2.1 Moduli space of abelian varieties

Let $g \ge 1$ be an integer. Let $D = \mathrm {diag}(d_1,\ldots ,d_g)$ with $d_1|\cdots |d_g$ be positive integers. Let $N \ge 1$ be an integer. Let $\mathbb {A}_{g,D}(N)$ be the moduli space of abelian varieties of dimension $g$, which are polarized of type $D$ equipped with level-$N$-structures.

It is well-known that $\mathbb {A}_{g,D}$ is a connected pure Shimura variety, associated with the connected pure Shimura datum $(\mathrm {GSp}_{2g,D},\mathfrak {H}_g^+)$; see [Reference Genestier and NgôGN06, § 1.2]. We hereby give a quick summary of this fact.

Let $V_{2g}$ be a $\mathbb {Q}$-vector space of dimension $2g$, and let

(2.1)\begin{equation} \Psi \colon V_{2g} \times V_{2g} \rightarrow \mathbb{G}_{a,\mathbb{Q}}, \quad (v_1,v_2) \mapsto v_1^{\!^{\intercal}} \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} v_2 \end{equation}

be a non-degenerate alternating form. Then the $\mathbb {Q}$-group $\mathrm {GSp}_{2g,D}$ is defined by

\begin{align*} \mathrm{GSp}_{2g,D} & =\{h \in \operatorname{GL}(V_{2g}) : \Psi(hv,hv') = \nu(h)\Psi(v,v')\text{ for some }\nu(h) \in \mathbb{G}_m\} \\ & =\biggl\{ h \in \operatorname{GL}_{2g} : h \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} h^{\!^{\intercal}} = \nu(h) \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix},~ \nu(h) \in \mathbb{G}_m \biggl\}. \end{align*}

Let $\mathrm {Sp}_{2g,D} = \mathrm {GSp}_{2g,D}^{\mathrm {der}}$.

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

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

Then $\mathrm {GSp}_{2g,D}(\mathbb {R})^+$ acts on $\mathfrak {H}_g^+$ by the formula

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

It is known that the action of $\mathrm {GSp}_{2g,D}(\mathbb {R})^+$ on $\mathfrak {H}_g^+$ thus defined is transitive.

The natural inclusion $\mathfrak {H}_g^+ \subseteq \{ Z = X + \sqrt {-1}Y \in M_{g \times g}(\mathbb {C}) : Z = Z^{\!^{\intercal }}\} \simeq \mathbb {C}^{g(g+1)/2}$ realizes $\mathfrak {H}_g^+$ as an open (in the usual topology) semi-algebraic subset of $\mathbb {C}^{g(g+1)/2}$. Hence this inclusion endows $\mathfrak {H}_g^+$ with a complex structure.

Let $\Gamma _{\mathrm {Sp}_{2g,D}}(N)= \{ h \in \mathrm {GSp}_{2g,D}(\mathbb {Z}): h \equiv I_{2g} \pmod {N} \}$. Then $\mathbb {A}_{g,D}(N) \simeq \Gamma _{\mathrm {Sp}_{2g,D}}(N) \backslash \mathfrak {H}_g^+$ as complex varieties. Thus we obtain a uniformization in the category of complex varieties

(2.2)\begin{equation} \mathfrak{H}_g^+ \rightarrow \mathbb{A}_{g,D}(N). \end{equation}

2.2 Universal abelian variety

Use the notation of the previous subsection. If furthermore $N\ge 3$, then $\mathbb {A}_{g,D}(N)$ is a fine moduli space and hence admits a universal family, which we call $\mathfrak {A}_{g,D}(N)$. We use $\pi \colon \mathfrak {A}_{g,D}(N) \rightarrow \mathbb {A}_{g,D}(N)$ to denote the natural projection.

The variety $\mathfrak {A}_{g,D}(N)$ is an example of a connected mixed Shimura variety (of Kuga type). We hereby give a construction of the connected mixed Shimura datum of Kuga type $(P_{2g,D,\mathrm {a}},\mathcal {X}_{2g,\mathrm {a}}^+)$ associated with $\mathfrak {A}_{g,D}(N)$.

Recall that $V_{2g}$ is a $\mathbb {Q}$-vector space of dimension $2g$. By abuse of notation we also use it to denote the $\mathbb {Q}$-vector group of dimension $2g$. Then the natural action of $\mathrm {GSp}_{2g,D}$ on $V_{2g}$ defines a $\mathbb {Q}$-group

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

It is not hard to see that the unipotent radical $\mathcal {R}_u(P_{2g,D,\mathrm {a}})$ of $P_{2g,D,\mathrm {a}}$ is $V_{2g}$, and the reductive part $P_{2g,D,\mathrm {a}}/\mathcal {R}_u(P_{2g,D,\mathrm {a}})$ of $P_{2g,D,\mathrm {a}}$ is $\mathrm {GSp}_{2g,D}$.

The space $\mathcal {X}_{2g,\mathrm {a}}^+$ is constructed as follows.

1. (i) As a set, $\mathcal {X}_{2g,\mathrm {a}}^+ = V_{2g}(\mathbb {R}) \times \mathfrak {H}_g^+$.

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

   (2.3)\begin{equation} (v,h) \cdot (v',x) = (v+hv',hx). \end{equation}
   This action is transitive.

3. (iii) Fix a Lagrangian decomposition of $V_{2g}(\mathbb {R}) = \mathbb {R}^{2g} \simeq \mathbb {R}^g \times \mathbb {R}^g$. The complex structure of $\mathcal {X}_{2g,\mathrm {a}}^+$ is the one given by the pullback of the following map:

   (2.4)\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 pair $(P_{2g,D,\mathrm {a}},\mathcal {X}_{2g,\mathrm {a}}^+)$ is a connected mixed Shimura datum as defined by Pink; see [Reference PinkPin05, Construction 2.9, Example 2.12] and [Reference PinkPin89, 2.25]. There is a natural morphism

\[ \tilde{\pi} \colon (P_{2g,D,\mathrm{a}},\mathcal{X}_{2g,\mathrm{a}}^+) \rightarrow (\mathrm{GSp}_{2g,D},\mathfrak{H}_g^+) \]

induced by the projection $P_{2g,D,\mathrm {a}} = V_{2g} \rtimes \mathrm {GSp}_{2g,D} \rightarrow \mathrm {GSp}_{2g,D}$. This is a Shimura morphism as defined by Pink [Reference PinkPin05, Definition 2.5] or [Reference PinkPin89, 2.3].

Let $\Gamma _{P_{2g,D,\mathrm {a}}}(N) = N V(\mathbb {Z}) \rtimes \Gamma _{\mathrm {Sp}_{2g,D}}(N)$. Then as complex varieties we have $\mathfrak {A}_g(N) \simeq \Gamma _{P_{2g,D,\mathrm {a}}}(N) \backslash \mathcal {X}_{2g,\mathrm {a}}^+$. This makes $\mathfrak {A}_{g,D}(N)$ a connected mixed Shimura variety. See [Reference PinkPin05, Example 2.12] or [Reference PinkPin89, 10.5, 10.9, 10.10]. We thus obtain a uniformization in the category of complex varieties

(2.5)\begin{equation} \mathcal{X}_{2g,\mathrm{a}}^+ \rightarrow \mathfrak{A}_{g,D}(N). \end{equation}

Now we have the following commutative diagram

(2.6)

where for simplicity of notation the vertical maps are the uniformizations we discussed above.

2.3 A Hodge-theoretic point of view on $\mathcal {X}_{2g,\mathrm {a}}^+$

Let us take a closer look at $\mathcal {X}_{2g,\mathrm {a}}^+$. We work with the category of analytic objects in this section. For future purpose, we make the situation slightly more general. Let $\Delta$ be a simply-connected complex space, and let $\pi _{\Delta } \colon \mathcal {A}_{\Delta } \rightarrow \Delta$ be a polarized family of abelian varieties of dimension $g \ge 1$. Use $\psi$ to denote this polarization.

Let $\underline {\mathbb {Z}}_{\Delta }$ (respectively $\underline {\mathbb {C}}_{\Delta }$) be the constant local system of $\mathbb {Z}$-rank $1$ (respectively of $\mathbb {C}$-dimension $1$) on $\Delta$. Let $\underline {\mathbb {Z}}_{\mathcal {A}_\Delta }$ (respectively $\underline {\mathbb {C}}_{\mathcal {A}_\Delta }$) be the constant local system of $\mathbb {Z}$-rank $1$ (respectively of $\mathbb {C}$-dimension $1$) on $\mathcal {A}_\Delta$.

