3.1 Defining the Calabi–Yaus

We will consider a family of Calabi–Yau varieties constructed as crepant resolutions of $n$-fold cyclic covers of projective space, due to Dolgachev and to Sheng, Xu and Zuo [Reference Sheng, Xu and ZuoSXZ13]. Most of the discussion holds for any $n \ge 2$, although the case $n = 2$ is completely classical and returns the Legendre family of elliptic curves; we therefore restrict to $n \ge 3$ simply for convenience. At a crucial point, we find that the cases $n = 3, 4, 6$ are distinguished, giving rise to ‘arithmetic’ families of Calabi–Yau varieties (e.g., these are exactly the cases for which the resulting $\mathcal {M}_{\mathrm {CY}}$ is Shimura); all three of these therefore display qualitatively different behavior and we note at the appropriate point where the condition $n \ne 3, 4, 6$ is crucial for the ‘nonarithmeticity’ phenomenon of the statement of our main Theorem 1.1.3.

The definition of the Calabi–Yau varieties goes through that of a curve; this construction has been studied by many authors, and we refer the reader to [Reference LooijengaLoo07, § 4.1]Footnote ⁸ for a more general version.

We first consider $2n$ points $x_1, \ldots, x_{2n}$ in $\mathbf {P}^1$, and the curve $C$ given by the $n$-fold cyclic cover of $\mathbf {P}^1$ branched at those $2n$ points; more precisely, we mean the cover determined by the same $n$-cycle monodromy about each branch point in the base, or simply the smooth, projective curve $C$ whose affine model is given by

(1)\begin{equation} C^{\circ} = \bigg\{y^n = \prod_{i = 1}^{2n} (x - x_i)\bigg\}. \end{equation}

By construction, $H^1(C, \mathbf {Q})$ has both a Hodge splitting and a $\mathbf {Z}/n$-action. More precisely, if we let $\zeta = e^{2\pi i/n}$, then $H^1(C, \mathbf {Q}(\zeta ))$ splits into eigenspaces for the $\mathbf {Z}/n$-action. Let $\mu \in \mathbf {Z}/n$ be the generator which acts on $C$ by $y \mapsto \zeta y$ in (1).

Then $H^1(C)[i]$ is a $\mathbf {Q}(\zeta )$-sub-Hodge structure of $H^1(C, \mathbf {Q}(\zeta ))$ in the sense that we have the decomposition

\[ H^1(C)[i] \underset{\mathbf{Q}(\zeta)}{\otimes} \mathbf{C} \simeq H^{1, 0}(C)[i] \oplus H^{0, 1}(C)[i] \]

given by the Hodge splitting of $H^1(C; \mathbf {C})$.

Proposition 3.1.2 The Hodge numbers of $H^1(C)[i]$ are given by $(2i - 1, 2(n-i)-1)$, for $1 \le i \le n - 1$; that is,

\[ \dim_{\mathbf{C}}H^{1,0}(C)[i]=2i-1, \quad \dim_{\mathbf{C}}H^{0,1}(C)[i]=2(n-i)-1. \]

We refer the reader to [Reference LooijengaLoo07, Lemma 4.2] for this computation. Note that the $i = 0$ eigenspace is trivial as any cohomology class in this space comes from $H^1(\mathbf {P}^1) \simeq 0$ via pullback.

Construction 3.1.3 It remains to introduce the Calabi–Yau $(2n-3)$-folds $X$. Here we follow the treatment in [Reference Sheng, Xu and ZuoSXZ13, Sections 2, 3]. For a collection of $2n$ hyperplanes $(H_1, \ldots, H_{2n})$ of $\mathbf {P}^{2n-3}$, we say that they are in general position if no $2n-2$ of them intersect at any point; that is, there are no unexpected intersections between the $H_i$. Then we may define an $n$-fold cyclic cover $X'$ of $\mathbf {P}^{2n-3}$ branched along these hyperplanes.

We give the rigorous construction here. For a line bundle $L$ on an arbitrary variety $Y$ and a positive integer $n$, consider the rank-$n$ vector bundle

\[ \mathcal{E}:= \mathcal{O}\oplus L^{\vee}\oplus \cdots \oplus (L^{\vee})^{\otimes n-1} \]

on $Y$; here $L^{\vee }$ denotes the dual of $L$. Now given a section of $L^{\otimes n}$, or equivalently a map

\[ (L^{\vee})^{\otimes n}\rightarrow\mathcal{O}, \]

we may define an algebra structure on $\mathcal {E}$ in the obvious way, and therefore we may form the variety

\[ X:= {\rm Spec}(\mathcal{E}), \]

and by construction $X$ admits a map to $Y$; in fact, this is a cyclic $n$-fold covering. In other words, a section $\sigma \in \Gamma (L^{\otimes n})$ defines a cyclic $n$-fold cover $X\rightarrow Y\!$.

Definition 3.1.4 For a collection of $2n$ points $p_1, \ldots p_{2n}$ on $\mathbf {P}^1$, we may consider $2n$ hyperplanes on $\mathbf {P}^{2n-3}$ as follows: recall that there is an isomorphism

(2)\begin{equation} {\rm Sym}^{2n-3}\mathbf{P}^1\cong \mathbf{P}^{2n-3}, \end{equation}

and also that, for each $i=1, \ldots, 2n$, the set of points of the form $\{p_i\}\times \mathbf {P}^1\times \cdots \times \mathbf {P}^1$ on the left-hand side of (2) gives a hyperplane on the right-hand side. Therefore, we obtain $2n$ hyperplanes $H_1, \ldots, H_{2n}$ on $\mathbf {P}^{2n-3}$, in general position.

We now apply the above construction to $Y=\mathbf {P}^{2n-3}$, $L=\mathcal {O}(2)$ and $\sigma \in \Gamma (\mathcal {O}(2n))$ such that the zero locus of $\sigma$ is precisely

\[ D:= \sum_{i=1}^{2n} H_i \subset \mathbf{P}^{2n-3} \]

to obtain a cyclic $n$-fold covering of $\mathbf {P}^{2n-3}$, which we denote by $X'$.

Note that a pleasant computation shows that the canonical bundle of the cyclic cover $X'$ defined above is trivial: indeed, using the formula $K_{X}=\pi ^*K_Y+R$ for a covering map $\pi :X\rightarrow Y$, where $R\subset X$ is the ramification divisor, we deduce that

\begin{align*} K_{X'}&=\pi^*((2-2n)H)+(2n)(n-1)H'\\ &=(n(2-2n)+2n(n-1))H'\\ &=0, \end{align*}

and so the canonical class of $X'$ is trivial; here $H$ denotes a hyperplane class in $\mathbf {P}^{2n-3}$ and $H'$ is the class of one of the $n$ components of the pullback of $H$.

Denote the moduli space of collections $(H_1, \ldots, H_{2n})$ of hyperplanes of $\mathbf {P}^{2n-3}$ in general position by $\mathcal {H}_n$. The following theorem was proved by Sheng, Xu and Zuo [Reference Sheng, Xu and ZuoSXZ13, Corollary 2.6].

