2.1 Weak Hodge loci of variations of mixed Hodge structure

Let S be a smooth quasi-projective algebraic variety and let ${\mathcal E}=\{{\mathbb E}_{\mathbb {Z}},W_{\bullet }, {\mathcal F}^{\bullet },\nabla \}$ be an admissible graded-polarizable variation of mixed Hodge structure over S.Footnote ² Let $({\mathcal G},{\mathcal D}_{{\mathcal G}})$ be the generic mixed Hodge datum, where ${\mathcal G}$ is the generic mixed Mumford-Tate group and ${\mathcal D}_{{\mathcal G}}$ is the mixed Mumford-Tate period domain. Let $({\mathcal H},{\mathcal D}_{{\mathcal H}})$ be the monodromy datum of the variation: here ${\mathcal H}$ is the geometric monodromy group of ${\mathcal E}$ , i.e., the neutral component of the $\mathbb {Q}$ -Zariski closure of the monodromy group associated to ${\mathbb E}_{\mathbb {Z}}$ . Then ${\mathcal G}$ and ${\mathcal H}$ are algebraic groups over $\mathbb Q$ and by a well-known theorem of Deligne and André [Reference AndréAnd92], ${\mathcal H}$ is a normal subgroup of ${\mathcal G}^{der}$ . Associated to this data, we have a holomorphic period map

$$\begin{align*}\Phi:S\rightarrow {\mathit \Gamma}\backslash {\mathcal D}_{{\mathcal G}}~,\end{align*}$$

where ${\mathit \Gamma }\subset {\mathcal G}(\mathbb Q)$ is an arithmetic sub-group containing the image of the monodromy of ${\mathcal E}$ . Let $(\overline {{\mathcal G}},\overline {{\mathcal D}}_{\overline {{\mathcal G}}})$ denote the quotient mixed Hodge datum where $\overline {{\mathcal G}}:={\mathcal G}/{\mathcal H}$ and $\overline {{\mathcal D}}_{{\mathcal G}/{\mathcal H}}$ is the mixed period domain associated to $\overline {{\mathcal G}}$ . Then the monodromy period domain ${\mathcal D}_{{\mathcal H}}$ is isomorphic to a fiber of the projection ${\mathcal D}_{{\mathcal G}}\rightarrow \overline {{\mathcal D}}_{\overline {{\mathcal G}}}$ and the period map $\Phi $ factors, up to a finite étale cover, as $\Phi =\iota \circ \Phi _{{\mathcal H}}$ where:

$$\begin{align*}S\xrightarrow{\Phi_{{\mathcal H}}} {\mathit \Gamma}_{{\mathcal H}}\backslash {\mathcal D}_{{\mathcal H}}{\hookrightarrow}{\iota} {\mathit \Gamma}\backslash {\mathcal D}_{{\mathcal G}}\xrightarrow{\pi} \overline{{\mathit \Gamma}} \backslash \overline{{\mathcal D}}_{\overline{{\mathcal G}}}~.\end{align*}$$

We refer to $\Phi _{{\mathcal H}}$ as the monodromy period map.

Let $Y\subseteq S$ be a closed irreducible subvariety of S. Then we have similarly a generic Hodge datum $({\mathcal G}_Y,{\mathcal D}_{{\mathcal G}_Y})$ , monodromy datum $({\mathcal H}_Y,{\mathcal D}^0_{Y})$ defined with respect to the normalization $Y^\nu $ of Y, and quotient mixed datum $(\overline {{\mathcal G}_Y},\overline {{\mathcal D}}_{\overline {{\mathcal G}_Y}})$ . The restriction $\Phi |_Y$ of the period map $\Phi $ to Y factors similarly, up to a finite cover, as the composition of the first two arrows in

$$\begin{align*}Y\rightarrow {\mathit \Gamma}_{{\mathcal H}_Y}\backslash {\mathcal D}^0_Y\hookrightarrow {\mathit \Gamma}_Y\backslash {\mathcal D}_{{\mathcal G}_Y}\xrightarrow{\pi_Y} \overline{{\mathit \Gamma}}_Y \backslash \overline{{\mathcal D}}_{\overline{{\mathcal G}_Y}}~.\end{align*}$$

The following result is proven in [Reference Klingler and OtwinowskaKO21] in the pure case, the proof generalizes easily to the mixed case:

Definition 2.2. Let Y be a weakly special subvariety. We say that Y is monodromically atypical if

$$\begin{align*}\mathrm{codim}_{\Phi(S)}\left(\Phi(Y)\right)<\mathrm{codim}_{{\mathcal D}_{{\mathcal H}}}{\mathcal D}^0_{Y}.\end{align*}$$

Our main tool in this paper is the following result established by Baldi and Urbanik [Reference Baldi and UrbanikBU24, Theorem 7.1]:

2.2 Brief review of normal functions

An admissible normal function on S with values in a polarizable VHS ${\mathcal V}$ (of negative weight) is an extension of admissibleFootnote ⁴ variations of MHS on S of the form

(1) $$ \begin{align} 0\to {\mathcal V}\to {\mathcal E}\to \mathbb{Z}(0)\to 0\,,\end{align} $$

where $\mathbb {Z}(0)$ is the constant VHS of rank 1 and weight 0. They constitute a group $\mathrm {ANF}_S({\mathcal V})$ under Baer sum.

