t,\]

which means $\mathcal{Y}\subseteq \mathcal{X}^{\mathrm{deg}}(t)$ .

To prove necessity, by the proof of Theorem 6.6, we can express $\mathcal{X}^{\deg}(t)$ as a finite union of Zariski-closed sets of the form $Z(\mathcal{A}_1,\mathcal{B}_1,N)$ . If $\mathcal{X}^{\deg}(t)=\mathcal{X}$ , then there is one $(\mathcal{A}_1,\mathcal{B}_1,N)\in \Sigma(\mathcal{X})$ with $Z(\mathcal{A}_1,\mathcal{B}_1,N)=\mathcal{X}$ . In particular, we have $\mathcal{X}\subseteq[N]^{-1}\mathcal{A}_1$ , which implies $\mathcal{A}_1=\mathcal{A}$ by assumption. Moreover, by definition of $Z(\mathcal{A}_1,\mathcal{B}_1,N)$ , the fiber dimension of $\varphi|_{\mathcal{X}}$ is greater than $\max \{0,g'-t\}$ at any point $x\in \mathcal{X}$ . Hence, we have

\[ \dim \varphi(\mathcal{X})<\dim \mathcal{X} -\max \{0,g'-t\}. \]

Thus, we are done.

We remark that the assumption above ensures that $\mathcal{X}$ is necessarily dominant over S, since the union of $\mathcal{A}|_{\pi(\mathcal{X})}$ with the zero section of $\mathcal{A}$ is a group subscheme of $\mathcal{A}$ . On the other hand, such an assumption is minor as in Pink’s Conjecture 1.4. In defining the degeneracy locus, the choice of the ambient abelian scheme is immaterial, and we may replace $\mathcal{A}$ with the smallest group subscheme of $\mathcal{A}|_{\pi(\mathcal{X})}$ containing $\mathcal{X}$ . This will yield an abelian subscheme $\mathcal{A}'$ of $\mathcal{A}|_{\pi(\mathcal{X})}$ if we further replace $\mathcal{X}$ with $[N]\mathcal{X}$ for some positive integer N. Then Theorem 6.7 applies in the setting $[N]\mathcal{X}\subseteq\mathcal{A}'$ .

6.2 The criterion

We keep our setup and notation from the beginning of the current section. Let $\widetilde{\mathcal{X}}\subseteq \mathrm{Lie}(\mathcal{A}/S)$ be a connected component of $\exp^{-1}(\mathcal{X})$ . We have the following diagram.

Let $\beta:\mathcal{B}_\Delta\rightarrow \mathbb{T}^{2g'}$ be a Betti map on $\mathcal{B}$ associated to the simply connected open neighborhood $\Delta\subseteq S$ . Consider the composition

\[\psi_\Delta: \mathrm{Lie}(\mathcal{A}_\Delta/\Delta)\xrightarrow{df}\mathrm{Lie}(\mathcal{B}_\Delta/\Delta)\xrightarrow{\exp}\mathcal{B}_\Delta\xrightarrow{\beta_\Delta}\mathbb{T}^{2g'}.\]

Note that $\psi_\Delta$ has complex analytic fibers.

Lemma 6.8. The volume form $\omega^{\wedge d}|_{\mathcal{X}}$ is nonzero at $x\in \mathcal{X}^{\mathrm{sm}}\cap \mathcal{A}_\Delta$ if and only if

\[ \psi_\Delta|_{\widetilde{\mathcal{X}}}:\widetilde{\mathcal{X}}\longrightarrow \mathbb{T}^{2g'} \]

is an immersion at $\tilde x$ , where $\tilde x\in \widetilde{\mathcal{X}}$ is a point above x.

Recall that a $C^\in fty$ -map $\varphi:\mathcal{M}\rightarrow \mathcal{N}$ between two differentiable manifolds is an immersion at p if the differential $d\varphi_p:T_p\mathcal{M}\rightarrow T_{\varphi(p)}\mathcal{N}$ is injective.