3.2 Hodge structures of Dolgachev Calabi–Yaus

Now we relate the curves constructed earlier as branched covers of $\mathbf {P}^1$ to the Calabi–Yau varieties constructed as covers of $\mathbf {P}^{2n-3}$. Recall from Theorem 3.1.5 that the cyclic cover $X'$ has a crepant resolution $X$ which is Calabi–Yau, and we have an isomorphism

\[ H^{2n-3}(X; \mathbf{Q}) \simeq H^{2n-3}(X'; \mathbf{Q}). \]

The right-hand side has a natural $\mathbf {Z}/n$-action since it arises as a cyclic cover, and therefore the left-hand side does as well. We may, therefore, decompose $H^{2n-3}(X)$ into eigenspaces as follows. As in § 3.1 we fix a generator $\mu \in \mathbf {Z}/n$.

Definition 3.2.1 We define

\[ H^{2n-3}(X)[i] \subset H^{2n-3}(X; \mathbf{Q}(\zeta)) \]

as the sub-vector space over $\mathbf {Q}(\zeta )$ on which $\mu$ acts by $\zeta ^i$.

Then we have the crucial relationship between the Hodge structures of $C$ and $X$.

Lemma 3.2.2 [Reference Sheng, Xu and ZuoSXZ13, Proposition 3.7]

We have the following isomorphism of Hodge structures:

\[ H^{2n-3}(X)[i] \simeq \bigwedge \! ^{2n-3} H^1(C)[i]. \]

Remark 3.2.3 Perhaps a more intrinsic way of phrasing the above lemma is that there is an isomorphism of $\mathbf {Q}$-Hodge structures with $\mathcal {A}: = \mathbf {Q}[X]/(X^n-1)$-action

\[ H^{2n-3}(X, \mathbf{Q})\cong \bigwedge_{\mathcal{A}}\! ^{2n-3}H^1(C, \mathbf{Q}). \]

In other words, we view $H^1(C, \mathbf {Q})$ as an $\mathcal {A}$-module, where the generator $X\in \mathcal {A}$ acts through $\mu \in \mathbf {Z}/n$, and then we take its $(2n-3)$th wedge power over $\mathcal {A}$.

We have the following statement.

Corollary 3.2.4 The Hodge structure $H^{2n-3}(X,\mathbf {Q})\otimes \mathbf {Q}(\zeta )$ decomposes as a sum

\[ H^{2n-3}(X,\mathbf{Q})\otimes \mathbf{Q}(\zeta)\cong \bigoplus_{i=1}^{n-1} V_i \]

where $V_i$ is a $\mathbf {Q}(\zeta )$-Hodge structure, concentrated only in Hodge degrees

\[ (p,q)=(2i-2, 2n-2i-1)\ \text{and}\ (2i-1, 2n-2i-2); \]

furthermore, the dimensions of these pieces of the Hodge decomposition are

\[ 2i-1 \ \text{and} \ 2n-2i-1, \]

respectively.

Proof. Indeed, we define the Hodge structures $V_i$ to be $\bigwedge ^{2n-3}H^1(C)[i]$ from Lemma 3.2.2. By Proposition 3.1.2 we may write the Hodge decomposition of $H^1(C)[i]$ as

\[ H^1(C)[i]\otimes \mathbf{C}\cong H^{1,0}\oplus H^{0,1}, \]

where we have omitted the dependence on $i$ on the right-hand side, and

\[ \dim H^{1,0}=2i-1, \quad \dim H^{0,1} = 2(n-i)-1. \]

For convenience let us pick a basis $\{e_i\}$ (respectively, $\{f_j\}$) for $H^{1,0}$ (respectively, $H^{0,1}$). Since the dimension of $H^1(C)[i]$ is $2n-2$, upon taking the $(2n-3)$th wedge power, the only nonzero elements obtained by wedging together the $e_i$ and $f_j$ must omit precisely one $e_i$ or one $f_j$. Therefore, the Hodge degrees of such an element are either

\[ (p,q)=(2i-2, 2(n-i)-1) \ \text{or} \ (2i-1, 2(n-i)-2); \]

furthermore, there are $2i-1$ (respectively, $2(n-i)-1$) choices of an $e_i$ (respectively, $f_j$) to omit, and therefore the Hodge numbers are $2i-1$ (respectively, $2n-2i-1$), as claimed.

Notation 3.2.5 We will sometimes use the following notation for bookkeeping when dealing with these Hodge numbers. We record the dimensions in a $(n-1)\times 2$ matrix

\[ \begin{pmatrix} \dim H^{1,0}(C)[1] & \dim H^{0,1}(C)[1]\\ \dim H^{1,0}(C)[2] & \dim H^{0,1}(C)[2]\\ \vdots & \vdots \\ \dim H^{1,0}(C)[n-1] & \dim H^{0,1}(C)[n-1] \end{pmatrix}. \]

For example, in the case $n=5$, it follows from Proposition 3.1.2 that the above matrix is

\[ \begin{pmatrix} 1 & 7 \\ 3 & 5\\ 5 & 3\\ 7 & 1 \end{pmatrix}. \]

As a consistency check let us see that the dimensions of $\mathcal {M}_{0,2n}$ and $\mathcal {H}_n$ agree. Indeed, each moduli space parametrizes hyperplanes inside a projective space modulo the action of a projective linear group, and the coincidence of the dimensions is the equality

\[ (2n)-(3)=(2n-3)(2n)-((2n-2)^2-1); \]

here the first term on each side of the equation is the number of moduli for the hyperplanes, while the second term is the dimension of the projective linear group.

3.3 The attractor condition for Dolgachev Calabi–Yau varieties

We are now finally in a position to study the attractor condition for Calabi–Yau varieties $X$ constructed as above. We first show that the attractor condition is equivalent to a condition on the periods of the associated curve $C$ as follows.

Note in the above that $H^{1,0}(C)[1]$ is defined as a subspace of $H^1(C,\mathbf {C})$ while $H^1(C)[1]$ is defined as a subspace of $H^1(C,\mathbf {Q}(\zeta ))$. In particular, as $\dim _{\mathbf {C}}H^{1,0}(C)[1] = 1$, the above attractor condition is also equivalent to the subspace $H^{1,0}(C)[1] \subset H^1(C)\otimes \mathbf {C}$ being defined over $\mathbf {Q}(\zeta )$.