They may be viewed as functions in the following way. There exist local lifts of $1\in \mathbb {Z}(0)$ : to a holomorphic section $e_F$ of ${\mathcal F}^0{\mathcal E}$ ; and to a section $e_{\mathbb {Z}}$ of ${\mathbb E}_{\mathbb {Z}}$ . Their difference $e_{\mathbb {Z}}-e_F$ lifts to a local section of ${\mathcal V}$ , with ambiguity (arising from choices of $e_F,e_{\mathbb {Z}}$ ) in ${\mathcal F}^0{\mathcal V}+{\mathbb V}_{\mathbb {Z}}$ . Passing to the generalized intermediate Jacobian bundle ${\mathcal J}({\mathcal V})={\mathcal V}/({\mathcal F}^0{\mathcal V}+{\mathbb V}_{\mathbb {Z}})$ , this construction produces a well-defined global holomorphic section $\nu \colon S\to {\mathcal J}({\mathcal V})$ . (The group structure is given by adding sections.) We write $\nu \in \mathrm {ANF}_S({\mathcal V})$ and ${\mathcal E}_{\nu }$ for the corresponding AVMHS.

By Griffiths transversality for ${\mathcal E}_{\nu }$ , a local lift $\tilde {\nu }$ of $\nu $ (to a section of ${\mathcal V}$ itself) has derivative $\nabla \tilde {\nu }$ a local section of $ {\mathcal F}^{-1}{\mathcal V}\otimes \Omega ^1_S$ (a property of $\nu $ known as quasi-horizontality). We can also analytically continue $\tilde {\nu }$ along a loop $\gamma \in \pi _1(S,s_0)$ and consider its monodromy $(\gamma -I)\tilde {\nu }\in V_{\mathbb C}$ (the fiber of ${\mathcal V}$ over $s_0$ ) whose image in $V_{\mathbb C}/(T_{\gamma }-I)V_{\mathbb C}$ is independent of the lift $\tilde {\nu }$ (with $T_{\gamma }$ given by the monodromy of the local system ${\mathbb V}_{\mathbb C}$ ). These notions of derivative and monodromy of $\nu $ are formalized by the infinitesimal and topological invariants $\delta \nu $ and $[\nu ]$ , defined as follows.

Consider the exact sequence of complexes of sheaves

$$\begin{align*}0\to {\mathcal F}^0\mathscr{C}^{\bullet}\oplus {\mathbb V}_{\mathbb{Z}}\to \mathscr{C}^{\bullet}\to \frac{\mathscr{C}^{\bullet}}{{\mathcal F}^0\mathscr{C}^{\bullet}\oplus {\mathbb V}_{\mathbb{Z}}}\to 0\end{align*}$$

where $\mathscr {C}^{\bullet }:=({\mathcal V}\otimes \Omega ^{\bullet }_S,\nabla )$ , ${\mathcal F}^0\mathscr {C}^{\bullet }:=({\mathcal F}^{-\bullet }{\mathcal V}\otimes \Omega ^{\bullet }_S,\nabla )$ , and ${\mathbb V}_{\mathbb {Z}}$ is placed in degree zero. Write $\mathrm {NF}_S({\mathcal V}):={\mathbb H}^0(S,\tfrac {\mathscr {C}^{\bullet }}{{\mathcal F}^0\mathscr {C}^{\bullet }\oplus {\mathbb V}_{\mathbb {Z}}})$ for the group of holomorphic quasi-horizontal sections of ${\mathcal J}({\mathcal V})$ .

Definition 2.6. (i) In the composition

(2)

$\delta \nu $ is the infinitesimal invariant of $\nu \in \mathrm {ANF}_S({\mathcal V})$ , and $[\nu ]$ the topological invariant of $\nu $ .

(ii) $\nu $ is said to be flat if $\delta \nu =0$ , and monodromically trivial (resp. monodromically torsion) if $[\nu ]$ is trivial (resp. torsion).

Sketch.

Working $\otimes \mathbb {Q}$ , if $[\nu ]=0$ then ${\mathbb E}_{\nu }={\mathbb V}\oplus \mathbb {Q}$ . If also $H^0(S,{\mathbb V})=\{0\}$ , then ${H^0(S,{\mathbb E}_{\nu })=\mathbb {Q}}$ and the Theorem of the Fixed Part [Reference Steenbrink and ZuckerSZ85, (4.19)] says the latter underlies a (constant) sub-AVMHS of ${\mathcal E}_{\nu }$ . Since it is of rank one, it can only be of type $(0,0)$ , splitting (1) and rendering $\nu =0$ . Thus $[\cdot ]$ is injective, whence (c) $\implies $ (a), when ${\mathbb V}$ has trivial fixed part.

The additional condition required for (b) $\implies $ (a) is to make the edge homomorphism injective. More precisely, the kernel of $\varepsilon $ in Remark 2.7 is $E_{\infty }^{1,0}=E_2^{1,0}=H^1(S,{\mathcal H}^0_{\nabla }{\mathcal F}^0\mathscr {C}^{\bullet })$ , and injectivity of $\nabla $ on ${\mathcal F}^0{\mathcal V}$ makes ${\mathcal H}^0_{\nabla }{\mathcal F}^0\mathscr {C}^{\bullet }$ vanish. One also needs that the part of the connecting homomorphism in (2) mapping $\mathrm {NF}_S({\mathcal V})\to {\mathbb H}^1({\mathcal F}^0\mathscr {C}^{\bullet })$ is injective, but this follows from ${\mathbb H}^0(\mathscr {C}^{\bullet }/{\mathbb V}_{\mathbb {Q}})\cong H^0(S,{\mathbb V}_{\mathbb {Q}})\otimes \mathbb {C}/\mathbb {Q}=\{0\}$ by the assumed triviality of the fixed part.

For the equivalence of (c) and (d), tracing through the definition shows that $[\nu ]$ is the image of $1\in H^0(S,\mathbb {Z}(0))$ under the connecting homomorphism for the extension

(3) $$ \begin{align} 0\to {\mathbb V}_{\mathbb{Z}}\to {\mathbb E}_{\nu,\mathbb{Z}}\to \mathbb{Z}(0)\to 0 \end{align} $$

of local systems. Finally, ${\mathcal H}$ is an extension of $\mathrm {Gr}\,{\mathcal H}$ by its unipotent radical U; (e) says that $U=\{1\}$ , and (d) is the same as ${\mathcal H}=\mathrm {Gr}\,{\mathcal H}$ , so (d) $\iff $ (e).

