2.1 Convention on abelian schemes

Let $g \ge 1$ be an integer. Let $S$ be an irreducible variety over $\mathbb {C}$ and let $\pi _S \colon \mathcal {A} \rightarrow S$ be an abelian scheme of relative dimension $g$. Since level structures are not important in this paper, we will use the following abuse of notation through the whole paper.

1. (i) We say that a subvariety $\mathcal {B}$ of $\mathcal {A}$ is an abelian subscheme of $\mathcal {A} \rightarrow S$ if there exists a finite covering $S' \rightarrow S$, inducing a morphism $\rho \colon \mathcal {A}' = \mathcal {A} \times _S S' \rightarrow \mathcal {A}$, such that $\mathcal {B} = \rho (\mathcal {B}')$ where $\mathcal {B}'$ is an abelian subscheme of $\mathcal {A}'/S'$ in the usual sense.Footnote ¹

2. (ii) We say that $\sigma$ is a section of $\mathcal {A} \rightarrow S$ if there exists a finite covering $S' \rightarrow S$, inducing a morphism $\rho \colon \mathcal {A}' = \mathcal {A} \times _S S' \rightarrow \mathcal {A}$, such that $\sigma = \rho \circ \sigma '$ where $\sigma ' \colon S' \rightarrow \mathcal {A}'$ is a section of $\mathcal {A}' \rightarrow S'$ in the usual sense.Footnote ² Denote by $\sigma (S) := (\rho \circ \sigma ')(S')$.

3. (iii) In (ii), call $\sigma$ a torsion section if $\sigma '(s')$ is a torsion point on $\mathcal {A}'_{s'}$ for each $s' \in S'(\mathbb {C})$; call $\sigma$ a constant section if $\sigma '$ is the composite of $S' \rightarrow C' \times S'$, $s' \mapsto (c',s')$, and the inclusion $C' \times S' \subseteq \mathcal {A}'$, where $C' \times S'$ is a constant abelian subscheme of $\mathcal {A}' \rightarrow S'$.

The following definition is convenient to study constant sections.

In particular, the zero section of any abelian scheme $\mathcal {A} \rightarrow S$ is an isotrivial abelian subscheme. If $\dim S = 0$, then $\mathcal {A} \rightarrow S$ is isotrivial.

Since the sum of two isotrivial abelian subschemes of $\mathcal {A} \rightarrow S$ is isotrivial, we can define the largest isotrivial abelian subscheme of $\mathcal {A} \rightarrow S$, which we denote by $\mathcal {C}$. Then any constant section of $\mathcal {A} \rightarrow S$ (defined above) has image in $\mathcal {C}$.

2.2 Modular map

When $g = 0$, let $\mathfrak {A}_0$ and $\mathbb {A}_0$ be a point. When $g \ge 1$. Let $D = \mathrm {diag}(d_1,\ldots ,d_g)$ be a $g \times g$ diagonal matrix with $d_1, \ldots , d_g$ positive integers such that $d_1|\cdots |d_g$.

For any integer $N \ge 3$, let $\mathbb {A}_{g,D}(N)$ denote the moduli space of abelian varieties of dimension $g$ with polarization of type $D$ and with level-$N$-structure. Then $\mathbb {A}_{g,D}(N)$ is a fine moduli space, and hence admits a universal family $\pi \colon \mathfrak {A}_{g,D}(N) \rightarrow \mathbb {A}_{g,D}(N)$. Since level structure is not important for our purpose, we shall drop the ‘$(N)$’ in the rest of the paper. When there is no ambiguity about the polarization type, we also drop the ‘$D$’ and simply use the notation $\pi \colon \mathfrak {A}_g \rightarrow \mathbb {A}_g$.

It is known that any abelian scheme $\pi _S \colon \mathcal {A} \rightarrow S$ can be equipped with a polarization of type $D$ for some $D = \mathrm {diag}(d_1,\ldots ,d_g)$, with $d_1, \ldots , d_g$ positive integers such that $d_1|\cdots |d_g$; see [Reference Genestier and NgôGN09, § 2.1]. Thus up to taking a finite covering of $S$ and the associated base change of $\mathcal {A} \rightarrow S$, we have the following modular map.

(2.1)

Then $\mathcal {A} \rightarrow S$ is isotrivial if and only if $\iota _S(S)$ is a point. In particular, if $S \subseteq \mathbb {A}_g$, then $\mathfrak {A}_g \times _{\mathbb {A}_g}S$ is isotrivial if and only if $S$ is a point.

3.1 Uniformizing space of $\mathbb {A}_g$

Let $\mathfrak {H}_g^+$ be the Siegel upper half space

\[ \{ Z = X + \sqrt{-1}Y \in M_{g \times g}(\mathbb{C}) : Z = Z^{{\intercal}},\ Y > 0\}. \]

It is well-known that the uniformization of $\mathbb {A}_g := \mathbb {A}_{g,D}$ in the category of complex varieties is given by

(3.1)\begin{equation} \textbf{u}_G \colon \mathfrak{H}_g^+ \rightarrow \mathbb{A}_g. \end{equation}

Let us take a closer look at this uniformization.

Let $\mathrm {Sp}_{2g}$ be the $\mathbb {Q}$-group

\[ \biggl\{ h \in \mathrm{GL}_{2g} : h \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} h^{\!^{\intercal}} = \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} \biggl\}, \]

and let $\mathrm {GSp}_{2g}$ be the image of $\mathbb {G}_{\mathrm {m}} \times \mathrm {Sp}_{2g}$ under the central isogeny $\mathbb {G}_{\mathrm {m}} \times \mathrm {SL}_{2g} \rightarrow \mathrm {GL}_{2g}$. Then $\mathrm {GSp}_{2g}(\mathbb {R})^+$, the connected component of $\mathrm {GSp}_{2g}(\mathbb {R})$ containing the identity, acts on $\mathfrak {H}_g^+$ by the formula

\[ \begin{pmatrix} A' & B' \\ C' & D' \end{pmatrix} Z = (A'Z+B')(C'Z+D')^{-1}, \quad \forall \begin{pmatrix} A' & B' \\ C' & D' \end{pmatrix} \in \mathrm{GSp}_{2g}(\mathbb{R})^+\text{ and }Z \in \mathfrak{H}_g^+. \]

It is known that the action of $\mathrm {GSp}_{2g}^{\mathrm {der}}(\mathbb {R}) = \mathrm {Sp}_{2g}(\mathbb {R})$ on $\mathfrak {H}_g^+$ thus defined is transitive, and the uniformization (3.1) is obtained by identifying $(\mathbb {A}_g)^{\mathrm {an}}$ with the quotient space $\Gamma _{\mathrm {GSp}_{2g}}\backslash \mathfrak {H}_g^+$ for a suitable congruence group $\Gamma _{\mathrm {Sp}_{2g}}$ of $\mathrm {Sp}_{2g}(\mathbb {Z})$.