Proof. Suppose $X$ satisfies the attractor condition. Then there exists $\gamma \in H^{2n-3}(X; \mathbf {Z})$ orthogonal to $H^{2n-4, 1}(X)$; equivalently, $\gamma$ is orthogonal to $H^{1, 2n-4}(X)$. Recall from the discussion above that $H^{1, 2n-4}(X)$ is contained within the $H^{2n-3}(X)[1]$ eigenspace. The distinct $\mu$-eigenspaces are certainly orthogonal under the intersection pairing, and if $\gamma _i \in H^{2n-3}(X)[i]$ denotes the summand of $\gamma$ under the decomposition of $H^{2n-3}(X; \mathbf {Q}(\zeta ))$, the above condition is equivalent to $\gamma _1$ orthogonal to $H^{1, 2n-4}(X)$. As the Hermitian pairing is perfect on $H^{1, 2n-4}(X)$, $\gamma$ cannot have any support within the said Hodge summand of

\[ H^{2n-3}(X)[1]_{\mathbf{C}} \simeq H^{1, 2n-4}(X) \oplus H^{0, 2n-3}(X), \]

and so we must have $\gamma _1 \in H^{0, 2n-3}(X)$. But $\gamma _1$ was defined as an element of the vector space $H^{2n-3}(X)[1]$, a vector space defined over $\mathbf {Q}(\zeta )$, and so both $H^{0, 2n-3}(X)$ and $H^{1, 2n-4}(X)$, as the orthogonal complement of $H^{0, 2n-3}(X)$ under the intersection pairing restricted to $H^{2n-3}(X)[1]$, are defined over $\mathbf {Q}(\zeta )$. But then

\begin{align*} H^{1, 2n-4}(X) &\simeq \bigwedge\!^{2n-2} H^{0, 1}(C)[1] \otimes H^{1, 0}(C)[1] \\ &\simeq (H^{0, 1}(C)[1])^{\vee} \otimes ( \det H^{0, 1}(C)[1] ) \otimes H^{1, 0}(C)[1] \\ &\simeq (H^{0, 1}(C)[1] )^{\vee} \otimes \det H^1(C)[1] \end{align*}

as a subspace of

\[ H^{2n-3}(X)[1] \simeq \bigwedge\!^{2n-3} H^1(C)[1] \simeq (H^1(C)[1])^{\vee} \otimes \det H^1(C)[1]. \]

Above, we use the notation $\det V = \bigwedge ^{\dim V} V$ and the isomorphism

\[ \bigwedge\!^{\dim V - 1} V \simeq V^{\vee} \otimes \det V. \]

In any case, we have that the decomposition

\[ H^{2n-3}(X)[1]_{\mathbf{C}} \simeq H^{1, 2n-4}(X) \oplus H^{0, 2n-3}(X) \]

is isomorphic to the decomposition

\begin{align*} &( (H^1(C)[1])^{\vee} \otimes \det H^1(C)[1] )_{\mathbf{C}}\\ &\quad \simeq ( (H^{0, 1}(C)[1])^{\vee} \otimes \det H^1(C)[1] ) \oplus ( (H^{1, 0}(C)[1] )^{\vee} \otimes \det H^1(C)[1] ) \end{align*}

induced from the Hodge splitting of $H^1(C)[1]_{\mathbf {C}}$. As $H^1(C)[1]$ and hence $\det H^1(C)[1]$ are defined over $\mathbf {Q}(\zeta )$, however, the condition that the first decomposition be defined over $\mathbf {Q}(\zeta )$ is equivalent to the condition that the second decomposition be defined over $\mathbf {Q}(\zeta )$, which in particular implies that there exists some $\omega \in H^{1, 0}(C)[1] \cap H^1(C)[1]$.

Conversely, given such an $\omega$, we have that the subspace $H^{1, 0}(C)[1] \subset H^1(C)[1]_{\mathbf {C}}$ is in fact defined over $\mathbf {Q}(\zeta )$ and hence so is $H^{0, 1}(C)[1]$ as its orthogonal complement; as above, the decomposition $H^{2n-3}(X)[1]_{\mathbf {C}} \simeq H^{1, 2n-4}(X) \oplus H^{0, 2n-3}(X)$ is then also defined over $\mathbf {Q}(\zeta )$. Then take some $\gamma _1 \in H^{0, 2n-3}(X)$ defined over $\mathbf {Q}(\zeta )$ so that, by construction, $\gamma _1$ is orthogonal to $H^{1, 2n-4}(X)$, and consider the Galois conjugates $\gamma _i$ under the action of ${\rm Gal}(\mathbf {Q}(\zeta )/\mathbf {Q})$ on $H^{2n-3}(X; \mathbf {Q}(\zeta ))$. These Galois conjugates will lie within the $H^{2n-3}(X)[i]$ eigenspaces for values of $i$ coprime to $n$ and thus be concentrated in Hodge summands away from the $(1, 2n-4)$ and $(0, 2n-3)$ summands, so that if we now define $\gamma = \sum _i \gamma _i$, the Galois-theoretic construction will give us $\gamma \in H^{2n-3}(X; \mathbf {Q})$ while its summand in the $H^{2n-3}(X)[1]$ eigenspace is still the original $\gamma _1$ we started with. As such, scaling $\gamma$ as necessary so it in fact lies in $H^{2n-3}(X, \mathbf {Z})$, we have produced some integral cohomology class orthogonal to $H^{1, 2n-4}(X)$, or equivalently $H^{2n-4, 1}(X)$, as desired.

3.4 Algebraicity of the associated curve

In this section we show that the algebraicity of the Dolagchev Calabi–Yau variety implies that of the curve associated to it. Therefore, to show that Dolgachev Calabi–Yau varieties provide counterexamples to the attractor conjecture it suffices to show that for the attractor varieties, the associated curves are not defined over $\bar {\mathbf {Q}}$.

Proof. It suffices to show that the pullback map induces an isomorphism

\[ \Gamma(X, \mathcal{L})\cong \Gamma(\mathbf{P}^{2n-3}, \mathcal{O}(1)). \]

Recall that we have the factorization

\[ X\xrightarrow{\sigma} X'\xrightarrow{\alpha} \mathbf{P}^{2n-3}, \]

where $\sigma$ is the crepant resolution from Theorem 3.1.5, and

\[ \alpha: X'\rightarrow \mathbf{P}^{2n-3} \]

denotes the $n$-fold covering of $\mathbf {P}^{2n-3}$ from Definition 3.1.4. We first show that

\[ \Gamma(X', \mathcal{L}')\cong \Gamma(\mathbf{P}^{2n-3}, \mathcal{O}(1)), \]

where $\mathcal {L}': = \alpha ^*\mathcal {O}(1)$. By construction of $X'$ (see Definition 3.1.4 and the paragraph preceding it),

\[ \alpha_*\mathcal{O}_{X'}\cong \mathcal{O}_{\mathbf{P}^{2n-3}}\oplus L^{\vee}\oplus \cdots \oplus (L^{\vee})^{\otimes n-1}, \]

where $L^{\vee }\cong \mathcal {O}(-2)$. Therefore,

\[ \alpha_*\mathcal{L}\cong \mathcal{O}_{\mathbf{P}^{2n-3}}(1)\oplus \mathcal{O}_{\mathbf{P}^{2n-3}}(-1)\oplus \cdots \oplus \mathcal{O}_{\mathbf{P}^{2n-3}}(-2n+3), \]

and hence $\Gamma (X', \mathcal {L}')=\Gamma (\mathbf {P}^{2n-3}, \mathcal {O}(1))$ as required. On the other hand, $X$ is obtained from $X'$ by blowing up along subvarieties of codimension at least two, and we claim

\[ \Gamma(X, \mathcal{L})\cong \Gamma(X', \mathcal{L}') \]

as well. Indeed, as $X$ and $X'$ fail to be isomorphic only in codimension two, this statement would follow from (algebraic) Hartogs’ lemma provided $X$ and $X'$ are both normal. That $X$ is normal follows from its smoothness, while $X'$ is normal as it is both $R_1$ and $S_2$ [Sta23]. Indeed, its singular set has codimension two, while it is $S_2$ given its construction as a hypersurface in a smooth ambient variety (the total space of a line bundle over $\mathbf {P}^{2n-3}$).