Then $R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta }$, which we define as the dual of $R^1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta }$, is a local system on $\Delta$ such that $(R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta })_s = H_1(\mathcal {A}_s,\mathbb {Z})$ for any $s \in \Delta$. It defines a variation of $\mathbb {Z}$-Hodge structures over $\Delta$ of type $\{(0,-1),(-1,0)\}$ in the following way: the $\mathcal {O}_{\Delta }$-sheaf $\mathcal {V}_{\Delta }=R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta } \otimes _{\underline {\mathbb {Z}}_{\Delta }} \mathcal {O}_{\Delta } \simeq \mathcal {H}^1_{\mathrm {dR}}(\mathcal {A}_{\Delta }/\Delta )^\vee$ is locally free, and has a locally free $\mathcal {O}_{\Delta }$-subsheaf $\mathcal {F}^0\mathcal {V}_{\Delta }$ defined by $(\mathcal {F}^0\mathcal {V}_{\Delta })_s = H^{0,1}(\mathcal {A}_s)^\vee$ for any $s\in \Delta$. From this we see that $\mathcal {V}_{\Delta }/\mathcal {F}^0\mathcal {V}_{\Delta } \simeq (\varOmega ^1_{\mathcal {A}_{\Delta }/\Delta })^\vee$. Note that $R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta }$ and $\mathcal {F}^0\mathcal {V}_{\Delta }$ are subsheaves of $\mathcal {V}_{\Delta }$, and

(2.7)\begin{equation} R_1(\pi_{\Delta})_*\underline{\mathbb{Z}}_{\mathcal{A}_\Delta} \cap \mathcal{F}^0\mathcal{V}_{\Delta} \text{ is trivial, namely is the constant sheaf }\underline{0}_{\Delta}. \end{equation}

Let $\textbf {V}_{\Delta }$ and $\textbf {F}^0\textbf {V}_{\Delta }$ be the vector bundles associated with the locally free sheaves $\mathcal {V}_{\Delta }$ and $\mathcal {F}^0\mathcal {V}_{\Delta }$. Then the discussion in the previous paragraph yields an exact sequence of holomorphic vector bundles on $\Delta$

(2.8)\begin{equation} 0 \rightarrow \textbf{F}^0\textbf{V}_{\Delta} \rightarrow \textbf{V}_{\Delta} \rightarrow \mathrm{Lie}(\mathcal{A}_{\Delta}/\Delta) \rightarrow 0. \end{equation}

Next we explain that there exists a real analytic diffeomorphism $\mathrm {Lie}(\mathcal {A}_{\Delta }/\Delta ) \simeq \mathbb {R}^{2g} \times \Delta$.

Note that $R_1(\pi _{\Delta })_*\underline {\mathbb {C}}_{\mathcal {A}_\Delta } = R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta } \otimes _{\underline {\mathbb {Z}}_{\Delta }} \underline {\mathbb {C}}_{\Delta }$ is the local system on $\Delta$ of the $\mathbb {C}$-dimension $2g$. There is a unique vector bundle with flat connection associated with it. Since $\Delta$ is simply connected, this vector bundle with flat connection is trivial. In other words, we have $\textbf {V}_{\Delta } \simeq \mathbb {C}^{2g} \times \Delta$ and $R_1(\pi _{\Delta })_*\underline {\mathbb {C}}_{\mathcal {A}_\Delta }$ is the subsheaf of $\mathcal {V}_{\Delta }$ consisting of constant sections (recall that $\mathcal {V}_{\Delta }$ is the sheaf of sections of the vector bundle $\textbf {V}_{\Delta }$ over $\Delta$). So the constant sheaf $R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta }$ is the subsheaf of $\mathcal {V}_{\Delta }$ consisting of the sections of $\textbf {V}_{\Delta }$ over $\Delta$ with image in $\mathbb {Z}^{2g} \times \mathcal {V}_{\Delta }$ (these sections are constant since $\mathbb {Z}^{2g}$ is discrete). So the inclusion of sheaves $R_1(\pi _{\Delta })_*\underline {\mathbb {Z}}_{\mathcal {A}_\Delta } \subseteq \mathcal {V}_{\Delta }$ corresponds to the natural inclusion $\mathbb {Z}^{2g} \times \Delta \subseteq \mathbb {C}^{2g} \times \Delta \simeq \textbf {V}_{\Delta }$. The intersection of $\mathbb {Z}^{2g} \times \Delta$ and $\textbf {F}^0\textbf {V}_{\Delta }$ in $\textbf {V}_{\Delta }$ is the zero section of $\textbf {V}_{\Delta } \rightarrow \Delta$ by (2.7).

Thus we obtain an injective map $\mathbb {Z}^{2g} \times \Delta \rightarrow \mathrm {Lie}(\mathcal {A}_{\Delta }/\Delta )$ as the composite

\[ \mathbb{Z}^{2g} \times \Delta \subseteq \mathbb{C}^{2g} \times \Delta \simeq \textbf{V}_{\Delta} \rightarrow \textbf{V}_{\Delta}/\textbf{F}^0\textbf{V}_{\Delta} \simeq \mathrm{Lie}(\mathcal{A}_{\Delta}/\Delta). \]

See (2.8) for the last equality. This injective map extends to

(2.9)\begin{equation} \mathbb{R}^{2g} \times \Delta \subseteq \mathbb{C}^{2g} \times \Delta \simeq \textbf{V}_{\Delta} \rightarrow \textbf{V}_{\Delta} / \textbf{F}^0\textbf{V}_{\Delta} \simeq \mathrm{Lie}(\mathcal{A}_{\Delta}/\Delta), \end{equation}

where the first inclusion is obtained by identifying $\mathbb {R}^{2g}$ as the real part of $\mathbb {C}^{2g}$. Note that apart from the first inclusion, every morphism in this composition is holomorphic.

Denote by

(2.10)\begin{equation} i_{\Delta} \colon \mathbb{R}^{2g} \times \Delta \rightarrow \mathrm{Lie}(\mathcal{A}_{\Delta}/\Delta) \end{equation}

the composite of the maps in (2.9). Then the following properties clearly hold true.

1. (i) The map $i_{\Delta }$ is a real analytic diffeomorphism. It is semi-algebraic if $\Delta$ has a semi-algebraic structure.

2. (ii) For any $a \in \mathbb {R}^{2g}$, the image $i_{\Delta }(\{a\} \times \Delta )$ is complex analytic.

3. (iii) Over each $s \in \Delta$, the induced map $(i_{\Delta })_s \colon \mathbb {R}^{2g} \rightarrow \mathrm {Lie}(\mathcal {A}_{\Delta }/\Delta )_s$ is a group homomorphism.

We end this subsection by relating it to the last subsection. Take $\Delta = \mathfrak {H}_g^+$ and $\mathcal {A}_{\Delta }$ to be the pullback of $\mathfrak {A}_{g,D}(N) \rightarrow \mathbb {A}_{g,D}(N)$ under the uniformization $\mathfrak {H}_g^+ \rightarrow \mathbb {A}_{g,D}(N)$ in (2.2). Let $\mathcal {X}_{2g,\mathrm {a}}^+ := \mathrm {Lie}(\mathcal {A}_{\Delta }/\Delta )$ for this choice. The complex space $\mathcal {X}_{2g,\mathrm {a}}^+$ thus obtained coincides with the one defined in § 2.2. Indeed, fix a Lagrangian decomposition $V_{2g}(\mathbb {R}) = \mathbb {R}^{2g} \simeq \mathbb {R}^g \times \mathbb {R}^g$ and write

(2.11)\begin{equation} \begin{array}{cccc} \mathcal{X}_{2g,\mathrm{a}}^+ \xleftarrow[i_{\mathfrak{H}_g^+}]{\sim} & V_{2g}(\mathbb{R}) \times \mathfrak{H}_g^+ = \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}

From this we obtain $\rho _g \colon \mathcal {X}_{2g,\mathrm {a}}^+ \rightarrow \mathbb {C}^g \times \mathfrak {H}_g^+$. Then it is not hard to see that $\rho _g$ is an isomorphism of complex spaces. Since $i_{\mathfrak {H}_g^+}$ is semi-algebraic, we can conclude that the $\mathcal {X}_{2g,\mathrm {a}}^+$ defined above is the same complex space as the one defined in § 2.2 in view of (2.4).

2.4 Mixed Shimura variety of Kuga type

In this subsection we recall the definition and some basic properties of mixed Shimura varieties of Kuga type. Let $\mathbb {S} = \mathrm {Res}_{\mathbb {C}/\mathbb {R}}\mathbb {G}_{m,\mathbb {C}}$ be the Deligne torus.

Let $(P,\mathcal {X}^+)$ be a connected mixed Shimura datum of Kuga type. The following results are proven by Pink; see [Reference PinkPin05, Fact 2.3(c)–(e)] or [Reference PinkPin89, 3.3, 9.24]. For any congruence subgroup $\Gamma \subseteq P(\mathbb {Q}) \cap P(\mathbb {R})_+$, where $P(\mathbb {R})_+$ is the stabilizer of $\mathcal {X}^+ \subseteq \operatorname {Hom}(\mathbb {S},P_{\mathbb {R}})$, the quotient $\Gamma \backslash \mathcal {X}^+$ is a complex analytic space with at most finite quotient singularities, and has a natural structure of a quasi-projective algebraic variety over $\mathbb {C}$. Moreover it is smooth if $\Gamma$ is sufficiently small.

Pink proved that every Shimura morphism of connected mixed Shimura varieties of Kuga type is algebraic. See [Reference PinkPin05, Facts 2.6] or [Reference PinkPin89, 3.4, 9.24].