Remark 2.12. In the geometric context, normal functions graph Abel-Jacobi mappings in families of cycles and higher cycles [Reference BlochBlo86]. Given a smooth projective family ${\mathcal X}\to S$ , with ${\mathcal V}\subset {\mathcal H}^n({\mathcal X}/S)(m)$ and $r=2m-n-1\geq 0$ , a primitiveFootnote ⁵ cycle $\mathfrak {Z}\in \mathrm {CH}^m({\mathcal X},r)\cong _{\otimes \mathbb {Q}}\mathrm {Gr}_{\gamma }^m K_{r}^{\text {alg}}({\mathcal X})$ produces a normal function $\nu _{\mathfrak {Z}}\in \mathrm {ANF}_S({\mathcal V})$ via

$$\begin{align*}\nu(s):=\mathrm{AJ}_{X_s}(\mathfrak{Z}|_{X_s})\in {\mathcal J}(V_s)=\frac{V_{s,\mathbb{C}}}{F^0V_{s,\mathbb{C}}+{\mathbb V}_{\mathbb{Z}}}\cong \mathrm{Ext}^1_{\text{MHS}}(\mathbb{Z}(0),V_s)\,.\end{align*}$$

When $r=0$ , ${\mathcal J}(V_s)$ is a compact complex torus known as an intermediate Jacobian; it is usuallyFootnote ⁶ not algebraic if the level of $V_s$ is larger than one. When $r>0$ , it is a noncompact complex torus. We refer to [Reference Kerr and PearlsteinKP11, Section 2] for more details.

By the main theorem of [Reference Brosnan and PearlsteinBP13], the zero locus of $\nu $ is an algebraic subvariety of S. Since the zero-locus of $m\nu $ is the m-torsion locus of $\nu $ , we get that the torsion locus with torsion order bounded (say, by N) is also algebraic. Thus the torsion locus of $\nu $ is a countable union of algebraic subvarieties. In the remainder of this paper we aim to improve this statement.

3.1 The main theorem

Let $\mathcal {M}$ be a smooth quasi-projective complex variety and let ${\mathcal V}=\{{\mathbb V}_{\mathbb Z},{\mathcal F}^\bullet ,\nabla \}$ be a non-trivial polarized $\mathbb Z$ -VHS of negative weight on $\mathcal {M}$ . Let G denote the generic Mumford-Tate group of ${\mathcal V}$ and $\phi \colon \mathcal {M}_{\mathbb {C}}^{\text {an}}\to X=\Gamma \backslash D=\Gamma \backslash G_{\mathbb {R}}/K$ be the induced period map. By construction the image $M:=\phi ({\mathcal M})$ is Hodge-generic in X; while X is in general not an algebraic variety, we know that M is one by [Reference Bakker, Brunebarbe and TsimermanBBT23a].

Let $(H,{\mathcal D}_H)$ be the monodromy data of ${\mathcal V}$ and let $\phi _H:{\mathcal M}\rightarrow X_H:=\Gamma _H\backslash {\mathcal D}_{H}$ be the monodromy period map. We assume that $X_H$ is a product of its simple factors, which we can always ensure by taking a finite étale cover of ${\mathcal M}$ .

Let ${\mathcal V}_0$ be the largest isotrivial summand of ${\mathcal V}$ which we will assume to be trivial, up to taking a finite étale cover of ${\mathcal M}$ . Then the variation ${\mathcal V}$ admits a decomposition:

(4) $$ \begin{align} {\mathcal V}={\mathcal V}_0\oplus \bigoplus_{\ell\in I} {\mathcal V}_\ell \end{align} $$

where for $\ell \in I$ , ${\mathcal V}_\ell $ are $\mathbb Q$ -simple non-isotrivial variations. Let $\nu \in \mathrm {ANF}_{\mathcal {M}}(\mathcal {V})$ be a non-torsion admissible normal function. Then $\nu $ admits a decomposition, after tensoring with $\mathbb Q$ , of the form $\nu _{\mathbb Q}=(\nu _\ell )_\ell $ which reflects the decomposition in (4). Let V be the $\mathbb Q$ -vector space underlying a fiber of ${\mathcal V}$ , which is also equal to $H^0(\widetilde {{\mathcal M}},\iota ^*{\mathbb V}_{\mathbb Q})$ , where $\iota :\widetilde {{\mathcal M}}\rightarrow {\mathcal M}$ is the universal cover.

We can now state the main theorem of this paper.

3.2 Proof of Theorem 3.2

The goal of this section is to prove Theorem 3.2. We will regard $\nu $ as a variation of mixed Hodge structure $\mathcal {E}_{\nu }\in \mathrm {Ext}^1_{\text {AVMHS}(\mathcal {M})}(\mathbb {Z}(0),{\mathcal V})$ . Since $\nu $ is generic, the geometric monodromy group ${\mathcal H}$ of $\mathcal {E}_{\nu }$ is an extension of H by V (by Corollary 2.10(i)); and thus the Mumford-Tate group ${\mathcal G}$ of ${\mathcal E}_{\nu }$ is an extension of G by V.