Hence, $\mathcal {L}$ and its linear system $X \to \mathbf {P}^{2n-3}$ are defined over $\bar {\mathbf {Q}}$, and so we learn that the ramification locus with irreducible components the $2n$ hyperplanes $H_1, \ldots, H_{2n}$ may be taken to be defined over $\bar {\mathbf {Q}}$ – that is, are defined over $\bar {\mathbf {Q}}$ after, possibly, an application of some $PGL_{2n-2}$ projective transformation to their original definition as corresponding to the points $x_i$. However, this condition is precisely the same as that the original $2n$ points may be taken to be defined over $\bar {\mathbf {Q}}$, that is, possibly after some $PGL_2$ transform, or equivalently that all their cross-ratios are in $\bar {\mathbf {Q}}$, and so it is then easy to reconstruct $C$ over $\bar {\mathbf {Q}}$. Indeed, the map from the $2n$ points to the $2n$ hyperplanes (or $2n$ points in a dual projective space) may be regarded as a morphism between (open loci of) $\mathrm {Sym}^{2n-3} \mathbf {P}^1 / S_3 \to \mathbf {P}^{2n-3} / S_{2n-2}$ (by using the simple $3$- or ($2n-1$)-transitivity of the $PGL_2$- and $PGL_{2n-2}$-actions, respectively) which is explicitly defined over $\bar {\mathbf {Q}}$. Indeed, one may write down this map in explicit coordinates; we refer the reader to [Reference Sheng, Xu and ZuoSXZ13, Claim 3.6].

3.5 Attractors are dense

In this section we show that the attractor points are Zariski dense in moduli space. This fact will be used when we apply the ZPC.

We define the auxiliary space

\[ \mathcal{M}_{0,2n}' := \{ (s, \omega)\,|\, s\in \mathcal{M}_{0,2n},\ \omega \in H^{1,0}(C_s)[1],\ \omega \neq 0\} \]

where we have denoted by $C_s$ the $n$-fold cover of $\mathbf {P}^1$ branched at the configuration of $2n$ points given by $s \in \mathcal {M}_{0,2n}$. Recall that $H^{1,0}(C_s)[1]$ is a one-dimensional $\mathbf {C}$-vector space and hence $\mathcal {M}_{0,2n}'$ is a $\mathbf {G}_m$-bundle over moduli space. Also let $\widetilde {\mathcal {M}}_{0,2n}'$ denote the universal cover of $\mathcal {M}_{0,2n}'$; on this universal cover we have a well-defined basis of the cohomology group $H^1(C, \mathbf {Q}(\zeta _n))[-1]$ which we denote by $\gamma _1, \ldots, \gamma _{2n-2}$.

We may now consider the so-called Schwarz map defined as follows:

\begin{align*} \pi: \widetilde{\mathcal{M}}_{0,2n} &\rightarrow \mathbf{C}^{2n-2}\\ (s, \omega) & \mapsto \bigg(\int_{\gamma_1}\omega, \ldots,\int_{\gamma_{2n-2}}\omega \bigg). \end{align*}

By Lemma 3.3.1 we have that a point $(s,\omega ) \in \widetilde {\mathcal {M}}_{0,2n}$ is an attractor point (more precisely, the point $s$ gives rise to an attractor Calabi–Yau and $\omega$ witnesses this) if and only if $\pi ((s,\omega ))$ has coordinates in $\mathbf {Q}(\zeta ) \subset \mathbf {C}$. Note that $\pi$ is a holomorphic local isomorphism; see, for example, [Reference LooijengaLoo07, Corollary 1.7]. So the image $\pi (\widetilde {\mathcal {M}}_{0,2n})$ contains some open ball inside $\mathbf {C}^{2n-2}$. Since $\mathbf {Q}(\zeta _n) \subset \mathbf {C}$ is dense, we have that the attractors are topologically dense, and hence Zariski dense as well. Note, moreover, that these attractor points are isolated, as each point of $\mathbf {Q}(\zeta )$ in $\mathbf {C}$ is isolated, and therefore we have the following proposition.

4.1 Algebraic attractors split off CM abelian varieties

In this section we show that if an attractor is algebraic, then the Jacobian of the corresponding curve $C$ must split off CM factors.

We make use of the following theorem of Shiga and Wolfart [Reference Shiga and WolfartSW95, Proposition 3], a consequence of the analytic subgroup theorem of Wüstholz.

Theorem 4.1.1 (Shiga and Wolfart)

Let $A$ be a simple abelian variety defined over $\bar {\mathbf {Q}}$ endowed with a nonzero differential $\Omega \in \Gamma (A, \Omega ^1_A)$ which is also defined over $\bar {\mathbf {Q}}$. For $\gamma \neq 0$, the period $\langle \Omega, \gamma \rangle$ is nonzero. Suppose that for any two classes $\gamma _1, \gamma _2 \in H^1(A; \mathbf {Z})$, the period ratios are algebraic:

(4)\begin{equation} \frac{\langle \Omega, \gamma_1 \rangle}{\langle \Omega, \gamma_2 \rangle} \in \bar{\mathbf{Q}}. \end{equation}

Then $A$ has complex multiplication. Moreover, if $K$ is the number field generated by the period ratios above then the CM field of $A$ is precisely $K$.

We now apply Theorem 4.1.1 to the case we are interested in. Let $X$ be a Dolgachev Calabi–Yau $(2n-3)$-fold as defined in § 3.1, and recall that $X$ is constructed from a curve $C$, which is a cyclic cover of $\mathbf {P}^1$ and is equipped with an automorphism $\mu$; the latter induces an action on ${\rm Jac}(C)$ which we denote by the same symbol.

Proof. First, note that if $X$ is defined over $\bar {\mathbf {Q}}$, then by Proposition 3.4.1 the same is true of $C$, and therefore of its Jacobian; let $C_{{\bar {\mathbf {Q}}}}$ be a model of $C$ over ${\bar {\mathbf {Q}}}$, namely, it is a curve over ${\bar {\mathbf {Q}}}$ such that $C_{{\bar {\mathbf {Q}}}}\otimes _{{\bar {\mathbf {Q}}}} \mathbf {C}\simeq C$, for some embedding ${\bar {\mathbf {Q}}} \hookrightarrow {} \mathbf {C}$. For brevity we write $A$ for the abelian variety ${\rm Jac}\ C$.

Now let $A_1$ denote the abelian variety, which is a summand of $A$ in the isogeny category, on which $\omega$ is supported; more precisely, we write $A\sim B_1\times B_2\times \cdots\times B_r$, and take $A_1$ to be the product of all the factors $B_i$ to which $\omega$ pulls back nontrivially. Note that since $\omega$ satisfies $\mu ^*\omega =\zeta \omega$, $\mu$ preserves $A_1$.