The morphism $\tilde {\pi }$ in (2.6) is an example of Shimura morphism of connected mixed Shimura data of Kuga type. It induces a Shimura morphism $\pi$ of connected mixed Shimura varieties of Kuga type.

Before going on, let us make the following remark. The notion ‘connected mixed Shimura subdatum’ exists a priori for any connected mixed Shimura datum $(P,\mathcal {X}^+)$, not necessarily of Kuga type. However, if $(P,\mathcal {X}^+)$ is of Kuga type, then all its connected mixed Shimura subdata a priori defined are again of Kuga type by reason of weight; see [Reference GaoGao17b, Proposition 2.9]. Hence our Definition 2.4(ii) is compatible with the usual convention.

The following fact is proven by Pink [Reference PinkPin89, 2.9].

In fact Pink proved that such a quotient exists for an arbitrary connected mixed Shimura datum (not necessarily of Kuga type), and then it is not hard to see that the resulting $(P,\mathcal {X}^+)/P_0$ is of Kuga type if $(P,\mathcal {X}^+)$ is.

Notation 2.5 We use the following notation. Let $M = \Gamma \backslash \mathcal {X}^+$ be a connected mixed Shimura variety of Kuga type associated with $(P,\mathcal {X}^+)$, and let $\mathbf {u} \colon \mathcal {X}^+ \rightarrow M$ be the uniformization.

We use $V$ to denote $\mathcal {R}_u(P)$, and $(G,\mathcal {X}_G^+)$ to denote $(P,\mathcal {X}^+)/V$. The quotient Shimura morphism is denoted by $\tilde {\pi } \colon (P,\mathcal {X}^+) \rightarrow (G,\mathcal {X}_G^+)$.

We use $M_G$ to denote the connected pure Shimura variety $\tilde {\pi }(\Gamma ) \backslash \mathcal {X}_G^+$, and use $\mathbf {u}_G \colon \mathcal {X}_G^+ \rightarrow M_G$ to denote the uniformization. Then $\tilde {\pi }$ induces a Shimura morphism $\pi \colon M \rightarrow M_G$. We thus have the following commutative diagram.

(2.12)

We gave an example of a connected mixed Shimura varieties of Kuga type, namely the universal abelian variety; see § 2.2. Specifically, (2.6) is a particular case of (2.12). But in fact, all connected mixed Shimura varieties of Kuga type arise from this case. More precisely, Pink proved the following result, which he called a reduction lemma.Footnote ³

Theorem 2.6 [Pin89, 2.26] Let $(P,\mathcal {X}^+)$ be a connected mixed Shimura datum of Kuga type such that $\dim V = 2g > 0$, where $V = \mathcal {R}_u(P)$. Then there exist a connected pure Shimura datum $(G_0,\mathcal {D}^+)$, a diagonal matrix $D = \mathrm {diag}(d_1,\ldots ,d_g)$ with $d_1|\cdots |d_g$ positive integers, and a Shimura morphism

\[ \tilde{\lambda} \colon (P,\mathcal{X}^+) \hookrightarrow (G_0,\mathcal{D}^+) \times (P_{2g,D,\mathrm{a}}, \mathcal{X}_{2g,\mathrm{a}}^+). \]

Taking the quotient of both sides by their unipotent radicals, we obtain a Shimura morphism $\tilde {\lambda }_G \colon (G,\mathcal {X}^+_G) \rightarrow (G_0,\mathcal {D}^+) \times (\mathrm {GSp}_{2g,D},\mathfrak {H}_g^+)$; see [Reference GaoGao17b, Proposition 2.9] for a proof.

Theorem 2.6 has the following immediate corollary.

Proof. If $V = 0$, then the conclusion certainly holds. Assume $\dim V = 2g > 0$. Then apply Theorem 2.6 to $(P,\mathcal {X}^+)$. We have the following cartesian diagram.

It is cartesian by comparing the dimension of the vertical fibers.

Now up to replacing $\Gamma$ by a subgroup of finite index, we may assume that $\tilde {\lambda }$ induces a closed immersion $M \rightarrow M_{G_0} \times \mathfrak {A}_g(N)$ by [Reference PinkPin89, 3.8(b)]. Thus we obtain a cartesian diagram

where the horizontal morphisms are closed immersions. Hence $\pi \colon M \rightarrow M_G$ is an abelian scheme of relative dimension $g$.

2.5 Realization of $\mathcal {X}^+$

Let $(P,\mathcal {X}^+)$ be a connected mixed Shimura datum of Kuga type. We recall the realization of $\mathcal {X}^+$ as in [Reference GaoGao17b, § 4].

We start with the dual $\mathcal {X}^\vee$ of $\mathcal {X}^+$; see [Reference PinkPin89, 1.7(a)] or [Reference MilneMil90, Chapter VI, Proposition 1.3].

Let $W$ be a faithful representation of $P$ and take any point $x_0 \in \mathcal {X}^+$. The weight filtration on $W$ is constant, so the Hodge filtration $x \mapsto \mathrm {Fil}_x^{\cdot }(W_{\mathbb {C}})$ gives an injective map $\mathcal {X}^+ \hookrightarrow \mathrm {Grass}(W)(\mathbb {C})$ to a certain flag variety. In fact, the injective map factors through

\[ \mathcal{X}^+{=} P(\mathbb{R})^+{/}\mathrm{Stab}_{P(\mathbb{R})^+}(x_0) \hookrightarrow P(\mathbb{C})/\exp(\mathrm{Fil}_{x_0}^0 \mathrm{Lie}P_{\mathbb{C}}) \hookrightarrow \mathrm{Grass}(W)(\mathbb{C}), \]

where the first injection is an open immersion; see [Reference PinkPin89, 1.7(a)] or [Reference MilneMil90, Chapter VI, (1.2.1)]. We define the dual $\mathcal {X}^\vee$ of $\mathcal {X}^+$ to be

\[ \mathcal{X}^\vee{=} P(\mathbb{C})/\exp(\mathrm{Fil}_{x_0}^0 \mathrm{Lie}P_{\mathbb{C}}). \]

Then $\mathcal {X}^\vee$ is clearly a connected smooth complex algebraic variety.

Example 2.9 Let us look at the example $(P_{2g,D,\mathrm {a}},\mathcal {X}_{2g,\mathrm {a}}^+)$. In this case, we can take $W$ be to a $\mathbb {Q}$-vector space of dimension $2g+1$ and identify $P_{2g,D,\mathrm {a}}$ with the following subgroup of $\operatorname {GL}_{2g+1}$:

\[ \begin{pmatrix} \mathrm{GSp}_{2g,D} & V_{2g} \\ 0 & 1 \end{pmatrix}. \]

It is then not hard to prove that the complex structure of $\mathcal {X}_{2g,\mathrm {a}}^+$ given by Proposition 2.8 coincides with the one given by (2.4).Footnote ⁴

The identification (2.4) can be generalized to arbitrary $(P,\mathcal {X}^+)$ of Kuga type in the following way. Use Notation 2.5. By [Reference WildeshausWil97, p. 6], there exists a Shimura morphism $\tilde {i} \colon (G,\mathcal {X}_G^+) \rightarrow (P,\mathcal {X}^+)$ such that $\tilde {\pi } \circ \tilde {i} = \mathrm {id}$. Then $\tilde {i}$ defines a Levi decomposition of $P = V\rtimes G$. Recall $\mathcal {X}^+ \subseteq \operatorname {Hom}(\mathbb {S},P_{\mathbb {R}})$. Define the bijective map

\[ \rho \colon V(\mathbb{R}) \times \mathcal{X}_G^+ \xrightarrow{\sim} \mathcal{X}^+, \quad (v,x) \mapsto \mathrm{int}(v) \circ \tilde{i}(x). \]

Under this identification, the action of $P(\mathbb {R})^+ = V(\mathbb {R}) \rtimes G(\mathbb {R})^+$ on $\mathcal {X}^+$ is given by the formula

\[ (v',h)\cdot (v,x) = (v' + hv , hx), \quad \forall (v',h) \in P(\mathbb{R})^+\text{ and }(v,x) \in \mathcal{X}^+. \]

It is proven that for the realization of $\mathcal {X}^+$ as an open (in the usual topology) semi-algebraic subset of $\mathcal {X}^\vee$, the identification $\rho$ above is semi-algebraic; see [Reference GaoGao17b, Proposition 4.3].

3.1 Review on the bi-algebraic system

Let $M = \Gamma \backslash \mathcal {X}^+$ be a connected mixed Shimura variety of Kuga type associated with $(P,\mathcal {X}^+)$. Let $\mathbf {u} \colon \mathcal {X}^+ \rightarrow M$ be the uniformization.

Recall the realization of $\mathcal {X}^+$ in § 2.5. By Proposition 2.8 $\mathcal {X}^+$ can be realized as an open (in the usual topology) semi-algebraic subset of a complex algebraic variety $\mathcal {X}^\vee$.

The following result is not hard to prove.

In view of this, we can introduce the following notation.