3.2 Uniformizing space of $\mathfrak {A}_g$

To obtain the uniformization of $\mathfrak {A}_g$, let us construct the following complex space $\mathcal {X}_{2g,\mathrm {a}}^+$:

1. (i) as a semi-algebraic space, $\mathcal {X}_{2g,\mathrm {a}}^+ = \mathbb {R}^{2g} \times \mathfrak {H}_g^+$;

2. (ii) the complex structure of $\mathcal {X}_{2g,\mathrm {a}}^+$ is the one given by

   (3.2)\begin{equation} \begin{array}{cccc} \mathcal{X}_{2g,\mathrm{a}}^+ = & \mathbb{R}^g \times \mathbb{R}^g \times \mathfrak{H}_g^+ & \xrightarrow{\sim} & \mathbb{C}^g \times \mathfrak{H}_g^+, \\ & (a,b,Z) & \mapsto & (Da+Zb, Z) \end{array}. \end{equation}

The uniformization of $\mathfrak {A}_g$ in the category of complex varieties is then given by

(3.3)\begin{equation} \textbf{u} \colon \mathcal{X}_{2g,\mathrm{a}}^+ \rightarrow \mathfrak{A}_g. \end{equation}

Similar to the discussion on $\textbf {u}_G$, there exists a $\mathbb {Q}$-group, which we call $P_{2g,\mathrm {a}}$ such that $P_{2g,\mathrm {a}}^{\mathrm {der}}(\mathbb {R})$ acts transitively on $\mathcal {X}_{2g,\mathrm {a}}^+$ and $\textbf {u}$ is obtained by identifying $(\mathfrak {A}_g)^{\mathrm {an}}$ with the quotient space $\Gamma \backslash \mathcal {X}_{2g,\mathrm {a}}^+$ for a suitable congruence subgroup $\Gamma = \mathbb {Z}^{2g} \rtimes \Gamma _{\mathrm {Sp}_{2g}}$ of $P_{2g,\mathrm {a}}^{\mathrm {der}}(\mathbb {Z})$. Let us briefly explain this.

Use $V_{2g}$ to denote the $\mathbb {Q}$-vector group of dimension $2g$. Then the natural action of $\mathrm {GSp}_{2g}$ on $V_{2g}$ defines a $\mathbb {Q}$-group

\[ P_{2g,\mathrm{a}} = V_{2g} \rtimes \mathrm{GSp}_{2g}. \]

The action of $P_{2g,\mathrm {a}}(\mathbb {R})^+$ on $\mathcal {X}_{2g,\mathrm {a}}^+$ is defined as follows: for any $(v,h) \in P_{2g,\mathrm {a}}(\mathbb {R})^+ = V_{2g}(\mathbb {R}) \rtimes \mathrm {GSp}_{2g}(\mathbb {R})^+$ and any $(v',x) \in \mathcal {X}_{2g,\mathrm {a}}^+$, we have

(3.4)\begin{equation} (v,h) \cdot (v',x) = (v+hv',hx), \end{equation}

where $\mathrm {GSp}_{2g}(\mathbb {R})^+$ acts on $\mathbb {R}^{2g}$ as above (3.2).

The natural projection of complex spaces $\tilde {\pi } \colon \mathcal {X}_{2g,\mathrm {a}}^+ \rightarrow \mathfrak {H}_g^+$ is equivariant with respect to the natural projection of groups $P_{2g,\mathrm {a}} \rightarrow \mathrm {GSp}_{2g}$. Hence by abuse of notation we also denote by $\tilde {\pi } \colon P_{2g,\mathrm {a}} \rightarrow \mathrm {GSp}_{2g}$.

3.3 Betti map

We define the Betti map in this section. We will start by defining the universal uniformized Betti map on $\mathcal {X}_{2g,\mathrm {a}}^+$, and then descend it to $\mathcal {A}_{\mathfrak {H}_g^+}$, the pullback of $\mathfrak {A}_g / \mathbb {A}_g$ under $\textbf {u}_G \colon \mathfrak {H}_g^+ \rightarrow \mathbb {A}_g$. Note that $\mathcal {A}_{\mathfrak {H}_g^+}$ is a family of abelian varieties over $\mathfrak {H}_g^+$.

Recall that $\mathcal {X}_{2g,\mathrm {a}}^+$ is defined to be $\mathbb {R}^{2g} \times \mathfrak {H}_g^+$ with the complex structure determined by (3.2). The universal uniformized Betti map $\tilde {b}$ is defined to be the natural projection

(3.5)\begin{equation} \tilde{b} \colon \mathcal{X}_{2g,\mathrm{a}}^+ \rightarrow \mathbb{R}^{2g}. \end{equation}

Then $\tilde {b}$ is semi-algebraic. For the complex structure on $\mathcal {X}_{2g,\mathrm {a}}^+$ given by (3.2), it is clear that $\tilde {b}^{-1}(r)$ is complex analytic for each $r \in \mathbb {R}^{2g}$.

Recall that $(\mathfrak {A}_g)^{\mathrm {an}} \simeq \Gamma \backslash \mathcal {X}_{2g,\mathrm {a}}^+$ as complex spaces for a suitable congruence subgroup $\Gamma = \mathbb {Z}^{2g} \rtimes \Gamma _{\mathrm {Sp}_{2g}}$ of $P_{2g,\mathrm {a}}^{\mathrm {der}}(\mathbb {Z})$. The family of abelian varieties $\mathcal {A}_{\mathfrak {H}_g^+}$ defined as at the beginning of this subsection can be identified with the quotient space $(\mathbb {Z}^{2g} \rtimes \{1\}) \backslash \mathcal {X}_{2g,\mathrm {a}}^+$. Now taking quotient by $\mathbb {Z}^{2g}$ on both sides of (3.5), we obtain the universal Betti map

(3.6)\begin{equation} b \colon \mathcal{A}_{\mathfrak{H}_g^+} \rightarrow \mathbb{T}^{2g}, \end{equation}

where $\mathbb {T}^{2g}$ denotes the real torus of dimension $2g$. By the discussion below (3.5), we have the following properties for $\tilde {b}$ and $b$.

1. (i) Both $\tilde {b}$ and $b$ are real-analytic, and $\tilde {b}$ is moreover semi-algebraic.

2. (ii) For each $r \in \mathbb {R}^{2g}$, resp. each $t \in \mathbb {T}^{2g}$, we have that $\tilde {b}^{-1}(r)$, resp. $b^{-1}(t)$, is complex analytic.

3. (iii) For each $\tau \in \mathfrak {H}_g^+$, the restriction $b|_{(\mathcal {A}_{\mathfrak {H}_g^+})_{\tau }}$ is a group isomorphism.

We summarize our notation regarding the uniformizations in the following diagram:

(3.7)

with the uniformization ${\mathbf {u}}$ from (3.3) and the uniformization ${\mathbf {u}}_G$ from (3.1).

5.1 Generically special subvarieties of sg type and bi-algebraic subvarieties

The goal of this subsection is to explain the relation between generically special subvarieties of sg type (see Definition 1.5) and bi-algebraic subvarieties of $\mathfrak {A}_g$.

Let us start with defining bi-algebraic subvarieties of $\mathfrak {A}_g$. Recall the uniformization $\mathbf {u} \colon \mathcal {X}_{2g,\mathrm {a}}^+ \rightarrow \mathfrak {A}_g$. By [Reference GaoGao17b, § 4] $\mathcal {X}_{2g,\mathrm {a}}^+$ can be embedded as an open, in the usual topology, semi-algebraic subset of a complex flag variety (hence algebraic) $\mathcal {X}_{2g,\mathrm {a}}^\vee$.

It is not hard to show that the intersection of two bi-algebraic subvarieties of $\mathfrak {A}_g$ is a finite union of irreducible bi-algebraic subvarieties of $\mathfrak {A}_g$. Hence for any subset $Z$ of $\mathfrak {A}_g$, there exists a smallest bi-algebraic subvariety $\mathfrak {A}_g$ which contains $Z$. We use $Z^{\mathrm {biZar}}$ to denote it. Note that $Z^{\mathrm {biZar}} \supseteq Z^{\mathrm {Zar}}$.

