Compact subvarieties of the moduli space of complex abelian varieties

If $\overline{\mathcal{V}}\supsetneq\mathcal{V}$ is a projective compactification of $\mathcal{V}$, such that the codimension of the boundary $\operatorname{codim}_{\overline{\mathcal{V}}}(\overline{\mathcal{V}}\setminus\mathcal{V})=d$, then by embedding $\overline{\mathcal{V}}\hookrightarrow{\mathbb{P}}^{N}$ and recursively choosing very general hypersurfaces $H_{1},\dots,H_{n-d+1}$ such that $\dim H_{1}\cap\dots\cap H_{i}\cap(\overline{\mathcal{V}}\setminus\mathcal{V})=(n-d)-i$ for any $i$, it follows that $H_{1}\cap\dots\cap H_{n-d+1}\cap\overline{\mathcal{V}}\subset\mathcal{V}$ is a compact subvariety, which moreover can be chosen to go through any given finite collection of points of $\mathcal{V}$. Thus the existence of such a compactification $\overline{\mathcal{V}}$ implies $\operatorname{mdim_{c,gen}}(\mathcal{V})\geq d-1$. We note, however, that in general there is no implication going the other way, as is easily seen by considering ${\mathbb{P}}^{a}\times{\mathbb{C}}^{b}\subset{\mathbb{P}}^{a+b}$ for various $a,b$.

The Satake compactification ${\mathcal{A}}_{g}^{*}={\mathcal{A}}_{g}\sqcup{\mathcal{A}}_{g-1}\sqcup\dots\sqcup{\mathcal{A}}_{0}$ of ${\mathcal{A}}_{g}$ has boundary of codimension $g$, which by the above implies $\operatorname{mdim_{c,gen}}({\mathcal{A}}_{g})\geq g-1$. Our first result is that this is sharp.

What we prove is in fact a more precise statement, showing that the maximal dimension of a compact subvariety $X\subsetneq{\mathcal{A}}_{g}$ containing a Hodge-generic point $p\in{\mathcal{A}}_{g}$ (which, as we recall below, is equivalent to saying that $p$ does not belong to any of the countably many proper Shimura subvarieties of ${\mathcal{A}}_{g}$, see also the end of Section 2.2) is equal to $g-1$. This will be the special easier case of 3.4, which establishes this statement for a Hodge-generic point $p\in{\mathcal{S}}$ in any non-compact Shimura subvariety ${\mathcal{S}}\subseteq{\mathcal{A}}_{g}$.

Turning to the question of maximal dimension of all compact subvarieties, the best known upper bound for ${\mathcal{A}}_{g}$ is the 20 year old result of Keel and Sadun [KS03], who proved $\operatorname{mdim_{c}}({\mathcal{A}}_{g})\leq\tfrac{g(g-1)}{2}-1$ for any $g\geq 3$. This was conjectured by Oort, and underscored the difference between ${\mathcal{A}}_{g}\otimes{\mathbb{C}}$ and ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$, as in finite characteristic there does exist a complete subvariety of ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ of codimension $g$ — the locus of ppav whose subscheme of $p$-torsion points is supported at zero.

We determine precisely the maximal dimension of compact subvarieties of ${\mathcal{A}}_{g}$, for all genera.

In Table 1 we give our results and the bound of Keel-Sadun, for comparison, in some small genera, and also for $g=100$ to underscore the difference of the growth rates.

What we actually prove is much stronger: we show that any compact subvariety of ${\mathcal{A}}_{g}$ of dimension at least $g$ is contained in the product of a $(k-1)$-dimensional compact subvariety of ${\mathcal{A}}_{k}$, for some $0\leq k<g$, and a compact Shimura subvariety of ${\mathcal{A}}_{g-k}$ (see 2.9 and 3.4). We then determine the maximal dimension of compact Shimura subvarieties of ${\mathcal{A}}_{g}$. In fact our proof and analyses of compact Shimura subvarieties of ${\mathcal{A}}_{g}$ allows us to describe all maximal-dimensional compact subvarieties of ${\mathcal{A}}_{g}$ — they are either Hodge-generic, or compact Shimura subvarieties of a specific type. See 5.10 for a precise statement, and 5.11 for very explicit examples of maximal-dimensional compact Shimura subvarieties of ${\mathcal{A}}_{g}$.