3.2 Statement of the Ax–Schanuel theorem

Let $M = \Gamma \backslash \mathcal {X}^+$ be a connected mixed Shimura variety of Kuga type associated with $(P,\mathcal {X}^+)$. Let $\mathbf {u} \colon \mathcal {X}^+ \rightarrow M$ be the uniformization. The typical case is when $M= \mathfrak {A}_{g,D}$ is the universal abelian variety.

Theorem 3.4 (Ax–Schanuel for mixed Shimura varieties of Kuga type) Let $\mathscr {Z}$ be a complex analytic irreducible subvariety of $\mathrm {graph}(\mathbf {u}) \subseteq \mathcal {X}^+ \times M$, and denote by $\tilde {Z}$ the image of $\mathscr {Z}$ under the natural projection $\mathcal {X}^+ \times M \rightarrow \mathcal {X}^+$. Then

\[ \dim \mathscr{Z}^{\mathrm{Zar}} - \dim \tilde{Z} \ge \dim \tilde{Z}^{\mathrm{biZar}}, \]

where $\mathscr {Z}^{\mathrm {Zar}}$ means the Zariski closure of $\mathscr {Z}$ in $M \times \mathcal {X}^+$. Moreover the equality holds if and only if $\tilde {Z}$ is a complex analytic irreducible component of $\tilde {Z}^{\mathrm {Zar}} \cap \mathbf {u}^{-1}(\mathbf {u}(\tilde {Z})^{\mathrm {Zar}})$.

We also state the weak Ax–Schanuel theorem for mixed Shimura varieties of Kuga type to make the statement more clear. It follows directly from Theorem 3.4.

Theorem 3.5 (Weak Ax–Schanuel for mixed Shimura varieties of Kuga type) Let $\tilde {Z}$ be a complex analytic irreducible subset of $\mathcal {X}^+$. Then

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

Moreover the equality holds if and only if $\tilde {Z}$ is a complex analytic irreducible component of $\tilde {Z}^{\mathrm {Zar}} \cap \mathbf {u}^{-1}(\mathbf {u}(\tilde {Z})^{\mathrm {Zar}})$.

Before moving on, we point out that the ‘Moreover’ part of Theorem 3.5 immediately follows from the main part and the intersection dimension inequality.

3.3 Geometric description of bi-algebraic subvarieties

By [Reference GaoGao17b, Theorem 8.1], bi-algebraic subvarieties of $\mathfrak {A}_{g,D}$ are precisely the weakly special subvarieties defined by Pink [Reference PinkPin05, Definition 4.1(b)]. The definition of weakly special subvarieties will be given in § 3.4. For the moment we present the geometric description of these weakly special subvarieties. Recall the projection $\pi \colon \mathfrak {A}_{g,D} \rightarrow \mathbb {A}_{g,D}$, which is proper. We have the following result.

Let us explain condition (ii) in more details. We have that $\pi (Y)$ is a closed irreducible subvariety of $\mathbb {A}_{g,D}$, and hence $\pi ^{-1}(\pi (Y)) = \mathfrak {A}_{g,D}|_{\pi (Y)}$ is an abelian scheme over $\pi (Y)$. Condition (ii) means there exists a finite covering $B' \rightarrow \pi (Y)$ such that under the base change $\mathfrak {A}' := B' \times _{\pi (Y)} \pi ^{-1}(\pi (Y))$ and the natural projection $p' \colon \mathfrak {A}' \rightarrow \pi ^{-1}(\pi (Y))$, we have $Y = p'(\mathfrak {B}+\sigma + \sigma _0)$, where $\mathfrak {B}$ is an abelian subscheme of $\mathfrak {A}' \rightarrow B'$, $\sigma$ is a torsion section of $\mathfrak {A}'\rightarrow B'$, and $\sigma _0$ is a constant section of (the largest constant abelian subscheme of) $\mathfrak {A}'\rightarrow B'$.

Proposition 3.7 is proven as a consequence of the following result.

Proposition 3.8 [Gao17a, Proposition 3.3] Let $B$ be an irreducible subvariety of $\mathbb {A}_{g,D}$. Then

\[ \begin{array}{c} \hskip-10pt\{\text{Up to taking a finite covering of }B,\text{ the translates of an abelian subscheme} \\ \hskip-10pt\text{of }\pi^{{-}1}(B) \rightarrow B\text{ by a torsion section and then by a constant section}\} \\ \qquad\qquad = \{\text{irreducible components of }\pi^{{-}1}(B) \!\cap\! F : F\text{ weakly special in } \mathfrak{A}_{g,D} \text{ with } B \subseteq \pi(F)\}. \end{array} \]

3.4 Review on weakly special subvarieties

Now let us give the definition of weakly special subvarieties following Pink. Let $(P,\mathcal {X}^+)$ and $M$ be as above Theorem 3.4.

Our formulation is slightly different from [Reference PinkPin05, Definition 4.1(b)], but it is not hard to show that they are equivalent; see [Reference GaoGao17b, § 5.1].

4.1 Fundamental set

Fix a Shimura embedding $(P,\mathcal {X}^+) \hookrightarrow (G_0,\mathcal {D}^+) \times (P_{2g,D,\mathrm {a}}, \mathcal {X}_{2g,\mathrm {a}}^+)$. Such an embedding exists by Theorem 2.6.

Let $\mathfrak {F}_{\mathrm {Sp}_{2g}}$ be a Siegel fundamental set for the action of $\mathrm {Sp}_{2g,D}(\mathbb {Z})$ on $\mathfrak {H}_g^+$. Let $\mathfrak {F}_{P_{2g,\mathrm {a}}} \subseteq \mathcal {X}_{2g,\mathrm {a}}^+$ be defined as $i_{\mathfrak {H}_g^+}( (-k,k)^{2g} \times \mathfrak {F}_{\mathrm {Sp}_{2g}} )$ for some $k \ge 1$, where

\[ i_{\mathfrak{H}_g^+} \colon V_{2g}(\mathbb{R}) \times \mathfrak{H}_g^+ \simeq \mathcal{X}_{2g,\mathrm{a}}^+ \]

is the real-algebraic map defined in (2.11). Then $\mathfrak {F}_{P_{2g,\mathrm {a}}}$ is a fundamental set for the action of $\mathbb {Z}^{2g} \rtimes \mathrm {Sp}_{2g,D}(\mathbb {Z})$ on $\mathcal {X}_{2g,\mathrm {a}}^+$.

Let $\mathfrak {F}_{\mathcal {D}^+}$ be a fundamental set for the action of $\Gamma _0$ on $\mathcal {D}^+$ as in [Reference Klingler, Ullmo and YafaevKUY16, Theorem 1.9]. Then $\mathfrak {F} = \mathfrak {F}_{\mathcal {D}^+} \times \mathfrak {F}_{P_{2g,\mathrm {a}}}$ is a fundamental set for $\mathbf {u} \colon \mathcal {D}^+ \times \mathcal {X}_{2g,\mathrm {a}}^+ \rightarrow S_0 \times \mathfrak {A}_g$. Moreover $\mathbf {u}|_{\mathfrak {F}}$ is definable in $\mathbb {R}_{\mathrm {an},\exp }$ by [Reference Peterzil and StarchenkoPS13].

It is possible to choose $\mathfrak {F}_{\mathrm {Sp}_{2g}}$, $k$, and $\mathfrak {F}_{\mathcal {D}^+}$ such that $\mathfrak {F}\cap \mathcal {X}^+$ is a fundamental set for $\mathbf {u} \colon \mathcal {X}^+ \rightarrow M$. Replace $\mathfrak {F}$ by $\mathfrak {F} \cap \mathcal {X}^+$, then $\mathbf {u}|_{\mathfrak {F}}$ is definable in $\mathbb {R}_{\mathrm {an},\exp }$. We may furthermore enlarge $\mathfrak {F}$ such that $\mathfrak {F}$, respectively $\mathfrak {F}_G:= \tilde {\pi }(\mathfrak {F})$, is open in $\mathcal {X}^+$, respectively in $\mathcal {X}_G^+$, in the usual topology.

4.2 Some results on the pure part

Let $\tilde {Z}$ be as in Theorem 4.1. In this subsection we summarize some known results on the pure part, which will be used in the proof of Theorem 5.2. Assume $\dim \tilde {\pi }(\tilde {Z}) > 0$. All distances and norms $\Vert \cdot \Vert _{\infty }$ below are as in [Reference Klingler, Ullmo and YafaevKUY16, § 5.1].

Fix $\tilde {z}_{0,G} \in \tilde {\pi }(\tilde {Z})$. All constants below depend only on $\mathcal {X}^+_G$, $\mathfrak {F}_G$, $\tilde {\pi }(\tilde {Z})$, and $\tilde {z}_{0,G}$.

For any $T > 0$, define

(4.1)\begin{equation} B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) = \{ \tilde{z}_G \in \mathcal{X}^+_G : d^{\mathrm{horz}}(\tilde{z}_G, \tilde{z}_{0,G}) < \log T \}, \end{equation}

where $d^{\mathrm {horz}}(\cdot , \cdot )$ is the $G^{\mathrm {der}}(\mathbb {R})^+$-invariant hyperbolic metric on $\mathcal {X}^+_G$.