Bi-algebraic subvarieties of $\mathfrak {A}_g$ are closely related to generically special subvarieties of sg type defined in Definition 1.5 by the following proposition.

Proposition 5.3 [Gao17a, Proposition 3.3] Let $B$ be an irreducible subvariety of $\mathbb {A}_g$. Denote by $\mathfrak {A}_g|_B = \pi ^{-1}(B)$. Then we have

\begin{align*} & \{\text{generically special subvarieties of sg type of } \mathfrak{A}_g|_B\} \\ &\quad = \{ \text{irreducible components of }(\mathfrak{A}_g|_B) \cap F : F \text{ irreducible bi-algebraic in } \mathfrak{A}_g \text{ with } B \subseteq \pi(F)\}. \end{align*}

The following equivalent form of Proposition 5.3 is more practical for our use.

5.2 Weak Ax–Schanuel for $\mathfrak {A}_g$

One of the most important tools we use to study the Betti map is the following weak Ax–Schanuel theorem for $\mathfrak {A}_g$ [Reference GaoGao20, Theorem 1.1 or Theorem 3.5].

Theorem 5.5 Let $\tilde {Z}$ be a complex analytic irreducible subset of $\mathcal {X}_{2g,\mathrm {a}}^+$. Then

\[ \dim \tilde{Z}^{\mathrm{Zar}} + \dim ({\mathbf{u}}(\tilde{Z}))^{\mathrm{Zar}} \ge \dim \tilde{Z} + \dim ({\mathbf{u}}(\tilde{Z}))^{\mathrm{biZar}}, \]

where $\tilde {Z}^{\mathrm {Zar}}$ means the smallest irreducible algebraic subset of $\mathcal {X}_{2g,\mathrm {a}}^+$ which contains $\tilde {Z}$.

5.3 A quick introduction to Shimura varieties

We gather some notation and facts on Shimura varieties. We will need the knowledge (only) for the proofs in §§ 7 and 8.

Recall that in § 3.1, we have associated a reductive $\mathbb {Q}$-group $\mathrm {GSp}_{2g}$ and a complex space $\mathfrak {H}_g^+$ to the moduli space $\mathbb {A}_g$. In other words, we have associated a pair $(\mathrm {GSp}_{2g}, \mathfrak {H}_g^+)$ to $\mathbb {A}_g$, where:

• • $\mathrm {GSp}_{2g}$ is a reductive $\mathbb {Q}$-group;

• • $\mathfrak {H}_g^+$ is a complex space on which $\mathrm {GSp}_{2g}(\mathbb {R})^+$ acts transitively;

• • as a complex space, $\mathbb {A}_g$ is the quotient of $\mathfrak {H}_g^+$ by a congruence subgroup of $\mathrm {GSp}_{2g}^{\mathrm {der}}(\mathbb {Z})$.

This pair $(\mathrm {GSp}_{2g},\mathfrak {H}_g^+)$ is a special case of pure Shimura datum, and the third bullet point makes $\mathbb {A}_g$ a pure Shimura variety. In general, a pure Shimura datum is a pair $(G,\mathcal {X}_G^+)$ such that $G$ is a reductive $\mathbb {Q}$-group and $\mathcal {X}_G^+$ is a complex space on which $G^{\mathrm {der}}(\mathbb {R})^+$ acts transitively (along with some other properties). A pure Shimura variety is a quotient space $\Gamma _G \backslash \mathcal {X}_G^+$ for some congruence subgroup $\Gamma _G$ of $G^{\mathrm {der}}(\mathbb {Z})$.

Next we turn to the universal abelian variety $\mathfrak {A}_g$. In § 3.2 we have associated with it a $\mathbb {Q}$-group $P_{2g,\mathrm {a}}$ and a complex space $\mathcal {X}_{2g,\mathrm {a}}^+$. Note that $P_{2g,\mathrm {a}}$ is not a reductive group. Nevertheless we have the following properties for the pair $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$:

• • $P_{2g,\mathrm {a}}$ is a $\mathbb {Q}$-group, whose unipotent radical is a vector group;

• • $\mathcal {X}_{2g,\mathrm {a}}^+$ is a complex space on which $P_{2g,\mathrm {a}}(\mathbb {R})^+$ acts transitively;

• • as a complex space, $\mathfrak {A}_g$ is the quotient of $\mathcal {X}_{2g,\mathrm {a}}^+$ by a congruence subgroup of $P_{2g,\mathrm {a}}^{\mathrm {der}}(\mathbb {Z})$.

This makes the pair $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$ a mixed Shimura datum of Kuga type, and the third bullet point makes $\mathfrak {A}_g$ a mixed Shimura variety of Kuga type. In general a mixed Shimura datum of Kuga type is a pair $(P,\mathcal {X}^+)$ such that $P$ is a $\mathbb {Q}$-group whose unipotent radical is a vector group, and $\mathcal {X}^+$ is a complex space on which $P(\mathbb {R})^+$ acts transitively (along with some other properties). A mixed Shimura variety of Kuga type is a quotient space $\Gamma \backslash \mathcal {X}^+$ for some congruence subgroup $\Gamma$ of $P^{\mathrm {der}}(\mathbb {Z})$.

It is worth pointing out that any pure Shimura datum is a mixed Shimura datum of Kuga type (such that the unipotent radical of the underlying group is trivial). Given two mixed Shimura data of Kuga type $(Q,\mathcal {Y}^+)$ and $(P,\mathcal {X}^+)$, a map $f \colon (Q,\mathcal {Y}^+) \rightarrow (P,\mathcal {X}^+)$ is called a Shimura morphism if $f$ is a group homomorphism on the underlying groups and is a holomorphic morphism on the underlying complex spaces, and that $f(q \cdot \tilde {y}) = f(q) \cdot f(\tilde {y})$ for any $q \in Q(\mathbb {R})^+$ and any $\tilde {y} \in \mathcal {Y}^+$. For a Shimura morphism $f$ of this form, we say that $(f(Q), f(\mathcal {Y}^+))$ is a mixed Shimura subdatum of $(P,\mathcal {X}^+)$. Applying the discussion to $(P,\mathcal {X}^+) = (P_{2g,\mathrm {a}} , \mathcal {X}_{2g,\mathrm {a}}^+)$, we get the definition of mixed Shimura subdata of $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$. Applying the discussion to $(G,\mathcal {X}_G^+) = (\mathrm {GSp}_{2g}, \mathfrak {H}_g^+)$, we get the definition of Shimura subdata of $(\mathrm {GSp}_{2g}, \mathfrak {H}_g^+)$.

It is known that for each mixed Shimura subdatum $(Q,\mathcal {Y}^+)$ of $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$, the unipotent radical of $Q$ is $V_{2g} \cap Q$ by weight reasons; see [Reference GaoGao17b, Proposition 2.9].

Define the special subvarieties of $\mathfrak {A}_g$ to be the subvarieties of the form ${\mathbf {u}}(\mathcal {Y}^+)$ where $\mathcal {Y}^+$ is the underlying space of some mixed Shimura subdatum $(Q,\mathcal {Y}^+)$ of $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$. Define the special subvarieties of $\mathbb {A}_g$ to be the subvarieties of the form ${\mathbf {u}}_G(\mathcal {Y}_G^+)$, where $\mathcal {Y}_G^+$ is the underlying space of some Shimura subdatum $(H, \mathcal {Y}_G^+)$ of $(\mathrm {GSp}_{2g}, \mathfrak {H}_g^+)$.