Proof. Note that $\ker (d\psi_\Delta)_{\tilde x}$ is exactly the lift of $\ker\omega_x$ . Thus,

\[ \ker(d\psi_\Delta|_{\widetilde{\mathcal{X}}})_{\tilde x}=\ker (d\psi_\Delta)_{\tilde x}\cap T_{\tilde x}\widetilde{\mathcal{X}} \]

is isomorphic to $\ker\omega_x\cap T_x\mathcal{X}$ . The equivalence follows by the discussion in § 5.2, since we know $\omega^{\wedge d}_x|_{\mathcal{X}}\neq 0$ precisely when $\ker\omega_x\cap T_x\mathcal{X}=0.$

Now we can give a necessary condition for the vanishing of the volume form.

Proof. For the first statement, by the o-minimal Chow’s theorem [Reference Peterzil and StarchenkoPS09, Corollary 4.5], it suffices to show that $\widetilde{\mathcal{W}}$ is definable in $\mathbb{R}_{\mathrm{an,exp}}$ (or any o-minimal structure). This follows from the algebraicity of $\widetilde S$ and local triviality of $\widetilde{\mathcal{W}}$ by Proposition 5.3 over the simply connected definable open subsets of $\widetilde S$ . For the second statement, Proposition 4.7 implies that the endomorphism groups of $\mathcal{A}$ and $\mathcal{A}^{\mathrm{biZar}}$ coincide. In particular, the $\mathbb{R}$ -homomorphism $\mathcal{A}\dashrightarrow \mathcal{B}$ can be extended to an $\mathbb{R}$ -homomorphism of abelian schemes $f_1:\mathcal{A}_1\dashrightarrow \mathcal{B}_1$ over $S^{\mathrm{biZar}}$ and we have

\[ \mathcal{W}=\ker(df)= \ker (df_1)|_S. \]

The conclusion follows.

For $x\in \mathcal{X}^{\mathrm{sm}}$ , the fiber of $\psi_\Delta|_{\widetilde{\mathcal{X}}}$ over $\psi_\Delta(\tilde x)$ is positive-dimensional by Lemma 6.8, and complex analytic. Let $\widetilde F$ be a connected component of the fiber containing $\tilde x$ and $F:=\exp(\widetilde F)\subseteq \mathcal{A}$ . We lift F to an analytic irreducible subset of $\mathbb{C}^g\times \mathfrak{H}_g$ by lifting $\Delta\subseteq S$ to $\widetilde\Delta\subseteq\widetilde S$ , and by slight abuse of notation, still denote it by $\widetilde F$ . By Corollary 4.9, we get

\[ \delta_{v}(F^{\mathrm{Zar}})\leq \mathrm{rel. dim}\widetilde{F}^{\mathrm{Zar}} -\dim \widetilde{F}. \]

But $\widetilde F$ is contained in $\widetilde{\mathcal{W}}+\sigma$ for some horizontal section $\sigma=(r_0,s)\in \mathbb{R}^{2g}\times \widetilde S\cong \mathbb{C}^g\times \widetilde S$ with $r_0$ fixed. Since $\widetilde{\mathcal{W}}$ is algebraic by Lemma 6.10, we get $\widetilde{F}^{\mathrm{Zar}}\subseteq \widetilde{\mathcal{W}}+\sigma$ . This implies

\[ \mathrm{rel. dim}\widetilde{F}^{\mathrm{Zar}}\leq \mathrm{rank} \mathcal{W}=g-g'. \]

Thus, $\delta_v(F^{\mathrm{Zar}})<g-g'$ and $x\in F^{\mathrm{Zar}}\subseteq \mathcal{X}^{\mathrm{deg}}(g-g')$ . Therefore $\mathcal{X}^{\mathrm{sm}}\subseteq\mathcal{X}^{\mathrm{deg}}(g-g')$ and by Theorem 6.6, we conclude that $\mathcal{X}^{\mathrm{deg}}(g-g')=\mathcal{X}$ .