Lemma 4.2 For each $\tilde {z}_G \in B^{\mathrm {horz}}(\tilde {z}_{0,G}, T) \subseteq \mathcal {X}^+_G$, we have

\[ \Vert \tilde{z}_G \Vert_{\infty} \le c_0 T. \]

Proof. For $\tilde {z}_G \in B^{\mathrm {horz}}(\tilde {z}_{0,G}, T) \subseteq \mathcal {X}^+_G$, there exists $g \in G(\mathbb {R})^+$ such that $g\cdot \tilde {z}_{0,G} = \tilde {z}_G$. Then by [Reference Klingler, Ullmo and YafaevKUY16, Lemma 5.4], we have

\[ \log \Vert g \Vert_{\infty} \le d^{\mathrm{horz}}(\tilde{z}_G, \tilde{z}_{0,G}) < \log T. \]

Hence we are done.

Now let us consider

(4.2)\begin{equation} L_G(T):=\{ \gamma_G \in \Gamma_G : \gamma_G \mathfrak{F}_G \cap B^{\mathrm{horz}}(\tilde{z}_{0,G}, T) \not= \emptyset \}. \end{equation}

Lemma 4.3 There exist two constants $c_1, c_2 >0$ such that

\[ H(\gamma_G) \le c_1 T^{c_2}\quad \text{ for any }\quad \gamma_G \in L_G(T). \]

All volumes below are hyperbolic volumes.

Lemma 4.4 There exist two constants $c_3 >0$ and $c_4 > 0$ such that

\[ \mathrm{vol}(\tilde{\pi}(\tilde{Z}) \cap B^{\mathrm{horz}}(\tilde{z}_{0,G},T)) \ge c_3 T^{c_4 \dim \tilde{\pi}(\tilde{Z})}, \quad \forall T \gg 0. \]

Lemma 4.5 There exists a constant $c_5 > 0$ such that

\[ \mathrm{vol}(\tilde{\pi}(\tilde{Z}) \cap \gamma_G\mathfrak{F}_G) \le c_5 \]

for all $\gamma _G \in \Gamma _G$.

6.1 Algebraic family associated with $\mathscr{B}$

Mok's idea to prove the Ax type transcendence results is to use algebraic foliations. In our situation, we wish to construct a family $\mathscr{F}'$ of varieties in $\mathcal {X}^+$ associated with $\mathscr {Z}$, such that $\mathscr{F}'$ is $\Gamma _0$-invariant for a suitable subgroup $\Gamma _0$ of $\Gamma$. This construction can be realized, for example, by using Hilbert schemes. Then $\mathbf {u}(\mathscr{F}')$ is a foliation on $M$. Next we wish to improve the algebraicity of $\mathbf {u}(\mathscr{F}')$ to make it into an algebraic subvariety. We present this process for our situation in this subsection. We point out that it is [Reference Mok, Pila and TsimermanMPT19, § 3] adapted to mixed Shimura varieties of Kuga type with some slight modifications.

Let $Y = \mathbf {u}(\tilde {Z})^{\mathrm {Zar}} = \mathrm {pr}_M(\mathscr {Z})^{\mathrm {Zar}}$. Let $\mathcal {X}^\vee$ be as in Proposition 2.8, then $\mathcal {X}^+$ is open (in the usual topology) semi-algebraic in $\mathcal {X}^\vee$. Let $\mathbf {H}$ be the Hilbert scheme of all subvarieties of $\mathcal {X}^\vee \times Y$ with the same Hilbert polynomial as $\mathscr{B}$, and let ${\boldsymbol {\mathscr{B}}} \rightarrow \mathbf {H}$ be the (modified) universal family, namely

\[ {\boldsymbol {\mathscr{B}}} = \{(\tilde{x},m,[W])\in \mathcal{X}^+ \times Y \times \mathbf{H} : (\tilde{x},m) \in W \} \hookrightarrow (\mathcal{X}^+ \times Y) \times \mathbf{H} \subseteq (\mathcal{X}^+ \times M) \times \mathbf{H}, \]

where $[W]$ means the point of $\mathbf {H}$ representing $W$. It is known that $\mathbf {H}$ is proper. Denote by

\[ \psi \colon {\boldsymbol {\mathscr{B}}} \rightarrow \mathcal{X}^{+} \times M \]

the natural projection. Then $\psi$ is proper since $\mathbf {H}$ is proper.

The action of $\Gamma$ on $\mathcal {X}^+ \times M$ induces an action of $\Gamma$ on ${\mathscr {B}}$ by

(6.1)\begin{equation} \gamma(\tilde{x}, m, [W]) = (\gamma\tilde{x}, m, [\gamma W]). \end{equation}

It is clear that $\psi$ is $\Gamma$-equivariant.

Recall that $\Delta = \mathrm {graph}(\mathcal {X}^+ \rightarrow M) \subseteq \mathcal {X}^+ \times M$. Define ${\boldsymbol {\mathscr {Z}}} \subseteq {\boldsymbol {\mathscr {B}}} \cap (\Delta \times \mathbf {H})$ to be

(6.2)\begin{equation} {\boldsymbol {\mathscr{Z}}} = \{(\tilde{\delta},[W]) \in \Delta \times \mathbf{H} : \tilde{\delta} \in \Delta \cap W, ~\dim_{\tilde{\delta}} (\Delta \cap W)\ge \dim \mathscr{Z}\}. \end{equation}

Then ${\boldsymbol {\mathscr {Z}}}$ is a closed complex analytic subset of ${\boldsymbol {\mathscr {B}}}$. Hence $\psi ({\boldsymbol {\mathscr {Z}}})$ is closed complex analytic in $\mathcal {X}^+ \times M$ since $\psi$ is proper.

Note that $\Delta$ is $\Gamma$-invariant. So ${\boldsymbol {\mathscr {Z}}}$ is $\Gamma$-invariant for the action (6.1). Hence we can define $\Gamma \backslash {\boldsymbol {\mathscr {Z}}}$, which is naturally a complex analytic variety. Moreover $\psi ({\boldsymbol {\mathscr {Z}}})$ is also $\Gamma$-invariant since $\psi$ is $\Gamma$-equivariant. We thus obtain the following commutative diagram.

(6.3)

Then $\bar {\psi }$ is proper since $\psi$ is proper.

We are now ready to prove the following result.

Proof. We claim that $\psi ({\boldsymbol {\mathscr {Z}}}) \cap (\mathfrak {F} \times M)$ is definable in $\mathbb {R}_{\mathrm {an},\exp }$. Since

\[ \psi({\boldsymbol {\mathscr{Z}}}) \cap (\mathfrak{F} \times M) = \psi({\boldsymbol {\mathscr{Z}}} \cap (\mathfrak{F} \times M \times \mathbf{H})), \]

it suffices to prove that ${\boldsymbol {\mathscr {Z}}} \cap (\mathfrak {F} \times M \times \mathbf {H})$ is definable. But

\begin{align*} {\boldsymbol {\mathscr{Z}}} \cap (\mathfrak{F} \times M \times \mathbf{H}) = \{(\tilde{x},m,[W]) \in \mathfrak{F} \times M \times \mathbf{H} : & ~ (\tilde{x},m) \in W,~\tilde{x} \in (\mathbf{u}|_{\mathfrak{F}})^{{-}1}(m), \\ & ~\dim_{(\tilde{x},m)} (\Delta \cap (\mathfrak{F} \times M) \cap W)\ge \dim \mathscr{Z} \}. \end{align*}

Hence ${\boldsymbol {\mathscr {Z}}} \cap (\mathfrak {F} \times M \times \mathbf {H})$ is definable since $\mathbf {u}|_{\mathfrak {F}}$ is definable.

Now since $\psi ({\boldsymbol {\mathscr {Z}}})$ is $\Gamma$-invariant, we have that

\[ (\mathbf{u},\mathrm{id})(\psi({\boldsymbol {\mathscr{Z}}})) = (\mathbf{u},\mathrm{id})( \psi({\boldsymbol {\mathscr{Z}}}) \cap (\mathfrak{F} \times M) ) \]

is closed complex analytic and definable in $M \times M$. Hence $(\mathbf {u},\mathrm {id})(\psi ({\boldsymbol {\mathscr {Z}}}))$ is closed algebraic by Peterzil and Starchenko's o-minimal Chow [Reference Peterzil and StarchenkoPS09, Theorem 4.5]; see also [Reference Mok, Pila and TsimermanMPT19, Theorem 2.2].

Proof. First, observe that $\psi ({\boldsymbol {\mathscr {Z}}}) \subseteq \Delta$. Hence $\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}) = (\mathbf {u},\mathrm {id})(\psi ({\boldsymbol {\mathscr {Z}}})) \subseteq M$ with $M$ viewed as a subvariety of $M \times M$ via the diagonal embedding. Next, we have $\psi ({\boldsymbol {\mathscr {Z}}}) \subseteq \psi ({\boldsymbol {\mathscr {B}}}) \subseteq \mathcal {X}^+ \times Y$ by definition of ${\boldsymbol {\mathscr {B}}}$. Hence $\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}) = (\mathbf {u},\mathrm {id})(\psi ({\boldsymbol {\mathscr {Z}}})) \subseteq \mathrm {pr}_2^{-1}(Y) \subseteq M \times M$ for the projection to the second factor $\mathrm {pr}_2 \colon M \times M \rightarrow M$. So $\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}) \subseteq M \cap \mathrm {pr}_2^{-1}(Y) = Y$ with $Y$ viewed as a subvariety of $M \times M$ via the diagonal embedding.