Let us end this subsection by the following proposition on the geometric meaning of special subvarieties of $\mathfrak {A}_g$ and beyond. Recall the notation $\tilde {\pi } \colon P_{2g,\mathrm {a}} \rightarrow \mathrm {GSp}_{2g}$ at the end of § 3.2 and the uniformizations (see (3.7)).

(5.1)

Proposition 5.6 Let $M$ be a special subvariety of $\mathfrak {A}_g$. Then $M_G := \pi (M)$ is a special subvariety of $\mathbb {A}_g$, and $M$ is the translate of an abelian subscheme of $\mathfrak {A}_g|_{M_G} \rightarrow M_G$ by a torsion section. Conversely all special subvarieties of $\mathfrak {A}_g$ are obtained in this way.

More precisely, if $M$ is associated with the mixed Shimura subdatum of Kuga type $(Q,\mathcal {Y}^+)$ of $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$ (namely $M = {\mathbf {u}}(\mathcal {Y}^+)$), then the relative dimension of $M \rightarrow M_G$ is

\[ g_Q := \tfrac{1}{2}\dim (V_{2g} \cap Q). \]

5.4 Quotient by a normal group

In §§ 7 and 8, we need the geometric interpretation of the operation of taking the quotient mixed Shimura varieties of Kuga type. We explain this in the current subsection.

The setting is as follows. Let $(Q,\mathcal {Y}^+)$ be a mixed Shimura subdatum of $(P_{2g,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$, and let $M = {\mathbf {u}}(\mathcal {Y}^+)$ be the associated special subvariety of $\mathfrak {A}_g$. Take a normal subgroup $N$ of $Q$. Pink [Reference PinkPin89, 2.9] constructed the quotient mixed Shimura datum $(Q,\mathcal {Y}^+)/N$ whose underlying group is $Q/N$.

By Proposition 5.6, $M_G := \pi (M)$ is a special subvariety of $\mathbb {A}_g$. Let $(G_Q, \mathcal {Y}_{G_Q}^+)$ be the pure Shimura subdatum of $(\mathrm {GSp}_{2g}, \mathfrak {H}_g^+)$ in Proposition 5.6, then $M_G = {\mathbf {u}}_G(\mathcal {Y}_{G_Q}^+)$.

Let $V_Q = V_{2g} \cap Q$ and $V_N = V_{2g} \cap N$. We explained above (5.1) that $V_Q$ is the unipotent radical of $Q$. Since $N \lhd Q$, group theory implies that $V_N$ is the unipotent radical of $N$. Let $G_N = N/V_N$, then $G_N$ is a normal subgroup of $G_Q$.

Proposition 5.6 says that $\pi |_M \colon M \rightarrow M_G$ itself is an abelian scheme of relative dimension $g_Q = \frac{1}{2}\dim (V_{2g} \cap Q)$. Let $\epsilon \colon M_G \rightarrow M$ be the zero section.Footnote ⁴ It induces a Levi decomposition $Q = V_Q \rtimes G_Q$ and a semi-algebraic isomorphism

\[ \mathcal{Y}^+ \simeq V_Q(\mathbb{R}) \times \mathcal{Y}_{G_Q}^+ \]

such that ${\mathbf {u}}(\{0\} \times \mathcal {Y}_{G_Q}^+) = \epsilon (M_G)$. In the rest of this subsection, we shall use this identification of $\mathcal {Y}^+$ with $V_Q(\mathbb {R}) \times \mathcal {Y}_{G_Q}^+$.

Deligne [Reference DeligneDel71, Rappel 4.4.3] proved that $\mathcal {Y}^+ \rightarrow \mathcal {Y}^+_{G_Q}$ is a variation of Hodge structure of type $(-1,0)+(0,-1)$.

We have $V_N \lhd Q$ since $N\lhd Q$ and $V_N$ is the unipotent radical of $N$. The reductive group $G_Q$, as a subgroup of $\mathrm {GSp}_{2g}$, acts on $V_{2g}$. Now $V_N \lhd Q$ implies that $V_N$ is a $G_Q$-submodule of $V_{2g}$, so $V_N$ is a sub-Hodge structure of $V_Q$. Thus $V_N(\mathbb {R}) \times \mathcal {Y}_{G_Q}^+ \rightarrow \mathcal {Y}^+_{G_Q}$ is a subvariation of Hodge structure of $\mathcal {Y}^+ \rightarrow \mathcal {Y}^+_{G_Q}$. Thus ${\mathbf {u}}(V_N(\mathbb {R}) \times \mathcal {Y}_{G_Q}^+)$ is an abelian subscheme of $M \rightarrow M_G$ by [Reference DeligneDel71, Rappel 4.4.3].

The quotient $\tilde {p}_N \colon (Q,\mathcal {Y}^+) \rightarrow (Q,\mathcal {Y}^+)/N$ can be constructed in two steps. First we take the quotient $(Q_0,\mathcal {Y}_0^+) := (Q,\mathcal {Y}^+)/V_N$, and then we take $(Q', \mathcal {Y}^{\prime +}) := (Q_0,\mathcal {Y}_0^+)/G_N$.

The geometric meaning of the quotient $\tilde {p}_0 \colon (Q,\mathcal {Y}^+) \rightarrow (Q_0,\mathcal {Y}_0^+) = (Q,\mathcal {Y}^+)/V_N$ is as follows. We have the following commutative diagram (where $\Gamma _0 = \tilde {p}_0(\Gamma \cap Q(\mathbb {R}))$):

(5.2)

such that $M_0 \rightarrow M_G$ is the abelian scheme obtained by taking the quotient of $M \rightarrow M_G$ by ${\mathbf {u}}(V_N(\mathbb {R}) \times \mathcal {Y}_{G_Q}^+)$. Let $g_N = \frac{1}{2} \dim V_N$, then the relative dimension of $M_0 \rightarrow M_G$ is $g_Q - g_N$.

To explain the quotient $\tilde {p}' \colon (Q_0, \mathcal {Y}_0^+) \rightarrow (Q', \mathcal {Y}^{\prime +}) = (Q_0,\mathcal {Y}_0^+)/G_N$, we need the following preliminary. We know that $G_N$ is a normal subgroup of $G_Q$. We take for granted that we can do the operation $(G_Q, \mathcal {Y}_{G_Q}^+)/G_N$ for pure Shimura datum, and this quotient gives a morphism $p_{G_N} \colon M_G \rightarrow M'_G$.Footnote ⁵

For each $y'_G \in M'_G$, the inverse image $(p_{G_N})^{-1}(y'_G)$ equals ${\mathbf {u}}_G(G_N(\mathbb {R})^+\tilde {y}_G)$ for some $\tilde {y}_G \in \mathcal {Y}_{G_Q}^+$. So the connected algebraic monodromy group of $(p_{G_N})^{-1}(y'_G)$ is $G_N^{\mathrm {der}}$. Since $N \lhd Q$, we have $G_N = N/V_N \lhd Q/V_N$. Hence $G_N$ acts trivially on $V_Q/V_N$. So $M_0|_{(p_{G_N})^{-1}(y'_G)} \rightarrow (p_{G_N})^{-1}(y'_G)$ is an isotrivial abelian scheme by Deligne's theorem of the fixed part [Reference DeligneDel71, Corollaire 4.1.2]. We have the following commutative diagram:

(5.3)

where $\pi '$ is an abelian scheme (of relative dimension $g_Q - g_N$) such that each closed fiber of $M_0|_{(p_{G_N})^{-1}(y'_G)} \rightarrow (p_{G_N})^{-1}(y'_G)$ is the abelian variety $(\pi ')^{-1}(y'_G)$. In other words, the lower box is an intermediate step of the following modular map.