For a general abelian scheme $\mathcal{A}/S$ , take a moduli sieve $\iota:\mathcal{A}\rightarrow \mathfrak{A}_g$ . If $\iota|_{\mathcal{X}}$ is not generically finite, then $\mathcal{X}^{\mathrm{deg}}(0)=\mathcal{X}$ and, hence, $\mathcal{X}^{\mathrm{deg}}(g-g')=\mathcal{X}$ . Assume without loss of generality that $\iota|_{\mathcal{X}}$ is quasi-finite. Consider the corresponding abelian schemes $\iota(\mathcal{A}),\iota(\mathcal{B})$ . It is easy to see that the particular choice of ample $\mathcal{L}_{\mathcal{B}}$ does not matter. Thus, we take a relatively ample $\mathcal{L}_{\iota(\mathcal{B})}$ on $\iota(\mathcal{B})$ and use $\mathcal{L}_{\mathcal{B}}:=\iota^*\mathcal{L}_{\iota(\mathcal{B})}$ . Then the corresponding Betti form on $\iota(\mathcal{A})$ pulls back to the Betti form $\omega$ on $\mathcal{A}$ . The measure $\omega^{\wedge d}|_{\mathcal{X}}\equiv 0$ implies the same for the corresponding measure on $\iota(\mathcal{X})$ . Thus, we get $\iota(\mathcal{X})^{\mathrm{deg}}(g-g')=\iota(\mathcal{X})$ . As in the proof of Theorem 6.5, we have that for a vertically optimal $\mathcal{Z}'$ of $\iota(\mathcal{X})$ , any irreducible component of $\iota|_{\mathcal{X}}^{-1}(\mathcal{Z}')$ is vertically optimal of the same vertical defect. From this we conclude that $\mathcal{X}^{\mathrm{deg}}(g-g')=\mathcal{X}$ .

7.1 Overview

The Yuan–Zhang theory, which builds upon classical Arakelov theory, provides a unified framework for intersection theory on line bundles over general quasi-projective varieties defined over a field. It begins with the category of metrized line bundles on the Berkovich analytic space associated with a quasi-projective variety over a field equipped with a set of valuations, and then trims this eventually to the subcategory of integrable adelic line bundles. Specifically, metrized line bundles that arise directly from geometric models, such as hermitian line bundles on projective arithmetic varieties, are referred to as model adelic line bundles. In general, adelic line bundles are conceived as limits of model adelic line bundles, with the limits taken in an appropriate boundary topology. The most important subcategory of integrable adelic line bundles consists of those adelic line bundles that are applicable in intersection theory. These are defined as the difference between two nef adelic line bundles, analogous to the construction of Lebesgue integrals, where nef adelic line bundles play a role similar to that of nonnegative functions. Furthermore, the association of the category of adelic line bundles to the quasi-projective variety has natural functoriality under morphisms of the underlying varieties, maintaining compatibility with all relevant substructures.

Now let us delve into more details. Let $\bar{K}:=(K,\Sigma)$ denote a pair consisting of a field K and a set $\Sigma$ of valuations on K in one of the following cases.

1. (1) Geometric: K is a field and $\Sigma$ consists of one element, the trivial valuation on K.

2. (2) Archimedean local: K is the field $\mathbb{C}$ of complex numbers and $\Sigma$ consists of one element, the usual norm on $\mathbb{C}$ .

3. (3) Arithmetic: K is a number field and $\Sigma$ is the set of all normalized valuations of K, including the trivial valuation.

Let X be a quasi-projective variety over the field K and let L be a usual line bundle on X. For a pair $\bar{K}=(K,\Sigma)$ as above, there is an associated Berkovich analytic space $X^{\mathrm{an}}=X^{\mathrm{an}}_{\Sigma}$ and the natural extension of L to $L^{\mathrm{an}}$ on $X^{\mathrm{an}}$ . A metrized line bundle is a pair $(L,\lVert\cdot\rVert)$ with a continuous metric $\lVert\cdot\rVert$ on $L^{\mathrm{an}}$ . We shall not give a precise definition of adelic line bundles here. Roughly speaking, an adelic line bundle extending L is a metrized line bundle which can be regarded as a suitable completion of L at various places in $\Sigma$ . If there is no ambiguity, such an extension is denoted by $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu$ . The group of isomorphism classes of adelic line bundles is denoted by $\mathrm{\widehat{Pic}}(X/\bar{K})$ , with group law given by tensor products and written additively. The subset consisting of nef (respectively, integrable) adelic line bundles is denoted by $\mathrm{\widehat{Pic}}_{\mathrm{nef}}(X/\bar{K})$ (respectively, $\mathrm{\widehat{Pic}}_{\mathrm{int}}(X/\bar{K})$ ). Given a morphism $f:X\rightarrow Y$ over K, the pullback map $f^*: \mathrm{\widehat{Pic}}(Y/\bar{K})\rightarrow \mathrm{\widehat{Pic}}(X/\bar{K})$ is naturally defined and preserves nefness and integrability.