By assumption we have $\mathscr {Z} \subseteq \psi ({\boldsymbol {\mathscr {Z}}})$. Hence $(\mathbf {u},\mathrm {id})(\mathscr {Z}) \subseteq (\mathbf {u},\mathrm {id})(\psi ({\boldsymbol {\mathscr {Z}}})) = \bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}})$. Applying the projection $\mathrm {pr}_1 \colon M \times M \rightarrow M$ to the first factor, we get

(6.4)\begin{equation} \mathbf{u}(\tilde{Z}) = \mathbf{u}(\mathrm{pr}_{\mathcal{X}^+}(\mathscr{Z})) \subseteq \mathrm{pr}_1(\bar{\psi}(\Gamma \backslash {\boldsymbol {\mathscr{Z}}})). \end{equation}

Taking Zariski closures of both sides, we get $Y \subseteq \mathrm {pr}_1(\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}))$. But $\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}) \subseteq Y$ with $Y$ viewed as a subvariety of $M \times M$ via the diagonal embedding. We are done.

6.2 The $\mathbb {Q}$-stabilizer and first steps towards the normality

In this subsection we relate $Y$ to the $\mathbb {Q}$-stabilizer of $\mathscr{B}$ via monodromy. We point out that it is [Reference Mok, Pila and TsimermanMPT19, § 3] adapted to the universal abelian variety. However, we need to carefully distinguish the Mumford–Tate group and the monodromy group in the mixed case, since the underlying group is no longer reductive (so that we do not have a decomposition $\mathcal {X}^+ = \mathcal {X}_1^+ \times \mathcal {X}_2^+$ as in the pure case, and are thus not able to replace $\mathcal {X}^+$ by an $N(\mathbb {R})^+$-orbit).

Let ${\boldsymbol {\mathscr {Z}}}_0$ be the complex analytic irreducible component of ${\boldsymbol {\mathscr {Z}}}$, which contains $\mathscr {Z} \times [\mathscr{B}]$. By Corollary 6.3, $\bar {\psi }(\Gamma \backslash {\boldsymbol {\mathscr {Z}}}_0) = Y$ since $Y$ is irreducible. Consider the following map

(6.5)\begin{equation} \pi_1(\Gamma\backslash{\boldsymbol {\mathscr{Z}}}_0) \xrightarrow{\bar{\psi}_*} \pi_1(Y) \rightarrow \Gamma. \end{equation}

Denote by $\Gamma _0$ the image of this map, and by $N$ the identity component of the $\mathbb {Q}$-Zariski closure (in $P$) of $\Gamma _0$. Then it is clear that $\Gamma _0 {\boldsymbol {\mathscr {Z}}}_0 \subseteq {\boldsymbol {\mathscr {Z}}}_0$ for the action $\Gamma$ on ${\boldsymbol {\mathscr {B}}}$ defined by (6.1).

Denote by $\theta \colon {\boldsymbol {\mathscr {B}}} \rightarrow \mathbf {H}$ the natural projection. Let ${\boldsymbol {\mathscr {F}}} = \theta ^{-1}(\theta ({\boldsymbol {\mathscr {Z}}}_0)) = \{(\tilde {x},m,[W]) : [W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0), ~ (\tilde {x},m) \in W\}$, then ${\boldsymbol {\mathscr {F}}} \subseteq {\boldsymbol {\mathscr {B}}}$ is the family of algebraic varieties parametrized by $\theta ({\boldsymbol {\mathscr {Z}}}_0) \subseteq \mathbf {H}$, with the fiber over each $[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0)$ being $W$. Then we have

\[ \Gamma_0 {\boldsymbol {\mathscr{F}}} \subseteq {\boldsymbol {\mathscr{F}}} \]

for the action $\Gamma$ on ${\boldsymbol {\mathscr {B}}}$ defined by (6.1); see below (6.5).

For any $[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0)$ and any complex analytic irreducible component $\mathscr {Z}'$ of its fiber $({\boldsymbol {\mathscr {Z}}}_0)_{[W]}$, we have $\mathbf {u}(\mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}'))^{\mathrm {Zar}} \subseteq Y$ as $\mathscr {Z}' \subseteq \psi ({\boldsymbol {\mathscr {Z}}}_0)$. Thus $\mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}')^{\mathrm {biZar}} \subseteq \tilde {Z}^{\mathrm {biZar}}$ as $Y^{\mathrm {biZar}} = \mathbf {u}(\tilde {Z})^{\mathrm {biZar}}$. Moreover for a very general $[W]\in \theta ({\boldsymbol {\mathscr {Z}}}_0)$, i.e. outside a countable union of proper closed subvarieties, there exists a component $\mathscr {Z}'$ of $({\boldsymbol {\mathscr {Z}}}_0)_{[W]}$ such that $\mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}')^{\mathrm {biZar}} = \tilde {Z}^{\mathrm {biZar}}$.

Let

\[ \Gamma_{{\boldsymbol {\mathscr{F}}}} = \{\gamma \in \Gamma : \gamma W \subseteq W,\ \forall [W] \in \theta({\boldsymbol {\mathscr{Z}}}_0)\}. \]

Then for a very general $[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0)$ (but may be different from above), we have

(6.6)\begin{equation} \mathrm{Stab}_{\Gamma}(W) = \Gamma_{{\boldsymbol {\mathscr{F}}}}. \end{equation}

Indeed, let $\mathbf {H}_{\gamma } := \{[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0) : \gamma W \subseteq W\}$ for each $\gamma \in \Gamma$. Then for any $\gamma \not \in \Gamma _{{\boldsymbol {\mathscr {F}}}}$, we have that $\mathbf {H}_{\gamma }$ is a proper subvariety of $\theta ({\boldsymbol {\mathscr {Z}}}_0)$. Thus it suffices to take $[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0) \setminus {\bigcup} _{\gamma \in \Gamma \setminus \Gamma _{{\boldsymbol {\mathscr {F}}}}} \mathbf {H}_{\gamma }$.

The two paragraphs above together imply: for a very general $[W]\in \theta ({\boldsymbol {\mathscr {Z}}}_0)$, we have $\mathrm {Stab}_{\Gamma }(W) = \Gamma _{{\boldsymbol {\mathscr {F}}}}$ and there exists a component $\mathscr {Z}'$ of $({\boldsymbol {\mathscr {Z}}}_0)_{[W]}$ such that $\mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}')^{\mathrm {biZar}} = \tilde {Z}^{\mathrm {biZar}}$.

It is known that $\mathscr{B}$ is a fiber of ${\boldsymbol {\mathscr {F}}}$. We claim that it suffices to handle the case where $\mathscr{B}$ is a very general fiber as in the previous paragraph. Indeed, take $[W] \in \theta ({\boldsymbol {\mathscr {Z}}}_0)$ very general. Let $\mathscr {Z}'$ be an irreducible component of $({\boldsymbol {\mathscr {Z}}}_0)_{[W]}$ with $\mathbf {u}(\mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}'))^{\mathrm {biZar}} = \tilde {Z}^{\mathrm {biZar}}$. We have $\dim \mathscr {Z}' \ge \dim \mathscr {Z}$ by definition of ${\boldsymbol {\mathscr {Z}}}$, and $\dim W = \dim \mathscr{B}$ by definition of the Hilbert scheme $\mathbf {H}$. Denote by $\mathscr{B}' = (\mathscr {Z}')^{\mathrm {Zar}}$, then $\mathscr{B}' \subseteq W$. Then $\dim \mathscr{B} - \dim \mathscr {Z} \ge \dim \mathscr{B}' - \dim \mathscr {Z}'$ and $\dim \tilde {Z}^{\mathrm {biZar}} = \dim \mathrm {pr}_{\mathcal {X}^+}(\mathscr {Z}')^{\mathrm {biZar}}$. So to prove the Ax–Schanuel theorem (Theorem 4.1) for $(\mathscr{B},\mathscr {Z})$, it suffices to prove it for $(\mathscr{B}', \mathscr {Z}')$. If $\dim \mathscr{B}' < \dim W = \dim \mathscr{B}$, then Proposition 6.1 follows from induction hypotheses and we are done. If $\dim \mathscr{B}' = \dim W$, then $\mathscr{B}' = W$ is a very general fiber of ${\boldsymbol {\mathscr {F}}}$ as we desire.

By (6.6), $(\Gamma _{{\boldsymbol {\mathscr {F}}}}^{\mathrm {Zar}})^{\circ }$ is the $\mathbb {Q}$-stabilizer of $\mathscr{B}$, which is the $H$ defined in (5.1). The ‘Moreover’ part of Lemma 6.4 implies $\mathscr{B} = N(\mathbb {R})^+\tilde {z} \times \mathbf {u}(N(\mathbb {R})^+\tilde {z})$, and hence $H$ is a subgroup of $N$.