On the other hand, since $\omega \in H^1(C, \mathbf {Q}(\zeta ))$, all the periods

\[ \langle \omega, \gamma\rangle, \quad \text{for} \ \gamma\in H^1(A_1, \mathbf{Q}), \]

are contained in $\mathbf {Q}(\zeta )$. Since $\dim _{\mathbf {C}}H^{1,0}(C)[1]=1$ by Proposition 3.1.2, and $H^{1,0}(C)[1]\simeq \Gamma (C_{{\bar {\mathbf {Q}}}}, \Omega ^1_{C_{{\bar {\mathbf {Q}}}}})[1]\otimes _{{\bar {\mathbf {Q}}}} \mathbf {C}$, there is a nonzero complex number $\lambda$ such that $\Omega : = \lambda \omega$Footnote ¹¹ is defined over $\bar {\mathbf {Q}}$ when considered as a differential on $C$ (or equivalently, $A$). Applying Theorem 4.1.1 to each simple isogeny factor of $A_1'$ of $A_1$, and the pullback of $\Omega$ to $A_1'$, we deduce that $A_1'$ has complex multiplication by a subfield of $\mathbf {Q}(\zeta )$, as required.

We summarize this section with the following theorem.

4.2 Prym varieties

In fact, while $\mathcal {M}_{0, 2n}$ does naturally map to the Shimura variety parametrizing $(n-1)^2$-dimensional (principally polarized) abelian varieties as above, for the application of the ZPC it is necessary to refine this map slightly, especially when $n$ is not a prime. The end result will be a map to a PEL-type Shimura variety instead of simply $\mathcal {A}_{(n-1)^2}$. Therefore, in this section we study the construction of Prym varieties, which are a certain quotient of the Jacobian.

Recall that, for $n\geq 2$, $C\rightarrow \mathbf {P}^1$ denotes the cyclic $n$-fold covering of $\mathbf {P}^1$ branched at $2n$ points, whose affine model is given in (1); moreover, there is an action of $\mathbf {Z}/n$ on $C$. Now suppose we have a divisor $n'$ of $n$, and let $\mathbf {Z}/n'\subset \mathbf {Z}/n$ denote the unique order- $n'$ subgroup of $\mathbf {Z}/n$; we fix a generator $\mu \in \mathbf {Z}/n$ as before, and further denote $\mu ': = (n/n')\mu$, which is a generator of this $\mathbf {Z}/n'$ subgroup.

Definition 4.2.1 For each $n'$ dividing $n$, define

\[ C':= C/(\mathbf{Z}/n'), \]

where $\mathbf {Z}/n'$ acts on $C$ via the inclusion $\mathbf {Z}/n'\subset \mathbf {Z}/n$. Also let

\[ \pi_{n'}: C\rightarrow C' \]

denote the natural quotient map. By further quotienting by a $\mathbf {Z}/(n/n')$, we also have a map $C'\rightarrow \mathbf {P}^1$, which is a cyclic $n/n'$-fold covering.

Proposition 4.2.2 Let $J_{n'}$ denote the cokernel of the pullback map

\[ \pi^*_{n'}:{\rm Jac}(C')\rightarrow {\rm Jac}(C). \]

Then its cohomology is given by

\[ H^1(J_{n'}, \mathbf{Q}(\zeta))\cong \bigoplus_i H^1(C, \mathbf{Q}(\zeta))[in']; \]

equivalently, the above is the sum of all $H^1(C, \mathbf {Q}(\zeta ))[j]$ where $j$ satisfies

\[ \zeta^{jn/n'}=1. \]

Remark 4.2.3 As a consistency check, we see that the above sum is over $i=n', \ldots, ({n}/{n'}-1)n'$, and so there are $(n/n'-1)$ nontrivial summands, as expected, since

\begin{align*} C'\rightarrow \mathbf{P}^1 \end{align*}

is now an $n/n'$-fold cyclic cover.

Proof. Applying the Riemann–Hurwitz formula to the covering map $C'\rightarrow \mathbf {P}^1$, we have

\[ 2-2g(C')=\frac{n}{n'}(2)-2n\bigg(\frac{n}{n'}-1\bigg), \]

and hence

\[ g(C')=(n-1)\bigg(\frac{n}{n'}-1\bigg). \]

Here $g(C')$ denotes the genus of $C'$. On the other hand, recall that the points of the Jacobian of a curve are the degree-zero divisors modulo rational equivalence, and therefore the image of the pullback map

\[ \pi_{n'}^*: {\rm Jac}(C')\rightarrow {\rm Jac}(C) \]

is invariant under the $\mathbf {Z}/n'$-action. On the other hand, $\pi _{n'}^*$ is injective, since if $D$ is a degree-zero divisor on $C'$ such that

\[ \pi^*_{n'}(D)=(f) \]

for some rational function $f$ on $C$, then $f$ is invariant under the Galois group $\mathbf {Z}/n'$ of the covering map $C\rightarrow C'$, and therefore $f$ descends to $C$. Therefore, the map on homology induced by $\pi ^*_{n'}$ is also injective, and lands inside the invariant subspace $H_1({\rm Jac}(C), \mathbf {Q}(\zeta ))^{\mathbf {Z}/n'}$ (using $\mathbf {Q}(\zeta )$-coefficients). By the genus computation above, this gives an isomorphism

\[ H_1({\rm Jac}(C'), \mathbf{Q}(\zeta))\cong H_1({\rm Jac}(C), \mathbf{Q}(\zeta))^{\mathbf{Z}/n'}. \]

Dualizing, we have $H^1({\rm Jac}(C'), \mathbf {Q}(\zeta ))$ being the coinvariants of the $\mathbf {Z}/n'$-action on $H^1({\rm Jac}(C), \mathbf {Q}(\zeta ))$, which gives the desired result; indeed, since the action of a generator $\mu ': = (n/n')\mu \in \mathbf {Z}/n'$ on $H^1({\rm Jac}(C), \mathbf {Q}(\zeta ))[j]$ is given by $\zeta ^{nj/n'}$, the coinvariants are given by

\[ \bigoplus_{i=1}^{n-1}H^1({\rm Jac}(C), \mathbf{Q}(\zeta))[i]/(\zeta^{ni/n'}-1)H^1({\rm Jac}(C), \mathbf{Q}(\zeta))[i], \]

and we see that the nontrivial summands are indexed by $i$ such that $\zeta ^{in/n'}=1$, as claimed.

We immediately deduce the following simple corollary.

Corollary 4.2.4 We denote by $\pi ^*_{n'}$ the map on cohomologies induced by $\pi ^*_{n'}$. Then the quotient of $H^1(C, \mathbf {Q}(\zeta ))$ by the images of $\pi ^*_{n'}$ for all proper divisors $n'$ (i.e. $n'\neq 1, n$) is precisely the sum

\[ \bigoplus_{i\in (\mathbf{Z}/n)^{\times}} H^1(C, \mathbf{Q}(\zeta))[i]. \]

The above direct sum decomposition can be refined integrally, or equivalently as a statement about abelian varieties.

Definition 4.2.5 We now define the abelian variety

\[ {\rm Prym}:= {\rm Jac}(C)/\sum_{n'}{\rm Im}(\pi^*_{n'}), \]

and refer to it as the Prym variety. Here the sum is over proper divisors $n'$ as above.

Corollary 4.2.6 The abelian variety ${\rm Prym}$ has endomorphisms by $\mathbf {Q}(\zeta )$, and its $\mathbf {Q}(\zeta )$-Hodge structure is given by

\[ \bigoplus_{i\in (\mathbf{Z}/n)^{\times}} H^1(C, \mathbf{Q}(\zeta))[i]. \]

As such, it has dimension $(n-1)\phi (n)$.