We end this subsection by summarizing the quotient $\tilde {p}_N \colon (Q,\mathcal {Y}^+) \rightarrow (Q,\mathcal {Y}^+)/N$ (after taking the uniformizations) in the following commutative diagram:

(5.4)

where each vertical arrow is an abelian scheme, with the left one of relative dimension $g_Q$ and the other three of relative dimension $g_Q - g_N$. Moreover $M_0|_{p_{G_N}^{-1}(b')}$ is isotrivial for any $b' \in M'_G$ by the discussion above (

). And for any $x' \in M'$, we have that $p_N^{-1}(x')$ is the translate of an abelian subscheme of $M|_{\pi (p_N^{-1}(x'))} \rightarrow \pi (p_N^{-1}(x'))$ of relative dimension $g_Q - (g_Q - g_N) = g_N$ by a constant section.

7.1 Weakly optimal subvariety

To prove Theorem 7.1, we need the following definition of weakly optimal subvarieties. The notion was first introduced by Habegger and Pila [Reference Habegger and PilaHP16] to study the Zilber–Pink conjecture for abelian varieties and product of modular curves.

Weakly optimal subvarieties of $X$ are closely related to $X^{\mathrm {deg}}(t)$ by the following lemmas.

Proof. The condition $\dim \langle Z \rangle _{\mathrm {sg}} - \dim \pi (Z) < \dim Z + t$ can be rewritten to be

\[ ((\dim \langle Z \rangle_{\mathrm{sg}} - \dim \pi(Z)) + \dim \pi(Z)^{\mathrm{biZar}} ) - \dim Z < \dim \pi(Z)^{\mathrm{biZar}} + t. \]

By Corollary 5.4, we have $\dim Z^{\mathrm {biZar}} = (\dim \langle Z \rangle _{\mathrm {sg}} - \dim \pi (Z)) + \dim \pi (Z)^{\mathrm {biZar}}$. Hence the inequality above becomes $\delta _{\mathrm {ws}}(Z) < \dim \pi (Z)^{\mathrm {biZar}} + t$.

Proof. For any $Z \subseteq Z' \subseteq X$ with $Z'$ irreducible closed in $X$, if $\delta _{\mathrm {ws}}(Z') \le \delta _{\mathrm {ws}}(Z)$, then we have

\[ \delta_{\mathrm{ws}}(Z') \le \delta_{\mathrm{ws}}(Z) < \dim \pi(Z)^{\mathrm{biZar}} + t \le \dim \pi(Z')^{\mathrm{biZar}} + t, \]

where the second inequality follows from Lemma 7.3 (applied to $Z$). Applying Lemma 7.3 to $Z'$, we get $\dim \langle Z' \rangle _{\mathrm {sg}} - \dim \pi (Z') < \dim Z' + t$. The maximality of $Z$ then implies $Z = Z'$. So $Z$ is weakly optimal.

Thus $X^{\mathrm {deg}}(t)$ is the union of some weakly optimal subvarieties. We will show that this union is finite. The key point is the following finiteness theorem concerning weakly optimal subvarieties [Reference GaoGao20, Theorem 1.4]. See § 5.3 for notation.

7.2 An auxiliary proposition

Let $X$ be as in Theorem 7.1.

Let $Z$ be a positive dimensional closed irreducible subvariety $X$ such that $\dim \langle Z \rangle _{\mathrm {sg}} - \dim \pi (Z) < \dim Z + t$ and is maximal for this property. Then $Z$ is weakly optimal by Lemma 7.4. So Theorem 7.5 gives a finite set $\Sigma$ and a $((Q,\mathcal {Y}^+),N) \in \Sigma$ such that $Z^{\mathrm {biZar}} = {\mathbf {u}}(N(\mathbb {R})^+\tilde {y})$ for some $\tilde {y} \in \mathcal {Y}^+$. Recall that $N$ is a connected normal subgroup of $Q$ whose reductive part is semi-simple. Note that $N \not = 1$ since $Z$ has positive dimension.

Consider the operation of taking quotient mixed Shimura datum $\tilde {p}_N \colon (Q,\mathcal {Y}^+) \rightarrow (Q,\mathcal {Y}^+)/N =: (Q', \mathcal {Y}^{\prime +})$ discussed in § 5.4 and the induced morphism on the corresponding mixed Shimura varieties of Kuga type.

(7.1)

We refer to (5.4) for the geometric meaning of $p_N \colon M \rightarrow M'$ and notation.

For any integer $h$, define

(7.2)\begin{equation} E_h = \{ x \in X : \dim_x (p_N|_{X\cap M})^{-1}(p_N(x)) > h\}. \end{equation}

Then $E_h$ is Zariski closed in $X$.

Proof. (i) Recall that $Z$ satisfies $\dim \langle Z \rangle _{\mathrm {sg}} - \dim \pi (Z) < \dim Z + t$. Therefore $\delta _{\mathrm {ws}}(Z) < \dim \pi (Z)^{\mathrm {biZar}} + t$ by Lemma 7.3. Hence

\[ \dim Z > \dim Z^{\mathrm{biZar}} - \dim \pi(Z)^{\mathrm{biZar}} - t. \]

By construction we know that $Z^{\mathrm {biZar}}$ is a fiber of $p_N$. Hence as a morphism, $Z^{\mathrm {biZar}} \rightarrow \pi (Z)^{\mathrm {biZar}}$ is an abelian scheme of relative dimension $g_N$ by the discussion below (5.4). So the inequality above becomes $\dim Z > g_N - t$. As $Z \subseteq X \cap M$, we have that $Z \subseteq E_{g_N - t}$ by definition of $E_{g_N - t}$.

(ii) For any $x \in E_{g_N - t}$, there exists a component $Y$ of $(p_N|_{X\cap M})^{-1}(p_N(x))$ containing $x$ such that $\dim Y > g_N - t$. Let $x' = p_N(x)$.

By (5.4) and the discussion below, $p_N^{-1}(x') \rightarrow \pi (p_N^{-1}(x'))$ is an abelian scheme of relative dimension $g_N$.

Proposition 5.3 says that as a morphism, $Y^{\mathrm {biZar}} \rightarrow \pi (Y)^{\mathrm {biZar}}$ is an abelian scheme. Since $p_N^{-1}(x')$ is bi-algebraic [Reference GaoGao17b, Corollary 8.3] and $Y \subseteq p_N^{-1}(x')$, we have $Y^{\mathrm {biZar}} \subseteq p_N^{-1}(x')$. Thus the relative dimension of the abelian scheme $Y^{\mathrm {biZar}} \rightarrow \pi (Y)^{\mathrm {biZar}}$ is at most $g_N$. So

\[ \dim Y > g_N-t \ge \dim Y^{\mathrm{biZar}} - \dim \pi(Y)^{\mathrm{biZar}} -t. \]

Hence $\delta _{\mathrm {ws}}(Y) < \dim \pi (Y)^{\mathrm {biZar}} + t$. Thus $\dim \langle Y \rangle _{\mathrm {sg}} - \dim \pi (Y) < \dim Y + t$ by Lemma 7.3, and hence $Y \subseteq X^{\mathrm {deg}}(t)$ by definition. By varying $x \in E_h$, we get the conclusion.