Consider the period map for ${\mathcal E}_\nu $ :

$$\begin{align*}\Phi\colon \mathcal{M}\to {\mathcal X}=\tilde{{\mathit \Gamma}}\backslash \mathcal{D}=\tilde{{\mathit \Gamma}}\backslash\mathcal{G}(\mathbb R)/\mathcal{K}~,\end{align*}$$

where the arithmetic group $\tilde {\Gamma }$ is an extension of $\Gamma $ by the translation group $V_{\mathbb {Z}}$ . Consider also the monodromy period map

$$\begin{align*}\Phi_{\mathcal H}\colon \mathcal{M}\to {\mathcal X}_{\mathcal{H}}={\mathit \Gamma}_{\mathcal H}\backslash \mathcal{D}_{\mathcal H}={\mathit \Gamma}_{\mathcal H}\backslash\mathcal{H}(\mathbb R)/\mathcal{K}_{{\mathcal H}}~.\end{align*}$$

The $\mathbb {Q}$ -vector space V admits an H-stable decomposition into irreducible factors ${V=V_0\oplus \bigoplus _{\ell \in I} V_\ell} $ which reflects (4), and where $V_0=V^{H}$ .

There is a natural projection $\pi \colon {\mathcal X}\to X$ with fibers generalized intermediate Jacobians $\{{\mathcal J}_x\}$ , as well as natural sections corresponding to torsion points in $V_{\mathbb {C}}/V_{\mathbb {Z}}$ preserved by $\Gamma $ . This projection is compatible with the decomposition from (4), since the intermediate Jacobian $\{{\mathcal J}_x\}$ decomposes as the product of the intermediate Jacobians associated to each summand in (4). This results in a decomposition $\pi \colon {\mathcal D}={\mathcal D}_0\times _{D} \prod _{\ell \in I}{\mathcal D}_\ell \rightarrow D$ and $\pi \colon {\mathcal X}={\mathcal X}_0\times _X\prod _{\ell \in I}{\mathcal X}_\ell \rightarrow X$ . Here the fiber of ${\mathcal D}_{\ell }\to D$ over $\varphi \in D$ is $V_{\ell ,\mathbb {C}}/F^0_{\varphi }V_{\ell ,\mathbb {C}}$ , and that of ${\mathcal X}_{\ell }\to X$ is the generalized Jacobian ${\mathcal J}(V_{\ell })$ . Since the period map $\phi _H$ factors as product of simple factors, the monodromy period map $\Phi _{\mathcal H}$ similarly factors as:

$$\begin{align*}\Phi_{\mathcal H}:\mathcal{M}\rightarrow {\mathcal J}(V_0)\times \prod_{\ell\in I}{\mathcal X}_\ell~,\end{align*}$$

where ${\mathcal J}(V_0)$ is the generalized Jacobian associated to the isotrivial variation ${\mathcal V}_0$ .

Let $Y\subset \mathcal {M}$ be an irreducible subvariety, and denote by $(\mathcal {G}_{Y},\mathcal {D}_{Y})$ the mixed Hodge data associated to $\mathcal {E}_{\nu }|_{Y}$ . Further, we write $\mathcal {H}_Y\trianglelefteq \mathcal {G}_Y^{\text {der}}$ and $\mathcal {D}_Y^0\subseteq \mathcal {D}_Y$ for the geometric monodromy group and the monodromy period domain, which is the orbit of a basepointFootnote ⁷ in $\mathcal {D}_Y$ by $\mathcal {H}_Y(\mathbb {R})^+\mathcal {H}^{\text {un}}_Y(\mathbb {C})$ .

Recall that Y is monodromically atypical for $\Phi $ if Y is weakly special and $\text {codim}_{\Phi _{\mathcal H}(\mathcal {M})}(\Phi _{\mathcal H}(Y))<\text {codim}_{\mathcal {D}_{\mathcal H}}(\mathcal {D}^0_Y)$ .

Now let $\{({\mathcal G}_i,{\mathcal D}_i,{\mathcal N}_i)\}_{i\in \Sigma }$ be the finite set of triples guaranteed for ${\mathcal E}_{\nu }$ by Theorem 2.3. Take $Y\subset {\mathcal M}$ to be a maximal monodromically atypical subvariety, i.e., monodromically atypical and not contained in a larger monodromically atypical subvariety. Then up to the action by ${\mathit \Gamma }$ ,Footnote ⁸ there exist $i\in \Sigma $ and $y\in {\mathcal D}_i$ such that ${\mathcal H}_Y={\mathcal N}_i$ and $\mathcal {D}_Y^0=\mathcal {N}_i(\mathbb R)\mathcal {N}_{i}^{u}(\mathbb C)\cdot y$ .

We consider the two possibilities for this triple:

1. (1) $G_i:=\mathcal {G}_i/\mathcal {G}_i\cap V$ has $G_i^{der}< G^{der}$ a proper subgroup for i in a subset $\Sigma _0$ of $\Sigma $ .

2. (2) $G_i^{der}=G^{der}$ .

We now turn our attention to case $(2)$ . We have the following lemma.

Proof. Since ${N}_i:={\mathcal N}_i/({\mathcal N}_i\cap V)$ is normal in $G^{der}$ and equals $H_Y$ , and $H_Y=H$ by factorwise positivity, we have $N_i=H_Y=H$ . Since $\mathcal {N}_i\cap V$ is $N_i$ -stable, it is of the formFootnote ⁹

$$\begin{align*}V^{\prime}_{0}\oplus\bigoplus_{\ell\in I'}V_\ell\subsetneq V_{0}\oplus\bigoplus_{\ell\in I}V_\ell~,\end{align*}$$

for a subset $I'\subset I$ and a $\mathbb Q$ sub-Hodge structure $V^{\prime }_0\subset V_0$ . Hence we conclude that ${\mathcal H}_Y$ is, up to an irrelevant torus part, an extension of H by $V^{\prime }_{0}\oplus \bigoplus _{\ell \in I'}V_\ell $ .