4.3 PEL Shimura varieties

Now that we have the necessary statements on the Prym construction from § 4.2, we can define the refined period map, whose target is a certain PEL-type Shimura variety. Denote by $V$ the $\mathbf {Q}$ subspace of $H^1(C, \mathbf {Q})$ such that

\[ V\otimes \mathbf{Q}(\zeta)=\bigoplus_{r\in (\mathbf{Z}/n)^{\times}}V[r], \]

where $V[r]\subset H^1(C, \mathbf {Q}(\zeta ))$ denotes the $\zeta ^r$ eigenspace for the action of $\mu \in \mathbf {Z}/n$; as before, $V$ has an action of $\mathbf {Q}(\zeta )$. By Corollary 4.2.6, we may identify $V$ with $H^1({\rm Prym}, \mathbf {Q})$.

Definition 4.3.1 The abelian variety ${\rm Prym}$ from § 4.2 furnishes us with an integral lattice

\[ V_{\mathbf{Z}}:= H^1({\rm Prym}, \mathbf{Z})\subset H^1({\rm Prym}, \mathbf{Q})=V, \]

equipped with a symplectic form $\Psi$. Let

\[ \mathbf{S}:= {\rm Res}_{\mathbf{C}/\mathbf{R}}\mathbf{G}_m \]

denote the Deligne torus, and let $\mathfrak {H}$ denote the space of homomorphisms

\[ h: \mathbf{S}\rightarrow {\rm GSp}(V_{\mathbf{R}}, \Psi), \]

which in turn define Hodge structures of type $(-1,0)+(0,-1)$ on $V_{\mathbf {Z}}$. The space $\mathfrak {H}$ is isomorphic to the Siegel upper half space of dimension

\[ \frac{(n-1)\phi(n)((n-1)\phi(n)+1)}{2}, \]

since it parametrizes abelian varieties of dimension $(n-1)\phi (n)$.

The Shimura datum $({\rm GSp}(V, \Psi ), \mathfrak {H})$ certainly defines a Shimura variety to which $\mathcal {M}_{0,2n}$ maps; we can describe its $\mathbf {C}$-points as follows. The integral structure on $V$ defines a maximal compact subgroup $K\subset {\rm GSp}(V, \Psi )(\mathbf {A})$, and then we have

\[ {\rm Sh}({\rm GSp}(V, \Psi), \mathfrak{H})(\mathbf{C})={\rm GSp}(V, \Psi)(\mathbf{Q})\backslash \mathfrak{H}\times {\rm GSp}(V, \Psi)(\mathbf{A}_f)/K; \]

here $\mathbf {A}$ (respectively, $\mathbf {A}_f$) denotes the ring of (respectively, finite) adeles. However, as we shall presently see, $\mathcal {M}_{0,2n}$ lands inside a smaller Shimura subvariety. In the following we essentially follow the treatment of [Reference MoonenMoo10, Section 3.3], the only difference being that we work with the Prym whereas Moonen considers the entire Jacobian.

We introduce notation to describe the real points of $G$ following Moonen. For each $r\in (\mathbf {Z}/n)^{\times }$, $V[r]\otimes _{\mathbf {Q}(\zeta )} \mathbf {C}\oplus V[-r]\otimes _{\mathbf {Q}(\zeta )} \mathbf {C}$ is canonically defined over $\mathbf {R}$, and we write $V_{\mathbf {R}, (\pm r)}$ for its descent to $\mathbf {R}$: namely, $V_{\mathbf {R}, (\pm r)}$ is the $\mathbf {R}$-vector space such that $V_{\mathbf {R}, (\pm r)}\otimes \mathbf {C} \simeq V[r]\otimes _{\mathbf {Q}(\zeta )} \mathbf {C}\oplus V[-r]\otimes _{\mathbf {Q}(\zeta )} \mathbf {C}$. The polarization defines a skew-hermitian form $\beta _{(\pm r)}$ on $V_{\mathbf {R}, (\pm r)}$. We refer the reader to [Reference MoonenMoo10, § 4.5] and the references therein for more details.

The Prym construction therefore gives us a map

\[ P: \mathcal{M}_{0,2n}\rightarrow {\rm Sh}. \]

We have the following result, which is analogous to the classical result that the Torelli map is an embedding, although we only require an infinitesimal version of this.

Proof. We will show equivalently that, for each $x\in \mathcal {M}_{0,2n}$, the codifferential map

\[ P^*: T_{P(x)}^*{\rm Sh} \rightarrow T_x^* \mathcal{M}_{0,2n} \]

is surjective. Here for a variety $X$ and a point $x\in X$ we denote by $T^*_xX$ the cotangent space to $X$ at $x$. First we identify the source and target of $P^*$ in terms of the geometric structures at hand.

Claim 4.3.5 We have the following identifications of the cotangent spaces:

(6)$$\begin{gather} T^*_{P(x)}{\rm Sh} \cong \bigoplus_{\substack{r\in (\mathbf{Z}/n)^{\times}\\ r< n/2}} H^0(C, \Omega^1_C)[r]\otimes H^0(C, \Omega^1_C)[n-r], \end{gather}$$

(7)$$\begin{gather}T_x^* \mathcal{M}_{0,2n} \cong H^0(C, (\Omega^1_C)^{\otimes 2})_{{\rm inv}} . \end{gather}$$

Here $H^0(C, \Omega ^1_C)[r]$ denotes the $\zeta ^r$ eigenspace of $H^0(C, \Omega ^1_C)$, and the subscript ${\rm inv}$ in (7) denotes the invariant part of the $\mathbf {Z}/n$-action.

Furthermore, under these identifications, the restriction of $P^*$ on each of the factors is given in (6) by the cup product map

\[ \cup: H^0(C, \Omega^1_C)[r]\otimes H^0(C, \Omega^1_C)[n-r]\rightarrow H^0(C, (\Omega^1_C)^{\otimes 2})_{{\rm inv}} \]

Let us first see how to conclude the proof of this lemma, assuming this claim. Note that the dimension of $H^0(C, (\Omega ^1_C)^{\otimes 2})_{{\rm inv}}$ is $2n-3$, since it has to be the dimension of $\mathcal {M}_{0,2n}$. We will now show that the restriction (which we continue to denote by $P^*$)

(8)\begin{equation} P^*: H^0(C,\Omega^1_C)[1]\otimes H^0(C,\Omega^1_C)[n-1]\rightarrow H^0(C, (\Omega^1_C)^{\otimes 2})_{{\rm inv}} \end{equation}

is in fact an isomorphism; certainly the dimensions of the source and target agree, and so it suffices to show this map is injective. But this is clear since the space $H^0(C,\Omega ^1_C)[1]$ is one-dimensional and spanned by $\omega$, say, and so anything in the kernel of the map (8) takes the form $\omega \otimes \eta$ for some $\eta \in H^0(C, \Omega ^1_C)[n-1]$. On the other hand, $\omega$ and $\eta$ are nonzero 1-forms on $C$, and the quadratic differential obtained by multiplying them together is certainly nonzero. This shows that the map (8) is injective, and therefore it is an isomorphism, as required. Therefore, it suffices to prove Claim 4.3.5.