7.3 Proof of Theorem 7.1

Now we are ready to prove Theorem 7.1. Let $X$ be as in Theorem 7.1. Let $\Sigma$ be the finite set in Theorem 7.5.

For any positive dimensional closed irreducible subvariety $Z$ of $X$ such that $\dim \langle Z \rangle _{\mathrm {sg}} - \dim \pi (Z) < \dim Z + t$ and is maximal for this property, we obtain some $((Q,\mathcal {Y}^+),N) \in \Sigma$ from which we can construct a Zariski closed subset $E_{g_N - t}$ of $X$ such that $Z \subseteq E_{g_N - t}$ by (i) of Proposition 7.6. Since $\Sigma$ is a finite set, we have finitely many such $E_{g_N - t}$. By definition of $X^{\mathrm {deg}}(t)$, we then have that $X^{\mathrm {deg}}(t)$ is contained in the union of these $E_{g_N - t}$, which is a Zariski closed subset of $X$.

Conversely by (ii) of Proposition 7.6, each such $E_{g_N - t}$ is contained in $X^{\mathrm {deg}}(t)$.

Hence $X^{\mathrm {deg}}(t)$ is the union of these $E_{g_N - t}$. This is a finite union with each member being a closed subset of $X$. Hence $X^{\mathrm {deg}}(t)$ is Zariski closed in $X$.

8.1 Auxiliary proposition

We start by proving the following auxiliary proposition.

Proposition 8.2 We have $X = X^{\mathrm {deg}}(t)$ if and only if there exist:

• • a special subvariety $M_*$ of $\mathfrak {A}_g$, associated with $(Q_*,\mathcal {Y}^+_*)$, which contains $X$;

• • a non-trivial connected normal subgroup $N_*$ of $Q_*$ whose reductive part is semi-simple;

such that the following condition holds: for the operation of taking quotient mixed Shimura datum $\tilde {p}_{N_*} \colon (Q_*,\mathcal {Y}_*^+) \rightarrow (Q_*,\mathcal {Y}_*^+)/N_*$ discussed in § 5.4 and the induced morphism on the corresponding mixed Shimura varieties of Kuga type

(8.2)

we have $\dim X - \dim p_{N_*}(X) > g_{N_*} - t$, where $g_{N_*} = \frac{1}{2} \dim (V_{2g} \cap N_*)$.

8.2 Theorem 8.1 in terms of mixed Shimura variety

In this subsection, we prove Theorem 8.1 in terms of mixed Shimura variety. Then we translate it into the desired geometric description in the next subsection.

Let $M$ be the smallest special subvariety of $\mathfrak {A}_g$, which contains $X$. Assume that $M$ is associated with the mixed Shimura subdatum of Kuga type $(Q,\mathcal {Y}^+)$ of $(P_{2g,\mathrm {a}},\mathcal {X}_{2g,\mathrm {a}}^+)$.

Proposition 8.3 Assume either $t \le 0$, or $t = 1$ and $M = \mathfrak {A}_g|_{\pi (M)}$. Then $X = X^{\mathrm {deg}}(t)$ if and only if there exists a non-trivial connected normal subgroup $N$ of $Q$ whose reductive part is semi-simple, such that the following condition holds: For the operation of taking quotient Shimura datum $\tilde {p}_N \colon (Q,\mathcal {Y}^+) \rightarrow (Q,\mathcal {Y}^+)/N =: (Q', \mathcal {Y}^{\prime +})$ discussed in § 5.4 and the induced morphism on the corresponding mixed Shimura varieties of Kuga type

(8.3)

we have $\dim X - \dim p_N(X) > g_N - t$, where $g_N = \frac{1}{2} \dim (V_{2g} \cap N)$.

Proof of Proposition 8.3 We use Proposition 8.2. First $\Leftarrow$ of Proposition 8.3 follows directly from $\Leftarrow$ of Proposition 8.2.

Let us prove $\Rightarrow$ of Proposition 8.3. Let $M_*$, $(Q_*,\mathcal {Y}^+_*)$, $N_*$ and $g_{N_*}$ be as in $\Rightarrow$ of Proposition 8.2.

Set $N$ to be the identity component of $Q \cap N_*$. Since $X \subseteq M_*$ and $M$ is the smallest special subvariety of $\mathfrak {A}_g$, which contains $X$, we have $M \subseteq M_*$. Hence we may assume $(Q,\mathcal {Y}^+) \subseteq (Q_*,\mathcal {Y}_*^+)$. Let $N = Q \cap N_*$, then $N \lhd Q$. Replacing $N$ by $N \cap \tilde {\pi }^{-1}(\tilde {\pi }(N)^{\mathrm {der}})$, we may and do assume that the reductive part of $N$ is semi-simple; here $\tilde {\pi } \colon P_{2g,\mathrm {a}} \rightarrow \mathrm {GSp}_{2g}$.

Let us temporarily assume

(8.4)\begin{equation} \dim p_{N_*}(X) = \dim p_N(X) \end{equation}

and finish the proof by showing that this $N$ can be taken as the desired connected normal subgroup of $Q$. We need to prove the following:

1. (i) $\dim X - \dim p_N(X) > g_N - t$ for $g_N = \frac{1}{2} \dim (V_{2g} \cap N)$;

2. (ii) $N$ is a non-trivial group.

For (i), we have $g_N \le g_{N_*}$ since $N < N_*$. So $\dim X - \dim p_N(X) = \dim X - \dim p_{N_*}(X) > g_{N_*} - t \ge g_N - t$.

For (ii), suppose $N$ is trivial. Then $p_N = \mathrm {id}_M$ and thus $\dim p_N(X) = \dim X$. So $\dim p_{N_*}(X) = \dim X$. Hence $p_{N_*}|_X$ is generically finite. So $X= E_{g_{N_*}-t}$ implies that $g_{N_*} - t < 0$. But then $g_{N_*} = 0$ and $t = 1$ as $t \le 1$. Hence $N_*$ is reductive. It can be viewed as a subgroup of $\mathrm {Sp}_{2g}$ via $\tilde {\pi } \colon P_{2g,\mathrm {a}} \rightarrow \mathrm {GSp}_{2g}$. Now $N_* \lhd Q_*$ implies that the subgroup $N_*$ of $\mathrm {Sp}_{2g}$ acts trivially on $V_{2g} \cap Q_*$. By our hypothesis on $X$ (note that $t = 1$ now) and Proposition 5.6, we have $M = \mathfrak {A}_g|_{\pi (M)}$. Hence $V_{2g} \cap Q = V_{2g}$. Thus $V_{2g} \cap Q_* = V_{2g}$ since $Q \subseteq Q_*$. But the only connected subgroup of $\mathrm {Sp}_{2g}$ acting trivially on $V_{2g}$ is $1$. Hence $N_*$ is trivial, contradicting to the choice of $N_*$. Thus $N$ is non-trivial. Hence this $N$ can be taken as the desired connected normal subgroup of $Q$.

Now it remains to prove (8.4). Each fiber of $\tilde {p}_N$ is of the form $N(\mathbb {R})^+\tilde {y}$ for some $\tilde {y} \in \mathcal {Y}^+$, and each fiber of $\tilde {p}_{N_*}$ is of the form $N_*(\mathbb {R})^+\tilde {y}_*$ for some $\tilde {y}_* \in \mathcal {Y}^+_*$. As $\mathcal {Y}^+ \subseteq \mathcal {Y}^+_*$, we can take $\tilde {y}_* = \tilde {y} \in \mathcal {Y}^+$. By definition of $N$, we have $N(\mathbb {R})^+\tilde {y} = Q(\mathbb {R})^+ \tilde {y} \cap N_*(\mathbb {R})^+ \tilde {y} = \mathcal {Y}^+ \cap N_*(\mathbb {R})^+\tilde {y}$.