There are also natural relations among three cases above. To introduce these, we need to be more careful with the notation for $\bar{K}$ . By slight abuse of notation, we simplify the pair $(K,\Sigma)$ in three cases as K (geometric), $\mathcal{O}_{\mathbb{C}}$ (archimedean local), $\mathcal{O}_K$ (arithmetic), respectively. There are natural localization maps from arithmetic to archimedean local ( $K=\text{number field}$ and $\sigma:K\rightarrow \mathbb{C}$ is an embedding)

\[\mathrm{\widehat{Pic}}(X/\mathcal{O}_K)\xrightarrow{\text{loc.}} \mathrm{\widehat{Pic}}(X_\sigma/\mathcal{O}_{\mathbb{C}}) \]

and from arithmetic/archimedean local to geometric ( $K=\text{number field/}\mathbb{C}$ )

\[\mathrm{\widehat{Pic}}(X/\mathcal{O}_K)\xrightarrow{\text{loc.}} \mathrm{\widehat{Pic}}(X/K) \]

given by forgetting the metrics over the irrelevant places. There is a base change map in the geometric setting ( $K=\text{number field}$ and $\sigma:K\rightarrow \mathbb{C}$ is an embedding)

\[\mathrm{\widehat{Pic}}(X/K)\rightarrow \mathrm{\widehat{Pic}}(X_\sigma/\mathbb{C}).\]

In the rest of the article, we shall only use the notation $\bar{K}$ if a uniform treatment is possible.

Intersection theory is developed in the geometric and arithmetic contexts.

1. (1) Geometric: In this case, adelic line bundles arise as limits of ordinary line bundles on projective models of X over K. The intersection number is obtained via this limiting process. This framework generalizes the traditional intersection theory of line bundles in algebraic geometry to a quasi-projective setting.

2. (2) Arithmetic: Here, adelic line bundles are limits of hermitian line bundles on projective models of X over $\mathcal{O}_K$ . The intersection number is similarly derived through a limiting process. This theory extends Arakelov’s intersection theory of arithmetic line bundles to a quasi-projective context.

Denote the dimension of a relevant model of X by d, whence in the geometric (respectively, arithmetic) case, we have $d=\dim X$ (respectively, $d=\dim X+1$ ). We call d the model dimension of X. By extending $\mathbb{R}$ -linearly, intersection theory gives a symmetric multi-linear pairing

\begin{align*} [\cdot]_X:\mathrm{\widehat{Pic}}_{\mathrm{int}}(X/\bar{K})^d_{\mathbb{R}} &\longrightarrow \mathbb{R}\nonumber\\ (\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1,\ldots,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d) &\longmapsto [\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots \mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_X.\end{align*}

The pairing satisfies the following basic properties.

1. (1) If $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1,\ldots,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d\in \mathrm{\widehat{Pic}}_{\mathrm{nef}}(X/\bar{K})$ , then $[\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots \mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_X\geq 0$ .

2. (2) When K is a number field and $\sigma:K\rightarrow \mathbb{C}$ is an embedding, geometric intersection theory on $\mathrm{\widehat{Pic}}_{\mathrm{int}}(X/K)$ agrees with that on $\mathrm{\widehat{Pic}}_{\mathrm{int}}(X_\sigma/\mathbb{C})$ through base change.

3. (3) Projection formula [Reference Yuan and ZhangYZ26, Proposition 4.1.2]: Let $f:X\rightarrow Y$ be a dominant morphism of quasi-projective varieties over K. Let $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1,\ldots,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d\in \mathrm{\widehat{Pic}}_{\mathrm{int}}(Y/\bar{K})$ . Then

   (7.1) \begin{equation} [f^*\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots f^*\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_X=\deg f\cdot [\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots \mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_Y.\end{equation}
   Here, if f is generically finite, $\deg f$ is the degree of the extension of the function fields of X and Y; otherwise, $\deg f=0$ .