Proof of Claim By construction, we have an embedding

\[ {\rm Sh} \rightarrow {\rm Sh}({\rm GSp}(V, \Psi), \mathfrak{H}), \]

where the right-hand side denotes the Shimura variety attached to the Shimura datum $({\rm GSp}(V, \Psi ), \mathfrak {H})$. The latter is the moduli space of abelian varieties of dimension $(n-1)\phi (n)$ equipped with a polarization of the fixed type specified by the polarization on the Prym variety. Therefore, the tangent space to ${\rm Sh}({\rm GSp}(V, \Psi ), \mathfrak {H})$ at $P(x)$ is given by

(9)\begin{equation} {\rm Sym}^2(t_{{\rm Prym}})\subset t_{{\rm Prym}}\otimes t_{{\rm Prym}^{\vee}}, \end{equation}

where for an abelian variety $A$ we denote by $t_A$ the tangent space at the origin, and by $A^{\vee }$ its dual abelian variety. On the left-hand side of the above we have also made the identification

\[ t_{{\rm Prym}}\cong t_{{\rm Prym}^{\vee}} \]

using the polarization on ${\rm Prym}$. Note that in (9) the right-hand side is the deformation space of ${\rm Prym}$ with no reference to polarizations. By Corollary 4.2.6, we may make the identification

\[ t_{{\rm Prym}}\cong \bigoplus_{i\in (\mathbf{Z}/n)^{\times}} H^1(C, \mathcal{O}_C)[i]; \]

for convenience we denote the right-hand side of this identification by $H^1(C, \mathcal {O}_C)_{{\rm prim}}$. Similarly, we define $H^0(C, \Omega ^1_C)_{{\rm prim}}$ to be the sum of the eigenspaces of $H^0(C, \Omega ^1_C)$ with eigenvalues primitive $n$th roots of unity.

On the other hand, as mentioned above, the right-hand side of (9) is the deformation space of the abelian variety ${\rm Prym}$ (without reference to a polarization), and hence there is a Kodaira–Spencer map

(10)\begin{equation} {\rm KS}: t_{{\rm Prym}}\otimes t_{{\rm Prym}^{\vee}}\rightarrow {\rm Hom}(H^0(C, \Omega^1_C)_{{\rm prim}}, H^1(C, \mathcal{O}_C)_{{\rm prim}}), \end{equation}

which is the natural isomorphism once we make the identifications

\[ t_{{\rm Prym}}\otimes t_{{\rm Prym}^{\vee}}\cong H^1(C, \mathcal{O}_C)_{{\rm prim}}^{\otimes 2} \]

and

\[ H^1(C, \mathcal{O})_{{\rm prim}}\cong H^0(C, \Omega^1_C)_{{\rm prim}}^{\vee}, \]

the latter of which is induced by Serre duality.

By definition, ${\rm Sh}$ is contained in the locus of ${\rm Sh}({\rm GSp}(V, \Psi ), \mathfrak {H})$ where the Hodge structure $H^1$ admits a $\mathbf {Q}(\zeta )$-action and a splitting

\[ H^1\otimes_{\mathbf{Q}} \mathbf{Q}(\zeta)\cong \bigoplus_{r\in (\mathbf{Z}/n)^{\times}}H^1[r] \]

with prescribed Hodge numbers. Therefore, the Kodaira–Spencer map (10) restricted to ${\rm Sh}$ must preserve the different eigenspaces. In other words, we have

(11)\begin{equation} {\rm KS}|_{{\rm Sh}}: T_{P(x)}{\rm Sh}\rightarrow \bigoplus_{r\in (\mathbf{Z}/n)^{\times}}{\rm Hom}(H^0(C, \Omega^1_C)[r], H^1(C, \mathcal{O}_C)[r]); \end{equation}

now since ${\rm KS}$ itself is an isomorphism, ${\rm KS}|_{{\rm Sh}}$ is injective at least. On the other hand, deformations in ${\rm Sh}$ are also required to preserve the polarization: concretely this means that each element in the image of ${\rm KS}|_{{\rm Sh}}$ is invariant under the involution which identifies ${\rm Hom}(H^0(C, \Omega ^1_C)[r], H^1(C, \mathcal {O}_C)[r])$ and ${\rm Hom}(H^0(C, \Omega ^1_C)[n-r], H^1(C, \mathcal {O}_C)[n-r])$ for each $r\in (\mathbf {Z}/n)^{\times }$.Footnote ¹² Therefore, the projection

(12)\begin{equation} {\rm KS}|_{{\rm Sh}}: T_{P(x)}{\rm Sh}\rightarrow \bigoplus_{\substack{r\in (\mathbf{Z}/n)^{\times}\\ r< n/2}}{\rm Hom}(H^0(C, \Omega^1_C)[r], H^1(C, \mathcal{O}_C)[r]). \end{equation}

is also injective. Since the dimensions of the two sides of (12) now agree, and the map is injective, it must in fact be an isomorphism.

Again using Serre duality, for each $r=1, \ldots, n$, we have

\[ H^0(C, \Omega^1_C)[r]\cong H^1(C, \mathcal{O}_C)[n-r]^{\vee}, \]

and using the remark above we may rewrite (12) as

(13)\begin{align} T_{P(x)}{\rm Sh} &\cong \bigoplus_{\substack{r\in (\mathbf{Z}/n)^{\times}\\ r< n/2}} (H^0(C, \Omega^1_C)[r])^{\vee}\otimes H^1(C, \mathcal{O}_C)[r] \end{align}

(14)\begin{align} &\cong \bigoplus_{\substack{r\in (\mathbf{Z}/n)^{\times}\\ r< n/2}} H^1(C, \mathcal{O}_C)[n-r]\otimes H^1(C, \mathcal{O}_C)[r]. \end{align}

Dualizing, we therefore deduce

\[ T^*_{P(x)}{\rm Sh} \cong \bigoplus_{\substack{r\in (\mathbf{Z}/n)^{\times}\\r< n/2}} H^0(C, \Omega^1_C)[r]\otimes H^0(C, \Omega^1_C)[n-r], \]

which proves (6), as claimed.

The identification (7) is well known; see, for example, [Reference Lange and OrtegaLO11, Proposition 4.1]. Furthermore, [Reference Lange and OrtegaLO11, Proposition 4.1] also shows that the codifferential of the map

\[ \mathcal{M}_{0,2n}\rightarrow {\rm Sh}({\rm GSp}(V, \Psi), \mathfrak{H}), \]

namely

\[ P^*:{\rm Sym}^2(H^0(C, \Omega^1_C)_{{\rm prim}})\rightarrow H^0(C, (\Omega^1_C)^{\otimes 2})_{{\rm inv}}, \]

is the cup product map followed by projection onto the invariant factor, and therefore the same is true for the codifferential of the map

\[ \mathcal{M}_{0,2n}\rightarrow {\rm Sh}, \]

as claimed. This concludes the proofs of all the statements in Claim 4.3.5.

This gives immediately the following corollary.