6.3 Normality of the $\mathbb {Q}$-stabilizer

The argument in this subsection is new compared to the proof of the pure Ax–Schanuel theorem.

Proof of Proposition 6.1 Recall the definable set $\Theta$ defined in (5.2)

\[ \Theta=\{p\in P(\mathbb{R}): \dim(p^{{-}1}\mathscr{B}\cap (\mathfrak{F} \times M) \cap \Delta)=\dim \mathscr{Z}\}, \]

where $\mathfrak {F}$ is the fundamental set for $\mathbf {u} \colon \mathcal {X}^+ \rightarrow M$ defined in § 4.1. By Remark 5.3, we may and do assume that every positive dimensional semi-algebraic block in $\Theta$ is contained in a left coset of $\operatorname {Stab}_{P(\mathbb {R})}(\mathscr{B})$.

We have $H(\mathbb {R})^+ \mathscr{B} \subseteq \mathscr{B}$ since $\mathscr{B}$ is algebraic in $\mathcal {X}^+ \times M$ and the action of $P(\mathbb {R})^+$ on $\mathcal {X}^+ \times M$ is semi-algebraic.

Let $V_H = V \cap H$ and $V_N = V \cap N$. Let $G_H = H/V_H$, $G_N = N/V_N$, and $G = P/V$. Then $G_H \lhd G_N \lhd G$ by Lemma 6.4 and Lemma 6.5. Hence $G_H \lhd G$ since any reductive group over $\mathbb {Q}$, in particular $G$, is an almost direct product of its center and almost simple normal subgroups.

Now we are left to prove the following two facts.

1. (i) $V_H$ is a $G$-submodule of $V$.

2. (ii) $G_H$ acts trivially on $V/V_H$.

We start with (ii). By Lemma 6.5, we have $H\lhd N$. It follows that $G_H$ acts trivially on $V_N/V_H$. Hence it suffices to prove that $G_H$ acts trivially on $V/V_N$. But Lemma 6.4 asserts that $N \lhd P$, and hence $G_N$ acts trivially on $V/V_N$. In particular, $G_H$ acts trivially on $V/V_N$. Thus we have established (ii).

Now we prove (i). The ‘Moreover’ part of Lemma 6.4 implies that $\mathscr{B} \subseteq \tilde {F} \times F$, where $\tilde {F} = N(\mathbb {R})^+\tilde {z}$ and $F = \mathbf {u}(\tilde {F})$. Consider $\tilde {X} = \mathrm {pr}_{\mathcal {X}^+}(\mathscr{B})$, then $\tilde {X} \subseteq \tilde {F}$. Applying the natural projection $\tilde {\pi } \colon (P,\mathcal {X}^+) \rightarrow (G,\mathcal {X}_G^+)$, we get $\tilde {X}_G \subseteq G_N(\mathbb {R})^+\tilde {z}_G$. Here we denote by $\tilde {X}_G = \tilde {\pi }(\tilde {X})$ and $\tilde {z}_G = \tilde {\pi }(\tilde {z})$.

For the convenience of readers, the rest of the proof of (i) is divided into three steps.

Step I. Any $\tilde {x}_G \in \mathcal {X}_G^+$ gives a morphism $\mathbb {C}^\times = \mathbb {S}(\mathbb {R}) \rightarrow G(\mathbb {R})$. If we endow $V(\mathbb {R})$ with the complex structure determined by $V(\mathbb {R}) \simeq \tilde {\pi }^{-1}(\tilde {x}_G)$, then a real subspace $V'$ of $V_{\mathbb {R}}$ is complex if and only if the following condition holds: $\tilde {x}_G(\sqrt {-1}) \cdot V' \subseteq V'$. Set for simplicity $J_{\tilde {x}_G} = \tilde {x}_G(\sqrt {-1})$. Thus under this complex structure on $V(\mathbb {R})$, we have the following.

1. – The smallest complex subspace of $V(\mathbb {R})$ containing $V_H(\mathbb {R})$ is $V_H(\mathbb {R}) + J_{\tilde {x}_G} V_H(\mathbb {R})$.

2. – The largest complex subspace of $V(\mathbb {R})$ contained in $V_H(\mathbb {R})$ is $V_H(\mathbb {R}) \cap J_{\tilde {x}_G} V_H(\mathbb {R})$.

Now each point in $\tilde {X}_G$ is of the form $g\tilde {z}_G$ for some $g \in G_N(\mathbb {R})^+$.

We claim that $J_{\tilde {x}_G} V_H(\mathbb {R})$ is independent of $\tilde {x}_G$ when $\tilde {x}_G$ varies in $G_N(\mathbb {R})^+\tilde {z}_G$. For any $g\in G_N(\mathbb {R})^+$, we have $J_{\tilde {x}_G}^{-1} g J_{\tilde {x}_G} \in G_N(\mathbb {R})^+$ since $G_N \lhd G$. Hence $g J_{\tilde {x}_G} = J_{\tilde {x}_G} g'$ for some $g' \in G_N(\mathbb {R})^+$. Note that $J_{g\tilde {x}_G} = gJ_{\tilde {x}_G} g^{-1}$. Thus

\[ J_{g\tilde{x}_G} V_H(\mathbb{R}) = gJ_{\tilde{x}_G} g^{{-}1} \cdot V_H(\mathbb{R}) \subseteq g J_{\tilde{x}_G} \cdot V_H(\mathbb{R}) = J_{\tilde{x}_G} g' \cdot V_H(\mathbb{R}) \subseteq J_{\tilde{x}_G} \cdot V_H(\mathbb{R}). \]

Here we used the fact that $V_H$ is stable under $G_N$ (since $H \lhd N$). This proves the claim.

In particular, $J_{\tilde {x}_G} V_H(\mathbb {R})$ is independent of $\tilde {x}_G$ for $\tilde {x}_G \in \tilde {X}_G$.

Step II. Next for any $\gamma _G \in \tilde {\pi }(\Theta ) \cap \Gamma _G \supseteq \{\gamma _G \in \Gamma _G : \gamma _G \mathfrak {F}_G \cap \tilde {Z}_G \not = \emptyset \}$,Footnote ¹⁰ let $\tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+$ be a complex analytic irreducible component of $\tilde {Z}|_{\gamma _G \mathfrak {F}_G} := \tilde {Z} \cap \tilde {\pi }^{-1}(\gamma _G \mathfrak {F}_G)$, and define the following definable set

\[ \Theta' = \{v \in V(\mathbb{R}) : \dim( ({-}v,\gamma_G^{{-}1}) \mathscr{B} \cap (\mathfrak{F} \times M) \cap \Delta ) = \dim \mathscr{Z} \} \subseteq V(\mathbb{R}). \]

Then $\Theta ' \cap \Gamma \supseteq \{\gamma _V \in \Gamma _V : (\gamma _V,\gamma _G)\mathfrak {F} \cap \tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+ \not = \emptyset \}$, and $(\Theta ', \gamma _G^{-1}) \subseteq \Theta$. Let $\Gamma _{V_H} = V_H(\mathbb {Q}) \cap \Gamma _V$.

Suppose $\{\gamma _V \in \Gamma _V : (\gamma _V,\gamma _G)\mathfrak {F} \cap \tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+ \not = \emptyset \}$ is not contained in a finite union of $\Gamma _{V_H}$-cosets. Identify $\mathcal {X}^+ \simeq V(\mathbb {R}) \times \mathcal {X}_G^+$, then $\mathfrak {F} = (-k,k)^{2g} \times \mathfrak {F}_G$ for some $k \ge 1$, where $g = \dim V$. Hence

\[ \{\gamma_V \in \Gamma_V : (\gamma_V,\gamma_G)\mathfrak{F} \cap \tilde{Z}|_{\gamma_G \mathfrak{F}_G}^+ \not= \emptyset\} = \{\gamma_V \in \Gamma_V : ( (\gamma_V + \gamma_G({-}k,k)^{2g}) \times \gamma_G \mathfrak{F}_G ) \cap \tilde{Z}|_{\gamma_G \mathfrak{F}_G}^+ \not= \emptyset\}. \]