When $f:X\rightarrow Y$ is clear in the context, and $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1,\ldots, \mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d\in \mathrm{\widehat{Pic}}_{\mathrm{int}}(Y/\bar{K})$ , we shall write $[\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots \mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_X$ for $[f^*\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_1\cdots f^*\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_d]_X$ .

Definition 7.1. Let K be a number field. The height function associated with an arithmetic adelic line bundle $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu\in \mathrm{\widehat{Pic}}_{\mathrm{int}}(X/\mathcal{O}_K)_{\mathbb{R}}$ is a function $h_{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}:X(K^{\mathrm{alg}})\rightarrow \mathbb{R}$ defined by

\[ h_{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}(x):=\frac{[\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu]_{x'}}{[\kappa(x'):K]},\]

where $x\in X(K^{\mathrm{alg}})$ is an algebraic point and $x'\in X$ is the scheme-theoretic point of x.

For instance, on a projective space $\mathbb{P}^N$ over a number field K, the nave adelic line bundle $\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{O}(1)\mkern-1.5mu}\mkern 1.5mu$ is the adelic metric on $\mathcal{O}(1)$ whose associated height function $h_{\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{O}(1)\mkern-1.5mu}\mkern 1.5mu}$ is the usual nave height function.

7.2 Volume and bigness

We work in the geometric/arithmetic setting for this subsection. Let $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu\in \mathrm{\widehat{Pic}}(X/\bar{K})$ and let $\lVert\cdot\rVert$ be the associated metric on $L^{\mathrm{an}}$ .

Definition 7.2. The set of effective sections, denoted by $\widehat{H}^0(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu)$ , is defined to be the subset of global sections $s\in H^0(X, L)$ such that $\lVert s^{\mathrm{an}}(x)\rVert\leq 1$ for any $x\in X^{\mathrm{an}}$ , where $s^{\mathrm{an}}$ is the section on $L^{\mathrm{an}}$ corresponding to s. Define the quantity

\begin{equation*} \hat h^0(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu):=\begin{cases} \dim_K \widehat H^0(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu) & \text{in a geometric setting,}\\ \log\#\widehat H^0(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu) & \text{in a arithmetic setting.} \end{cases} \end{equation*}

The existence of nontrivial effective sections has the following implication on height.

Proof. Let $x\in X(K^{\mathrm{alg}})$ be a point with $s(x)\neq0$ and let $x'\in X$ be the scheme-theoretic point of x. By definition, we have

\[ mh_{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}(x)=\frac{[m\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu]_{x'}}{[\kappa(x'):K]}=\frac{1}{[\kappa(x'):K]}\sum_{v\in \Sigma_K}\sum_{y\in x'_v}-[\kappa(y):K_v]\log\lVert s(y)\rVert \]

where $\Sigma_K$ denotes all the places of K and $x'_v$ is the finite set of points of $X^{\mathrm{an}}$ given by the image of $x'\times_{\mathrm{Spec} K} \mathrm{Spec} K_v$ in $X_{K_v}$ . Then we have $h_{\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu}(x)\geq0$ since $\lVert s(y) \rVert\leq 1$ .

Definition 7.4. The volume of $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu\in \mathrm{\widehat{Pic}}(X/\bar{K})$ is defined as

\[\widehat{\mathrm{vol}}(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu):=\lim_{m\rightarrow \in fty}\frac{d!}{m^d}\widehat h^0(X,m\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu), \]

where the limit exists by [Reference Yuan and ZhangYZ26, Theorem 5.2.1]. More generally, the volume of a $\mathbb{Q}$ -adelic line bundle $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu\in \mathrm{\widehat{Pic}}(X/\bar{K})_{\mathbb{Q}}$ is defined as ${1}/{N^d}\widehat{\mathrm{vol}}(X,N\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu)$ for any $N\in \mathbb{N}$ such that $N\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu\in \mathrm{\widehat{Pic}}(X/\bar{K})$ .

If $\widehat{\mathrm{vol}}(X,\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu) \gt 0$ , we say $\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu$ is big.

For integrable adelic line bundles, volume is closely related to intersection theory in the following way by [Reference Yuan and ZhangYZ26, Theorem 5.2.2].