4.4 Special subvarieties

Now that we have defined the relevant PEL-type Shimura variety ${\rm Sh}$ to which $\mathcal {M}_{0,2n}$ maps naturally, we can rephrase Theorem 4.1.3 in terms of special subvarieties of ${\rm Sh}$. In this section we assume the validity of the attractor conjecture, so that the conclusion of Theorem 4.1.3 holds.

Let $X$ be an attractor Dolgachev Calabi–Yau variety corresponding to $x\in \mathcal {M}_{0,2n}$, with corresponding curve $C_x$Footnote ¹³ which comes with an automorphism $\mu$, as in § 3.1. Recall that from Theorem 4.1.3, the attractor conjecture implies that ${\rm Jac}\ C_x$ has an isogeny factor $\tilde {A_x}$,Footnote ¹⁴ which has CM and is invariant under the action $\mu$, and such that the distinguished differential $\omega$ is supported on $\tilde {A}_x$. Let $A_x$ denote the image of $\tilde {A_x}$ under the map ${\rm Jac}\ C_x\rightarrow {\rm Prym}_x$, where the latter map is as in § 4.3. Since there are only finitely many choices of $\tilde {A}_x$ up to isogeny,Footnote ¹⁵ the same is true for $A_x$. Note that the quotient map ${\rm Jac}\ C_x\rightarrow {\rm Prym}_x$ is compatible with the action of $\mu$, and hence $A_x$ is also acted on by $\mu$.

Let $V(A_x)$ and $V_x$ denote the first cohomology groups $H^1(A_x, \mathbf {Q})$ and $H^1({\rm Prym}_x, \mathbf {Q})$, respectively. We have a splitting of $\mathbf {Q}$-Hodge structures of the form $V_x=V(A_x)\oplus V'_x$, corresponding to the splitting up to isogeny of ${\rm Prym}_x$; here $V_x'$ denotes the $\mathbf {Q}$-Hodge structure of the quotient ${\rm Prym}_x/A_x$. Note that both $V(A_x)$ and $V_x'$ have actions by $\mathbf {Q}(\zeta )$ induced by the action of $\mu$, and the decomposition is compatible with these actions. Recall from Definition 4.3.2 that $G$ acts naturally on $V_x$.

Recall that in Definition 4.3.2 we defined special subvarieties associated to subgroups of $G$ through which the homomorphism on the Deligne torus may factor. From Theorem 4.1.3 we immediately deduce the following slight refinement.

Proposition 4.4.6 We have

(15)\begin{equation} {\rm codim}_{{\rm Sh}} {\rm Sh}_x \ge 2n - 3, \end{equation}

with equality if and only if $\phi (n)=2$.

Proof. The subvariety ${\rm Sh}_x$ is itself a unitary Shimura variety, whose dimension is governed by the signatures of $V'_x$; see below for an example demonstrating this. We have that

\[ \dim {\rm Sh}_x=\sum_{r\in (\mathbf{Z}/n)^{\times}/\pm} (2r-1-d_r)(2n-1-2r-d_{-r}). \]

Comparing toFootnote ¹⁸

\[ \dim {\rm Sh}=\sum_{r\in (\mathbf{Z}/n)^{\times}/\pm} (2r-1)(2n-1-2r), \]

we see that the difference between the $r=1$ terms of the two above expressions is $2n-3$, which gives us the desired inequality. Equality holds if and only if all the other terms in the two above expressions agree, and since from Proposition 4.4.6 we have $d_r+d_{-r}\geq 1$ for all $r$, this occurs if and only if there are no other terms, that is, $\phi (n)=2$.

Example 4.4.8 We give an example to illustrate the numerology that is at play here, for $n=5$, which is the smallest value for which the Dolgachev Calabi–Yau varieties give counterexamples to the attractor conjecture. In this case, since $5$ is a prime number, the Prym variety is simply the Jacobian of $C$. For example, if we fix the numerical data $d_1=d_2=1$, $d_3=d_4=0$, then we may write schematically the splitting of the Hodge structure as

(16)\begin{equation} \begin{pmatrix} 1 & 7\\ 3 & 5\\ 5 & 3\\ 7 & 1 \end{pmatrix}= \begin{pmatrix} 1 & 0\\ 1 & 0\\ 0 & 1\\ 0 & 1\\ \end{pmatrix}+\begin{pmatrix} 0 & 7\\ 2 & 5\\ 5 & 2\\ 7 & 0 \\ \end{pmatrix}. \end{equation}

In the equation above we follow Notation 3.2.5, and on the right-hand side we have written the dimension matrices of the summands of this splitting: the last term is the signatures of the summand $V_x'$. In this case, the special subvariety ${\rm Sh}_x$ has dimension $10$, while the ambient Shimura variety ${\rm Sh}$ has dimension $22$.

4.5 A brief digression: the arithmetic cases of $n = 3, 4, 6$ and connections to tilings of the sphere

Here we observe that the attractor conjecture works remarkably well in the cases when the Calabi–Yau moduli space does happen to be a Shimura variety, and point out a connection to the tilings of the sphere by polygons due to Engel and Smillie [Reference Engel and SmillieES18].

In fact, we mention an intriguing question for the interested reader: Engel and Smillie in the above paper study a very precise generating function of the attractor points in the arithmetic $n = 3, 4, 6$ cases in terms of a mock modular form; it is reasonable to ask if there may be any analogous (but presumably more complicated) behavior in the nonarithmetic cases.

6.1 A conjectural construction

We first recall an alternative description of the Dolgachev Calabi–Yau varieties. Recall that we have the curve $C$ whose affine model is

\[ C^{\circ}=\bigg\{y^n=\prod_{i=1}^{2n}(x-x_i)\bigg\}. \]

Then (see [Reference Sheng, Xu and ZuoSXZ13]) the cyclic cover $X'$ of $\mathbf {P}^{2n-3}$ in Definition 3.1.4 is given by

\[ C^{2n-3}/S_{2n-3}\ltimes N', \]

where $N'$ is the kernel of the sum map

\[ (\mu_n)^{2n-3}\rightarrow \mu_n. \]

In the above, each $\mu _n$ factor acts via

\[ x\mapsto x,\quad y \mapsto \zeta y, \]

for $\zeta$ a primitive $n$th root of unity.

We now consider a special one-parameter family $\mathcal {N} \subset \mathcal {M}_{CY}$ given by

\[ x_1=a, x_2=\zeta a, \ldots, x_n=\zeta^{n-1} a,\ y_1=1, \ldots, y_n=\zeta^{n-1}, \]

for some parameter $a$. For each such curve $C$ we can further quotient by the $\mu _n$ acting by

\[ x\mapsto \zeta x, \quad y\mapsto y; \]

let us denote this quotient by $Z$, and the total space of this family over $\mathcal {N}$ by $\mathcal {Z}$.

Note that this new action commutes with the action of the group $S_{2n-3}\ltimes N'$ above.

6.2 Zilber–Pink for one-parameter families

Assuming Conjecture 6.1.2, the same argument as in the case of the Dolgachev Calabi–Yau varieties yields counterexamples to the attractor conjecture, again conjectural on the ZPC. However, in the case of one-parameter families inside Shimura varieties, the ZPC is known to hold conditional on certain purely arithmetic statements as a result of work of Orr [Reference OrrOrr20, Theorem 3.4].