Denote by $\tilde {X}$ a complex analytic irreducible component of $\mathbf {u}^{-1}(X)$, which is contained in $\mathcal {Y}^+$. Then by the previous paragraph, each fiber of $\tilde {p}_N|_{\tilde {X}}$ is of the form $N(\mathbb {R})^+\tilde {y} \cap \tilde {X}$, and each fiber of $\tilde {p}_{N_*}|_{\tilde {X}}$ is of the form $N_*(\mathbb {R})^+\tilde {y} \cap \tilde {X}$. The last sentence of last paragraph says $N(\mathbb {R})^+\tilde {y} \cap \tilde {X} = \mathcal {Y}^+ \cap N_*(\mathbb {R})^+\tilde {y} \cap \tilde {X}$, which furthermore equals $N_*(\mathbb {R})^+\tilde {y} \cap \tilde {X}$ since $\tilde {X} \subseteq \mathcal {Y}^+$. Thus the fibers of $\tilde {p}_N|_{\tilde {X}}$ and $\tilde {p}_{N_*}|_{\tilde {X}}$ have the same dimensions. Hence the fibers of $p_N|_X$ and $p_{N_*}|_X$ have the same dimensions. Thus (8.4) holds. Now we are done.

8.3 Proof of Theorem 8.1

Let $M$ be the smallest special subvariety of $\mathfrak {A}_g$, which contains $X$. Assume that $M$ is associated with the mixed Shimura subdatum of Kuga type $(Q,\mathcal {Y}^+)$ of $(P_{2g,\mathrm {a}},\mathcal {X}_{2g,\mathrm {a}}^+)$. Denote by $V_Q = V_{2g} \cap Q$ and $G_Q = \tilde {\pi }(Q)$ the reductive part of $Q$. Let $g_Q = \frac{1}{2}\dim V_Q$.

Special subvarieties of $\mathfrak {A}_g$ are described in Proposition 5.6. Hence the smallest special subvariety $M$ of $\mathfrak {A}_g$, which contains $X$ can be described as follows:

• • let $M_G$ be the smallest special subvariety of $\mathbb {A}_g$, which contains $\pi (X)$;

• • let $M$ be the translate of an abelian subscheme of $\mathfrak {A}_g|_{M_G} \rightarrow M_G$ by a torsion section which contains $X$, minimal for this property.

Then $M \rightarrow M_G$ itself is an abelian scheme of relative dimension $g_Q$.

$\Rightarrow$ of Theorem 8.1. Apply $\Rightarrow$ of Proposition 8.3. Then we obtain a non-trivial connected normal subgroup $N$ of $Q$ whose reductive part is semi-simple, such that for the quotient Shimura morphism $p_N \colon M \rightarrow M'$, we have ($g_N = \frac{1}{2}\dim (V_{2g}\cap N)$)

(8.5)\begin{equation} \dim X - \dim p_N(X) > g_N - t. \end{equation}

Recall the geometric meaning of $p_N$ (5.4):

The morphism $p_0$ is taking the quotient of $M \rightarrow M_G$ by the abelian subscheme $\operatorname {Ker}(p_0)^{\circ }$ (which has relative dimension $g_N$). For each $b' \in M_G'$, the abelian scheme $M_0|_{p_{G_N}^{-1}(b')} \rightarrow p_{G_N}^{-1}(b')$ is isotrivial.

Take $\mathcal {B} = \operatorname {Ker}(p_0)^{\circ } \cap \mathcal {A}_X$. Then $\mathcal {B}$ is an abelian subscheme of $\mathcal {A}_X \rightarrow B$ of relative dimension $g_N$, namely $g_{\mathcal {B}} = g_N$. Now we can construct the maps in (8.1). We have $\mathcal {A}_X = M|_B$, $p_{\mathcal {B}} = p_0|_{\mathcal {A}_X}$, $\pi _{/\mathcal {B}} = \pi _0|_{M_0|_B}$, $\iota _{/\mathcal {B}} = (i \circ p')|_{M_0|_B}$ and $\iota _{/\mathcal {B},G} = (i_G \circ p_{G_N})|_B$. Thus

\[ \dim (\iota_{/\mathcal{B}} \circ p_{\mathcal{B}})(X) \le \dim p_N(X) < \dim X - g_N + t = \dim X - g_{\mathcal{B}} +t, \]

where the first inequality follows from $X \subseteq M|_B$ and the second inequality follows from (8.5). Thus it suffices to prove that $\iota _{/\mathcal {B}} \circ p_{\mathcal {B}}$ is not generically finite.

Suppose $\iota _{/\mathcal {B}} \circ p_{\mathcal {B}}$ is generically finite. Then $g_N = 0$, $p_{\mathcal {B}} = \mathrm {id}_{\mathcal {A}_X}$ and $\iota _{/\mathcal {B}}$ is generically finite. Hence $\dim (\iota _{/\mathcal {B}} \circ p_{\mathcal {B}})(X) = \dim X$. But then $\dim X < \dim X - g_{\mathcal {B}} + t = \dim X + t$. When $t \le 0$ this cannot hold. Hence $t = 1$ and $\mathcal {A}_X = \mathfrak {A}_g|_B$ by our hypothesis. Hence $M = \mathfrak {A}_g|_{M_G}$.

Since $g_N = 0$, we have $p_0 = \mathrm {id}_M$ and $M_0 = M = \mathfrak {A}_g|_{M_G}$. For each $b' \in M_G'$, the abelian scheme $\mathfrak {A}_g|_{p_{G_N}^{-1}(b')} \rightarrow p_{G_N}^{-1}(b')$ is isotrivial. So $\dim p_{G_N}^{-1}(b') = 0$; see the end of § 2. Hence $p_{G_N}$ is generically finite, and so is $p'$. Thus $\dim M = \dim M'$. This contradicts our choice of $N$ (non-trivial connected).

$\Leftarrow$ of Theorem 8.1. Before moving on, let me point out that if $t \le 0$, then this implication follows rather easily from Proposition 6.1 and Theorem 7.1 because we can translate it into studying the generic rank of the Betti map. However the argument below works also for $t =1$.

Hodge theory says that: (1) every abelian subscheme of $\mathcal {A}_X \rightarrow B$ is the intersection of an abelian subscheme of $M \rightarrow M_G$ with $\mathfrak {A}_g|_B$ (in particular $\mathcal {A}_X = M \cap \mathfrak {A}_g|_B = M|_B$ and $g_X = g_Q$); (2) the abelian subschemes of $M \rightarrow M_G$ are in one-to-one correspondence to $G_Q$-submodules of $V_Q$. See [Reference DeligneDel71, 4.4.1–4.4.3].

Assume that $\mathcal {B} = \mathfrak {B} \cap \mathfrak {A}_g|_B$ where $\mathfrak {B}$ is an abelian subscheme of $M \rightarrow M_G$, and that $\mathfrak {B}$ corresponds to the $G_Q$-submodule $V_N$ of $V_Q$. Let $g_N = \frac{1}{2}\dim V_N$, then $\mathfrak {B} \rightarrow M_G$ has relative dimension $g_N$. Hence $g_{\mathcal {B}} = g_N$.

Taking the quotient, we get the following.