Let $p_V \colon \mathcal {X}^+ \simeq V(\mathbb {R}) \times \mathcal {X}^+_G \rightarrow V(\mathbb {R})$, then $p_V((\gamma _V + \gamma _G(-k,k)^{2g}) \times \gamma _G \mathfrak {F}_G)$ is contained in a Euclidean ball of radius at most $kH(\gamma _G)$. But $\tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+$ is connected by choice. Thus if the set above is not contained in a finite union of $\Gamma _{V_H}$-cosets, then we get for free that it contains $\ge T$ elements in $\Gamma _V \setminus \Gamma _{V_H}$ of height at most $H(\gamma _G)T$ (for all $T\gg 0$).Footnote ¹¹ Hence by Pila and Wilkie [Reference PilaPil11, Theorem 3.6], there exist two constants $c,\epsilon > 0$ with the following property: for each $T \gg 0$, $\Theta '$ contains a semi-algebraic block $B'$,which is not in any coset of $V_H(\mathbb {R})$ and which contains $\ge c T^{\epsilon }$ elements in $\Gamma _V$ outside $\Gamma _{V_H}$ of height at most $T$. Recall our assumption that every positive dimensional semi-algebraic block in $\Theta$ is contained in a left coset of $\operatorname {Stab}_{P(\mathbb {R})}(\mathscr{B})$. In particular, $(B', \gamma _G^{-1}) \subseteq (\gamma _V' , \gamma _G^{-1}) \cdot \operatorname {Stab}_{P(\mathbb {R})}(\mathscr{B})$ for some $\gamma _V' \in B' \cap \Gamma _V$. Hence $(-\gamma _V' + \gamma _G B', 1) \subseteq \operatorname {Stab}_{P(\mathbb {R})}(\mathscr{B})$; the argument is similar to the end of Proposition 5.1 (right above Remark 5.3). But then

\[ ( (-\gamma_V' + \gamma_G B') \cap \Gamma_V, 1 )\subseteq \operatorname{Stab}_{P(\mathbb{R})}(\mathscr{B}) \cap \Gamma \subseteq H(\mathbb{Q}). \]

By letting $T \rightarrow \infty$ and varying $B'$ accordingly, we see that this inclusion cannot hold since each $B' \subseteq V(\mathbb {R})$ thus obtained is not contained in any coset of $V_H(\mathbb {R})$.

Thus $\{\gamma _V \in \Gamma _V : (\gamma _V,\gamma _G)\mathfrak {F} \cap \tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+ \not = \emptyset \}$ is contained in a finite union of $\Gamma _{V_H}$-cosets. Hence for the map $\bar {\mathbf {u}} \colon \mathcal {X}^+ \rightarrow \Gamma _V \backslash \mathcal {X}^+$,Footnote ¹² we have that $\bar {\mathbf {u}}(\tilde {Z}|_{\gamma _G \mathfrak {F}_G}^+)$ is closed in $(\Gamma _V \backslash \mathcal {X}^+)|_{\gamma _G \mathfrak {F}_G}$ in the usual topology.

In the following paragraph we consider the usual topology. As $\gamma _G\mathfrak {F}_G$ is an open subset of $\mathcal {X}^+_G$ for each $\gamma _G \in \Gamma _G$, each $\tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+$ is an open subset of $\tilde {Z}$. In particular, there exists a $\tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+$ as above such that $\dim \tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+ = \dim \tilde {Z}$. Consider the closure $W$ of $\bar {\mathbf {u}}(\tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+)$ in $\Gamma _V \backslash \mathcal {X}^+$. By the last paragraph, we have $W^{\circ } \subseteq \bar {\mathbf {u}}(\tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+) \subseteq W$ where $W^{\circ }$ is the interior of $W$. Now let $\tilde {Z}'$ be the complex analytic irreducible component of $\bar {\mathbf {u}}^{-1}(W)$ that contains $\tilde {Z}|_{\gamma _G\mathfrak {F}_G}^+$. Then we have $\tilde {Z}' \subseteq \tilde {Z}$ and $\bar {\mathbf {u}}(\tilde {Z}') = W$ is closed in $\Gamma _V\backslash \mathcal {X}^+$.

Set $\mathscr {Z}' := \mathrm {graph}(\tilde {Z}' \rightarrow \mathbf {u}(\tilde {Z}'))$. Then $\dim \mathscr {Z}' = \dim \mathscr {Z}$ because $\dim \tilde {Z}' = \dim \tilde {Z}$. By analytic continuation, we then have $(\mathscr {Z}')^{\mathrm {Zar}} = \mathscr {Z}^{\mathrm {Zar}} = \mathscr{B}$ and $(\tilde {Z}')^{\mathrm {Zar}} = \tilde {Z}^{\mathrm {Zar}} = \tilde {X}$.

Step III. Now let

\[ \Gamma' = \text{Im} ( \pi_1(\bar{\mathbf{u}}(\tilde{Z}')) \rightarrow \pi_1(\Gamma_V \backslash \mathcal{X}^+) = \Gamma_V ) \subseteq \Gamma_V. \]

Then $\Gamma '$ stabilizes $\tilde {Z}'$ for the action of $\Gamma _V$ on $\mathcal {X}^+$. Recall that $P(\mathbb {R})^+$ acts on $\mathcal {X}^+ \times M$ via its action on the first factor. So $\Gamma '$ stabilizes $\mathscr {Z}'$. Hence $(\Gamma ')^{\mathrm {Zar}}(\mathbb {R}) \subset V(\mathbb {R})$ stabilizes $\mathscr{B} = (\mathscr {Z}')^{\mathrm {Zar}}$. On the other hand, $H$ is defined to be the $\mathbb {Q}$-stabilizer of $\mathscr{B}$. So $(\Gamma ')^{\mathrm {Zar}} \subseteq H \cap V = V_H$.

Let us take a closer look at this. The identification $\pi _1(\Gamma _V \backslash \mathcal {X}^+) = \Gamma _V$ is realized via (2.10)

\[ i_{\mathcal{X}_G^+}^{{-}1} \colon \mathcal{X}^+{\simeq} V(\mathbb{R}) \times \mathcal{X}_G^+ \]

and hence $\Gamma _V \backslash \mathcal {X}^+ \simeq (\Gamma _V\backslash V(\mathbb {R})) \times \mathcal {X}_G^+$. We have $\tilde {Z}' \subseteq \tilde {X} = (\tilde {Z}')^{\mathrm {Zar}} \subseteq \tilde {F} = N(\mathbb {R})^+\tilde {z}$. Since $V_H \lhd N$, we can take the quotient of $\tilde {F} = N(\mathbb {R})^+\tilde {z}$ by $V_H(\mathbb {R})$ and get a real manifold. Call this quotient $q$. Set $\Gamma _{V_N} = \Gamma _V \cap N(\mathbb {Q})$ and $\Gamma _{V_H} = \Gamma _V \cap H(\mathbb {Q})$. Then we obtain the following commutative diagram.

Here all the isomorphisms in the diagram are compatible with the $i_{\mathcal {X}_G^+}^{-1}$ above. For now on, by abuse of notation we no longer write $i_{\mathcal {X}_G^+}^{-1}(\cdot )$.

Since $\Gamma ' \subseteq V_H(\mathbb {Q})$, we have that $[q](\bar {\mathbf {u}}(\tilde {Z}'))$ is contained in $\{t\} \times \tilde {F}_G$ for some $t \in (\Gamma _{V_N}/\Gamma _{V_H})\backslash (V_N/V_H)(\mathbb {R})$. But then $q(\tilde {Z}') \subseteq \{v'\} \times \tilde {F}_G$ for some $v' \in (V_N/V_H)(\mathbb {R})$. Thus there exists $v \in V(\mathbb {R})$ such that

(6.7)\begin{equation} \tilde{Z}' - (\{v\} \times \tilde{Z}'_G) \subseteq V_H(\mathbb{R}) \times \tilde{F}_G. \end{equation}

But $i_{\mathcal {X}_G^+}(\{v\} \times \tilde {Z}'_G)$ is complex analytic in $\mathcal {X}^+$ by property (ii) of $i_{\mathcal {X}_G^+}$ below (2.10). So the left-hand side of (6.7) is complex analytic.

We have proven in Step I that over each $\tilde {x}_G \in \tilde {F}_G$, the largest complex subspace of $V(\mathbb {R})$ contained in $V_H(\mathbb {R})$ is $V_H(\mathbb {R}) \cap J_{\tilde {x}_G} V_H(\mathbb {R})$ and $J_{\tilde {x}_G} V_H(\mathbb {R})$ is independent of the choice of $\tilde {x}_G$. Hence (6.7) implies

\[ \tilde{Z}' - (\{v\} \times \tilde{Z}'_G) \subseteq \big(V_H(\mathbb{R}) \cap J_{\tilde{x}_G} V_H(\mathbb{R})\big) \times \tilde{F}_G \]

since the left-hand side is complex analytic. Taking the Zariski closures, we getFootnote ¹³

\[ \tilde{X} - (\{v\} \times \tilde{X}_G) \subseteq \big(V_H(\mathbb{R}) \cap J_{\tilde{x}_G} V_H(\mathbb{R})\big) \times \tilde{F}_G. \]

But $V_H(\mathbb {R})$ stabilizes $\tilde {X}$, so $V_H(\mathbb {R}) = V_H(\mathbb {R}) \bigcap J_{\tilde {x}_G} V_H(\mathbb {R})$. But then $V_H(\mathbb {R})$ is complex for the complex structure of $V(\mathbb {R}) \simeq \mathcal {X}^+_{\tilde {x}_G}$ for any $\tilde {x}_G \in \tilde {F}_G$. Since $\tilde {Z}$ is assumed to be Hodge generic in $(P,\mathcal {X}^+)$, we may take $\tilde {x}_G$ such that $\mathrm {MT}(\tilde {x}_G) = G$. Then we see that $V_H$ is a $G$-module.