The group $\mathcal {G}_i$ is on the other hand an extension of G by $W_0\oplus \bigoplus _{\ell \in I_1}V_\ell $ for some $I_1\subseteq I,\, W_0\subset V_0$ a sub Hodge structure. Since ${\mathcal H}_Y={\mathcal N}_i$ is a subgroup of $\mathcal {G}_i^{der}$ , $I'\subseteq I_1$ and $V^{\prime }_0\subseteq W_0$ . Let $U=W_0/V^{\prime }_0\oplus \bigoplus _{\ell \in I_1\backslash I'}V_\ell $ . Then U and ${\mathcal H}_Y$ are both normal subgroups of $\mathcal {G}_i$ with $\mathcal {G}_i^{der}/{\mathcal H}_Y\simeq U$ , whence $\mathcal {G}_i^{der}$ is an almost direct product of ${\mathcal H}_Y$ and U. This implies that $H_Y=H$ acts trivially on U, and hence that $U\subset V_0$ as a sub-Hodge structure. Thus $I_1=I'$ .

Let ${\mathcal D}_{W_0}\rightarrow D$ , ${\mathcal D}_{V_0/W_0}\overset {\pi _0}{\to } D$ , and ${\mathcal D}_{\ell }\overset {\pi _{\ell }}{\to } D$ be the pre-intermediate-JacobianFootnote ¹⁰ fibrations associated to $W_0$ , $V_0/W_0$ , and $V_{\ell }$ . Then there are torsion sections $\sigma _i^{(0)}$ of $\pi _0$ , and $\sigma _i^{(\ell )}$ of $\pi _{\ell }$ for $\ell \in J:=I_1^c$ , such that

$$\begin{align*}{\mathcal D}_i={\mathcal D}_{W_0}\times_D \sigma_i^{(0)} (D_{V_0/W_0})\times_D \prod_{\ell \in I_1}{\mathcal D}_\ell\times_D \prod_{\ell\in J}\sigma_i^{(\ell)}(D_\ell)\,.\end{align*}$$

Writing ${\mathcal X}_i$ for the image of ${\mathcal D}_i$ in ${\mathcal X}$ , $\Phi ^{-1}({\mathcal X}_i)\subset {\mathcal M}$ is a union of (finitely many) components of the torsion locus of the projection $\nu _i=\nu _{V_0/W_0}+\sum _{\ell \in J}\nu _\ell $ of our normal function $\nu $ . By the finiteness result of Theorem 2.3, only finitely many such projections $\nu _i$ and sections $\sigma _i=\{\sigma _i^{(0)},\{\sigma _i^{(\ell )}\}\}$ appear (up to translation by $V_{\mathbb {Z}}$ ). In particular, we can bound the torsion order of $\nu _i$ uniformly in i, say by m.

As for ${\mathcal D}_Y^{0}$ , it is determined up to the choice of a point $y\in {\mathcal J}(U)$ , the intermediate Jacobian of the constant Hodge structure U. As we will be primarily interested in the situation where y is torsion, we make that assumption from now on. In other words, we suppose that Y is special. Notice that Theorem 2.3 does not say anything about the boundedness of the torsion orders of such points, but we are going to prove that this torsion order is indeed bounded.

Notice first that if either J is nonempty or $V_0/W_0\neq \{0\}$ , then Y is contained in the strict subvariety where $m\nu _i$ vanishes and we can skip the next discussion. Otherwise, assume that $V_0=W_0$ and $J=\emptyset $ , and thus that ${\mathcal G}_i={\mathcal G}$ . Denote the projection of the monodromy period map to ${\mathcal J}(U)$ by $\Phi _0:\mathcal {M}\rightarrow {\mathcal J}(U)$ . Since Y is assumed to be (monodromically) atypical, and the only ‘drop’ from ${\mathcal G}$ to ${\mathcal H}_Y={\mathcal N}_i$ is U, Y must be an atypical fiber of $\Phi _0$ over a torsion point.

Suppose U has an irreducible sub-Hodge structure $U'$ which is not of weight -1 level 1 or weight -2 level 0. The topological invariant associated to the projection $\nu _{U'}$ is computed (in the sense of Remark 2.11) by the composition $\pi _1({\mathcal M})\twoheadrightarrow H_1({\mathcal M},\mathbb {Z})\to H_1({\mathcal J}(U'),\mathbb {Z})\cong U^{\prime }_{\mathbb {Z}}$ induced by $\Phi _0'$ , the Zariski closure of whose image is (the underlying $\mathbb {Q}$ -vector subspace of) a sub-HS $U''\subset U'$ . On the other hand, we have $F^{-1}U_{\mathbb {C}}'\subsetneq U_{\mathbb {C}}'$ ; and the differential $d\Phi _0'\colon T{\mathcal M}\to T{\mathcal J}(U')\cong U_{\mathbb {C}}'/F^0U_{\mathbb {C}}'$ factors through $F^{-1}U_{\mathbb {C}}'/F^0U_{\mathbb {C}}'$ by Griffiths transversality, which forces $U^{\prime \prime }_{\mathbb {C}}\subset F^{-1}U^{\prime }_{\mathbb {C}}$ . Since $U'$ is irreducible, this forces $U''=0$ , in contradiction to the assumed genericity of $\nu $ .

Hence we may assume that all the Hodge structures appearing in U have either weight -1 level 1 or weight -2 level 0. Then ${\mathcal J}(U)$ is a semi-abelian variety, and it is an algebraic variety. Assume for the sake of contradiction that the image of $\Phi _0$ contains infinitely many torsion points, whose preimages are atypical. By [Reference Bakker, Brunebarbe and TsimermanBBT23b], the image of $\Phi _0$ is an algebraic subvariety and by the solution to the Manin-Mumford conjecture [Reference RaynaudRay83, Reference Pila and ZannierPZ08], the closure of the torsion locus in ${\Phi }_0(\mathcal M)$ is a translate of an algebraic subgroup B.

Lemma 3.6. The morphism $\Phi _0$ is dominant.