In particular, $\pi _{/\mathfrak {B}}$ is an abelian scheme of relative dimension $g_Q-g_N$. It induces the following modular map.

By definition of $\iota _{/\mathfrak {B},G}$, $(M/\mathfrak {B})|_{\iota _{/\mathfrak {B},G}^{-1}(a)}$ is an isotrivial abelian scheme for each $a \in \mathbb {A}_{g_Q-g_N}$.

Modular interpretation of Shimura varieties implies that $\iota _{/\mathfrak {B},G}$ is a Shimura morphism, namely $\iota _{/\mathfrak {B},G}$ is induced by some $\tilde {\iota }_{/\mathfrak {B},G} \colon (G_Q,\tilde {\pi }(\mathcal {Y}^+)) \rightarrow (\mathrm {GSp}_{2(g_Q-g_N)}, \mathfrak {H}_{g_Q-g_N}^+)$. Denote by $H$ the kernel of $\tilde {\iota }_{/\mathfrak {B},G}$ on the underlying groups, then $H \lhd G_Q$. Replace $H$ by $H^{\mathrm {der}}$, then $H \lhd G_Q^{\mathrm {der}}$.

Each fiber of $\tilde {\iota }_{/\mathfrak {B},G}$ on the underlying spaces is of the form $H(\mathbb {R})^+\tilde {y}_G$ for some $\tilde {y}_G \in \tilde {\pi }(\mathcal {Y}^+)$. Hence each fiber of $\iota _{/\mathcal {B},G}$ is of the form $\mathbf {u}_G(H(\mathbb {R})^+\tilde {y}_G)$ for some $\tilde {y}_G \in \tilde {\pi }(\mathcal {Y}^+)$, where $\mathbf {u}_G \colon \mathfrak {H}_g^+ \rightarrow \mathbb {A}_g$ is the uniformization. Thus the connected algebraic monodromy group of $\iota _{/\mathfrak {B},G}^{-1}(a)$ is $H$ for each $a \in \mathbb {A}_{g_Q-g_N}$. By Deligne's theorem of the fixed part [Reference DeligneDel71, Corollaire 4.1.2], $H$ acts trivially on $V_Q/V_N \simeq V_{2(g_Q-g_N)}$ because $(M/\mathfrak {B})|_{\iota _{/\mathfrak {B},G}^{-1}(a)}$ is isotrivial.

The zero section of the abelian scheme $M \rightarrow M_G$ gives rise to a Levi decomposition $Q = V_Q \rtimes G_Q$. Let $N = V_N \rtimes H$. Then $N$ is a connected normal subgroup of $Q$ whose reductive part is semi-simple. Moreover the quotient morphism $p_N$ is precisely $\iota _{/\mathcal {B}} \circ p_{\mathfrak {B}}$, namely the geometric interpretation (5.4) for this $N$ becomes the following.

(8.6)

Note that (8.1) is precisely (

) restricted to $B \subseteq M_G$.

If $N = 1$, then $p_N = \mathrm {id}_M$. This contradicts $\iota _{/\mathcal {B}} \circ p_{\mathcal {B}}$ being not generically finite. Hence $N$ is non-trivial.

Note that $X \subseteq M|_B$. Thus the inequality in $\Leftarrow$ of Theorem 8.1 becomes $\dim p_N(X) < \dim X - g_N + t$. Now it suffices to apply $\Leftarrow$ of Proposition 8.3 to this $N$.

9.1 Proof of Theorem 1.8

Fix $t \in \mathbb {Z}$. We proved that $\iota (X)^{\mathrm {deg}}(t)$ is Zariski closed in $\iota (X)$ for each $t$ in Theorem 7.1. Thus Theorem 1.8 follows immediately from the below lemma.

Proof. First let us make the following observation: for any $Y \subseteq \mathcal {A}$ irreducible, we have that $\langle Y \rangle _{\mathrm {sg}}$ is an irreducible component of $\iota ^{-1}(\langle \iota (Y) \rangle _{\mathrm {sg}})$. So

(9.2)\begin{equation} \dim \langle \iota(Y) \rangle_{\mathrm{sg}} - \dim \pi(\iota(Y)) = \dim \langle \iota(Y) \rangle_{\mathrm{sg}} - \dim \iota_S(\pi_S(Y)) = \dim \langle Y \rangle_{\mathrm{sg}} - \dim \pi_S(Y) \end{equation}

$\text {Case } t+r \le 0\text { or }r=0$. For $\subseteq$, take $Y \subseteq X^{\mathrm {deg}}(t)$ irreducible such that $\dim Y > 0$ and

\[ \dim \langle Y \rangle_{\mathrm{sg}} - \dim \pi_S(Y) < \dim Y + t. \]

As $\iota |_X$ has relative dimension $r$, we have $\dim \iota (Y) \ge \dim Y - r$. So (9.2) implies

\[ \dim \langle \iota(Y) \rangle_{\mathrm{sg}} - \dim \pi(\iota(Y)) = \dim \langle Y \rangle_{\mathrm{sg}} - \dim \pi_S(Y) < \dim Y + t \le \dim \iota(Y) + (t+r). \]

If $t+r \le 0$, then the inequality above further implies $0 < \dim \iota (Y)$. If $r=0$, then $\iota$ is generically finite and hence $\dim \iota (Y) = \dim Y >0$. Hence in either case, we have $\iota (Y) \subseteq \iota (X)^{\mathrm {deg}}(t+r)$. This proves $\subseteq$.

For $\supseteq$, conversely take $Y' \subseteq \iota (X)^{\mathrm {deg}}(t+r)$ irreducible such that $\dim Y' >0$ and

\[ \dim \langle Y' \rangle_{\mathrm{sg}} - \dim \pi(Y') < \dim Y' + t+r. \]

Let $Y$ be an irreducible component of $\iota |_X^{-1}(Y')$. Then $Y' = \iota (Y)$. So $\dim Y \ge \dim Y' + r >0$ by (9.1). By (9.2) we have then

\[ \dim \langle Y \rangle_{\mathrm{sg}} - \dim \pi_S(Y) = \dim \langle Y' \rangle_{\mathrm{sg}} - \dim \pi(Y') < \dim Y' + t+r = \dim Y + t. \]

Hence $Y \subseteq X^{\mathrm {deg}}(t)$. This proves $\supseteq$.

Case t + r > 0 and r > 0. Let $x \in X(\mathbb {C})$. Let $Y$ be the irreducible component of $\iota |_X^{-1}(\iota (x))$, which contains $x$. Then $\dim Y \ge r > 0$.

For $B:= \pi _S(\iota ^{-1}(\iota (x))) = \iota _S^{-1}(\pi (\iota (x)))$, we have that $\iota ^{-1}(\iota (x))$ is a constant section of $\mathcal {A}|_B \rightarrow B$. Thus $\dim \pi _S(Y) = \dim Y \ge r$, and $\langle Y \rangle _{\mathrm {sg}} = Y$. Hence

\[ \dim \langle Y \rangle_{\mathrm{sg}} - \dim \pi_S(Y) = \dim Y - \dim Y = 0 < r -r+1 \le \dim Y + (-r+1). \]

So $Y \subseteq X^{\mathrm {deg}}(-r+1)$. Thus $X \subseteq X^{\mathrm {deg}}(-r+1)$.

But $t \ge -r+1$ since $t+r>0$. So $X^{\mathrm {deg}}(-r+1) \subseteq X^{\mathrm {deg}}(t)$ by definition. So we are done.