Proof. By contradiction, assume that it is not. Since $B\subset \overline {\Phi _0({\mathcal M})}$ , then B is a strict algebraic subgroup of ${\mathcal J}(U)$ and $Z={\Phi }^{-1}_0(B)$ is a special subvariety of $\mathcal {M}$ . We claim that Z is atypical: indeed $\overline {\Phi _0(Z)}=B$ where the closures are taken in ${\mathcal J}(U)$ . Therefore:

$$\begin{align*}\mathrm{codim}_{\Phi_0({\mathcal M})} (\Phi_0(Z))=\mathrm{codim}_{\overline{\Phi_0({\mathcal M})}}( \overline{\Phi_0(Z)})=\mathrm{codim}_{\overline{\Phi_0({\mathcal M})}}( B)<\mathrm{codim}_{{\mathcal J}(U)}(B)~.\end{align*}$$

Hence ${\Phi }^{-1}_0(B)$ is a (monodromically) atypical subvariety of $\mathcal {M}$ that strictly contains Y, which contradicts the maximality of Y. This proves the lemma.

We are now ready to finish the proof of Lemma 3.5. By Lemma 3.6, the morphism $\Phi _0$ is dominant. Since Y is the fiber over a torsion point y and it is atypical, we have the inequality:

$$\begin{align*}\dim(\Phi_0({\mathcal M}))=\mathrm{codim}_{\Phi_0({\mathcal M})}(\Phi_0(Y))<\mathrm{codim}_{{\mathcal J}(U)}(\{y\})=\dim ({\mathcal J}(U))~,\end{align*}$$

which contradicts the fact that $\Phi _0$ is dominant.

Thus $\Phi _0({\mathcal M})$ contains only finitely many torsion points and we can bound the torsion order in the factor ${\mathcal J}(U)$ . We conclude that Y is a component of the torsion locus of a normal function $\nu _i=\nu _{W_0/V^{\prime }_0}$ and where the possible torsion orders are bounded (say, by m again). This concludes the proof of Lemma 3.5.

We have now produced by Lemma 3.5 the finite collection $(\nu _i)_i$ of projections of $\nu $ in the statement of Theorem 3.2 and we can finish the proof. Let $\mathscr {T}\subset {\mathcal M}$ denote the union of the zero loci of $m\nu _i$ . By [Reference Brosnan and PearlsteinBP13] and our genericity assumption on $\nu $ , this is a proper algebraic subvariety.

Finally, let T be a factorwise positive-dimensional component of $\mathrm {TL}(\nu )$ . On T, the monodromy period map factors through some torsion section $\sigma \colon D_{{\mathcal H}}\to {\mathcal D}_{{\mathcal H}}$ of codimension $\mathrm {rk}({\mathcal J}({\mathcal V}))$ . Now if (i) holds, then (invoking Griffiths transversality once more) the rank of $\delta \tilde {\nu }|_p\colon T_p{\mathcal M}\to {\mathcal F}^{-1}V_p/{\mathcal F}^0V_p$ at a point $p\in T$ , and hence $\text {codim}_{{\mathcal M}}(T)$ , is strictly less than $\mathrm {rk}({\mathcal J}({\mathcal V}))$ . If (ii) holds, then $\text {codim}_{{\mathcal M}}(T)<\dim ({\mathcal M})\leq \mathrm {rk}({\mathcal J}({\mathcal V}))$ . So in either case $\text {codim}_{{\mathcal M}}(T)<\mathrm {rk}({\mathcal J}({\mathcal V}))\leq $ monodromy codimension of T, which makes T monodromically atypical for ${\mathcal E}_{\nu }$ . Let Y be a maximal monodromically atypical subvariety containing T, and assume Y is not contained in $\mathscr {Z}$ . Then the analysis above shows that ${\mathcal D}_Y^0$ is determined by a torsion point, since it contains T, and that there exists i such that Y is a component of $\mathrm {TL}(\nu _i)$ with torsion order bounded by m. That is, $Y\subset \mathscr {T}$ . This concludes the proof of Theorem 3.2.

We now state some corollaries that follow from the previous theorem.

If the variation ${\mathcal V}$ is $\mathbb Q$ -simple and non-isotrivial, then notice that in the proof of Theorem 3.2, the only possibility for $\nu _i$ is $\nu $ , and if $\nu $ is generically non-torsion, then it is generic by Corollary 2.10(iii). Hence we get the following result.

3.3 Analytic density and equidistribution of the torsion locus

We assume in this section that ${\mathcal V}$ has weight $-1$ and level $1$ , that all its $\mathbb Q$ -factors are non-isotrivial, and $H=G^{der}$ . We continue to assume that the period map for ${\mathcal V}$ is quasi-finite. Let $\nu \in \mathrm {ANF}_{\mathcal {M}}({\mathcal V})$ be a generic normal function in the sense of Definition 2.9.

To analyze the density properties of the torsion locus of $\nu $ , we need to understand better the local properties of the normal function $\nu $ , namely its generic rank which we will now define.

Let ${\mathcal J}({\mathcal V})\rightarrow \mathcal {M}$ be the family of intermediate Jacobians of which $\nu $ is a section. For $U\subset \mathcal {M}$ an open simply connected subset, the family of Jacobians ${\mathcal J}({\mathcal V})\rightarrow U$ admits a $C^{\infty }$ -trivialization ${\mathcal J}(V_{\mathbb R})\times U\rightarrow U$ where ${\mathcal J}(V_{\mathbb R})$ is a compact torus. Projecting to the first factor and lifting to $V_{\mathbb R}$ yields the Betti map $\nu _{\mathbb R}:U\rightarrow V_{\mathbb R}$ .

For $s\in U$ , we can define the rank of $\nu $ at s as the rank of the differential of $\nu _{\mathbb R}$ at s. Denoting by $\widetilde {\mathcal {M}}$ the universal cover of $\mathcal {M}$ , the previous discussion produces a map $\nu _{\mathbb R}:\widetilde {\mathcal {M}}\rightarrow V_{\mathbb R}$ . The fibers of $\nu _{\mathbb R}$ are complex analytic subvarieties of $\widetilde {{\mathcal M}}$ by the same argument as in [Reference GaoGao20a, Section 4, (ii) page 2479].

In our setting, the rank of the Betti map was fully studied in [Reference GaoGao20a], and a key ingredient was to use the mixed Ax-Schanuel Theorem. The following result is a particular case of [Reference GaoGao20a, Theorem 1.4. (ii)]. Using this idea, we reproduce this result in a Hodge theoretic setting to make our paper more self-contained. The proof aligns more closely with [Reference GaoGao20b, Theorem 9.1]. We refer also to [Reference Gao and ZhangGZ24, Theorem 3.1] for a more general result in weight $-1$ and to [Reference Khelifa and UrbanikKU23] for a purely Hodge theoretic approach.

Proof. Let d be the complex dimension of $\mathcal {M}$ . Obviously the generic rank $2k$ of $\nu $ cannot be larger than $\mu $ . By way of contradiction, assume that this rank is smaller than $\mu $ (hence that $k<d$ and $k<\tfrac {1}{2}\dim _{\mathbb {Q}}(V)$ ). Let $\widetilde {\Phi }:\widetilde {\mathcal {M}}\rightarrow {\mathcal D}$ be the lift of the period map associated to the variation of mixed Hodge structure ${\mathcal E}_\nu $ . Every point $x\in {\mathcal J}(V_{\mathbb R})$ defines a holomorphic section $\sigma _x:D\rightarrow {\mathcal D}$ and it is clear that $\tilde {\Phi }^{-1}(\sigma _x(D)\cap \widetilde {\Phi }(\widetilde {{\mathcal M}}))=\nu _{\mathbb R}^{-1}(\{x\})$ .

Since $\nu _{\mathbb R}$ has generic rank $2k$ , we can find an analytic open subset $U\subset \widetilde {\mathcal {M}}$ such that the fibers of $\nu _{\mathbb R}$ above $\nu _{\mathbb R}(U)$ are complex analytic of complex dimension $d-k$ . Furthermore, since the Hodge locus of $\widetilde {\mathcal {M}}$ is a countable union of strict analytic subvarieties, we can find a point $y\in U$ which is Hodge generic by Baire’s theorem. Let $x=\nu _{\mathbb R}(y)\in V_{\mathbb R}$ ; then it satisfies the following properties:

1. (1) the fiber $Y_0=\nu _{\mathbb R}^{-1}(x)$ is complex analytic of complex dimension $d-k$ ( $>0$ by the assumption on k);

2. (2) $Y_0$ is Hodge generic (for ${\mathcal E}_{\nu }$ ) inside $\widetilde {\mathcal {M}}$ .

Let $\Delta ={\mathcal D}\times _{\Gamma \backslash {\mathcal D}}\mathcal {M}$ and $W=D_x\times \mathcal {M}$ . Then $Y_0$ is a complex analytic component of the intersection $W\cap \Delta $ in ${\mathcal D}\times \mathcal {M}$ and we have:

$$\begin{align*}\mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(Y_0)=\dim({\mathcal D})+k,\mspace{20mu}\!\! \mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(W)=\frac{1}{2}\dim_{\mathbb Q} V,\!\!\mspace{20mu}\mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(\Delta)=\dim({\mathcal D})\,.\end{align*}$$

Hence (by our assumption on k) the following inequality is satisfied:

$$\begin{align*}\mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(Y_0)<\mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(W)+\mathrm{codim}_{{\mathcal D}\times\mathcal{{M}}}(\Delta)~.\end{align*}$$

By the Ax–Schanuel theorem for variations of mixed Hodge structures [Reference Gao and KlinglerGK24, Reference Chung Tak ChiuChi25], we conclude that the projection of $Y_0$ to ${\mathcal M}$ is contained in a strict weakly special subvariety ${\mathcal Z}$ . That is, the period map sends ${\mathcal Z}$ into (a quotient of) a proper weak Mumford-Tate subdomain of ${\mathcal D}$ , and so the geometric monodromy group ${\mathcal H}_{{\mathcal Z}}$ of ${\mathcal E}_{\nu }|_{{\mathcal Z}}$ is a proper subgroup of ${\mathcal G}^{der}$ . But since $Y_0$ is Hodge generic, the generic Mumford-Tate group of ${\mathcal E}_{\nu }|_{{\mathcal Z}}$ is equal to ${\mathcal G}$ ; and because $Y_0$ is positive-dimensional (and $G^{ad}$ is simple), we have that $H_{\mathcal Z}=G^{der}$ . As $\mathcal {H}_{\mathcal Z}$ is a normal subgroup of ${\mathcal G}$ and $G^{der}$ acts non-trivially on every factor of V, we conclude that ${\mathcal H}_{\mathcal Z}={\mathcal H}={\mathcal G}^{der}$ , which is a contradiction. This proves the desired result.

Let $\psi :V_{\mathbb Z}\times V_{\mathbb Z}\rightarrow \mathbb Z$ be the polarization form and let $\beta $ be the corresponding harmonic $(1,1)$ form on ${\mathcal J}(V_{\mathbb R})$ . Then $\beta ^r$ determines a volume form on ${\mathcal J}(V_{\mathbb R})$ , where $r=\frac {1}{2}\dim _{\mathbb Q} V$ and let $\mathrm {vol}({\mathcal J}(V_{\mathbb R}))$ be the volume of ${\mathcal J}(V_{\mathbb R})$ with respect to $\beta ^r$ , which is also the covolume of the lattice $V_{\mathbb Z}$ in $V_{\mathbb R}$ Footnote ¹² . It is well known that the torsion points on ${\mathcal J}(V_{\mathbb R})$ are equidistributed with respect to $\beta ^r$ ; that is, for every open subset $U\subset {\mathcal J}(V_{\mathbb R})$ with measure-zero boundary, we have

$$\begin{align*}\left|\{z\in U|\, n\cdot z=0\}\right|\underset{n\rightarrow \infty}{\sim}\frac{n^{2r}}{ \mathrm{vol}({\mathcal J}(V_{\mathbb R}))}\cdot \int_{U}\beta^r .\end{align*}$$

The pullback form $\omega _{Betti}=\nu ^{*}\beta $ to $\widetilde {\mathcal {M}}$ is translation invariant and descends to a form on $\mathcal {M}$ , also known as the Betti form.

Theorem 3.12. We keep the same assumptions as Theorem 3.11 and we assume furthermore that ${\mathcal V}$ has rank $2r\leq 2\dim (\mathcal {M})=2d$ . Then $\mathrm {TL}(\nu )$ is analytically dense in $\mathcal {M}$ and equidistributed with respect to the Betti form $\omega _{Betti}$ , i.e., for every $U\subset \mathcal {M}$ relatively compact subset with zero measure boundary and differential form $\alpha $ , we have:

(5) $$ \begin{align} \int_{\{x\in U|\, n\cdot\nu(x)=0\}}\alpha \underset{n\rightarrow \infty}{\sim} \frac{n^{2r}}{ \mathrm{vol}({\mathcal J}(V_{\mathbb R}))}\cdot\int_{U}\alpha\wedge \omega_{Betti}^r. \end{align} $$

4.1 The torsion locus for the Ceresa cycle

For a curve C of genus g and $s\in C$ a complex base point, we have the Ceresa cycle $Z_C$ and associated normal function $\nu $ as described in the introduction. In the following we work with a level $\ell \geq 3$ structure (in the automorphic sense), so that the period map is 2:1 and stacky issues do not enter. By the ‘Main Theorem’ below we shall mean Theorem 3.2, though the reader can also take it to mean Theorem 1.3 (which also applies in each instance).

4.2 Normal functions over locally symmetric spaces

In general there are no nontrivial admissible normal functions on (most) locally symmetric varieties as a consequence of Raghunathan’s theorem, cf. [Reference Keast and KerrKK18]. Instead, one passes to a finite branched cover (or, deleting the branch locus, an étale neighborhood) and considers normal functions there with values in the base change of a Hermitian homogeneous VHS ${\mathcal V}=\widetilde {{\mathcal V}}^{\lambda }\{\tfrac {a}{2}\}$ over $X=\Gamma \backslash D=\Gamma \backslash G_{\mathbb {R}}/K$ . Such variations arise from an irreducible finite-dimensional representation of $G_{\mathbb {R}}$ of highest weight $\lambda $ (which we assume has descent to $\mathbb {Z}$ ); for G of Cartan type $A_n$ , $D_{\text {odd}}$ , or $E_6$ , the generalized half-twist $\{\tfrac {a}{2}\}$ is needed to make the Hodge numbers $(p,q)$ integral. See [op. cit.] for details. Of course, we can always apply a Tate twist to put ${\mathcal V}$ into negative weight.

4.3 The Fano normal function

Similar in spirit to the Ceresa normal function is the Fano normal function $\nu $ over the (10-dimensional) moduli space ${\mathcal M}$ of cubic threefolds. Given a threefold ${\mathbb T}$ and fixing a line $L_0\subset {\mathbb T}$ , the Abel-Jacobi images $\mathrm {AJ}_{{\mathbb T}}(L-L_0)\in {\mathcal J}(H^3({\mathbb T})(2))=:J({\mathbb T})$ (over all lines $L\subset {\mathbb T}$ ) sweep out a surface $F\subset J({\mathbb T})$ . Setting $F^-:=\imath (F)$ , the cycle $Z_{{\mathbb T}}:=F-F^-$ is homologous to zero, and its Abel-Jacobi class $\mathrm {AJ}_{J({\mathbb T})}(Z_{{\mathbb T}})\in {\mathcal J}(H^5_{pr}(J({\mathbb T}))(3))$ (resp. the Hodge structure $H^5_{pr}(J({\mathbb T})(3))$ ) defines $\nu $ (resp. the variation of Hodge structures ${\mathcal V}$ ) at $[{\mathbb T}]\in {\mathcal M}$ . Then $\phi \colon {\mathcal M}\to {\mathcal A}_5$ is an embedding [Reference Clemens and GriffithsCG72, (0.11)], ${\mathcal V}$ is irreducible of level 5 by [Reference Collino, Carlos Naranjo and Pietro PirolaCNP12, Remark 5.14](whence ${\mathcal F}^{-1}{\mathcal V}\neq {\mathcal V}$ ), and it is shown in [Reference Collino, Carlos Naranjo and Pietro PirolaCNP12, Theorem 1.1] that $\nu \in \mathrm {ANF}_{{\mathcal M}}({\mathcal V})$ is non-torsion. So the conclusion of Theorem 1.3 holds.

Even less is known about the torsion locus for the Fano normal function than for Ceresa. However, if we extend $\nu $ to the closure $\overline {{\mathcal M}}$ of $\phi ({\mathcal M})$ , then $\overline {{\mathcal M}}$ contains the hyperelliptic Jacobian locus $H_5$ and the extended normal function is torsion (in fact zero) on this locus [op. cit.].

4.4 Fields of definition of the torsion locus

In this section, we would like to determine the field of definition of the ‘isolated’ components of the torsion locus of a normal function. We first recall a general result about local systems on algebraic varieties.

Let S be a smooth complex quasi-projective algebraic variety and let ${\mathbb V}$ be a complex local system on S. For any subvariety $Y\subset S$ , we denote by ${\mathcal H}_Y$ the monodromy group of Y.

This leads at once to the main result of this subsection.