Recall that the Torelli map sending a smooth complex curve to its Jacobian is an injection $J:{\mathcal{M}}_{g}\hookrightarrow{\mathcal{A}}_{g}$ of the coarse moduli spaces (and is 2:1 onto its image as a map of stacks, but this does not matter for discussing compact subvarieties). Its image is contained in the moduli space ${\mathcal{A}}_{g}^{\rm ind}:={\mathcal{A}}_{g}\setminus({\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1}\cup{\mathcal{A}}_{2}\times{\mathcal{A}}_{g-2}\cup\dots)$ of indecomposable abelian varieties, and determining $\operatorname{mdim_{c}}({\mathcal{A}}_{g}^{\rm ind})$ and $\operatorname{mdim_{c,gen}}({\mathcal{A}}_{g}^{\rm ind})$ would also be very interesting, see Section 6.1.

The Torelli morphism extends to a proper morphism $J:{\mathcal{M}}_{g}^{\rm ct}\to{\mathcal{A}}_{g}$ from the moduli space of curves of compact type. As a corollary of their bound $\operatorname{mdim_{c}}({\mathcal{A}}_{3})\leq 2$ (and the theorem of Diaz [Dia84]), Keel and Sadun [KS03] deduce the bound $\operatorname{mdim_{c}}({\mathcal{M}}_{g}^{\rm ct})\leq 2g-4$ for any $g\geq 3$. Our results also have implications for ${\mathcal{M}}_{g}^{\rm ct}$.

The point of the difference between these numbers is that $J$ is injective at a generic point of ${\mathcal{M}}_{g}^{\rm ct}$, but not along $\partial{\mathcal{M}}_{g}^{\rm ct}$, as it sends a stable curve to the product of Jacobians of its irreducible components, forgetting the location of the nodes.

To obtain the first equality, for any $g\geq 3$ we construct a compact subvariety of ${\mathcal{M}}_{g}^{\rm ct}\setminus{\mathcal{M}}_{g}$ of dimension $\lfloor 3g/2\rfloor-2$, starting from a compact curve in ${\mathcal{M}}_{2}^{\rm ct}$.

The second inequality implies the bound on the maximal dimension of compact subvarieties of ${\mathcal{M}}_{g}^{\rm ct}$ that intersect ${\mathcal{M}}_{g}$, and follows from B (in fact B also improves Keel and Sadun’s bound $\operatorname{mdim_{c}}({\mathcal{M}}_{g}^{\rm ct})\leq 2g-4$ to $\operatorname{mdim_{c}}({\mathcal{M}}_{g}^{\rm ct})\leq\lfloor(\lfloor g/2\rfloor)^{2}/4\rfloor$ for $g\leq 28$).

It is tempting to conjecture that in fact $\operatorname{mdim_{c}}(J({\mathcal{M}}_{g}^{\rm ct}))\leq g-1$ for all $g\geq 2$, which would then imply $\operatorname{mdim_{c}}({\mathcal{M}}_{g}^{\rm ct})=\lfloor 3g/2\rfloor-2$ for all $g\geq 2$. Notice that Krichever [Kri12] claimed exactly such a bound, but unfortunately that proof was never completed.

The inspiration for the proof of A is the following. For contradiction, assume that $X\subsetneq{\mathcal{A}}_{g}$ is a compact subvariety of dimension at least $g$. Note that for a fixed elliptic curve $E$, the codimension of the locus $[E]\times{\mathcal{A}}_{g-1}\subsetneq{\mathcal{A}}_{g}$ is equal to $g$. If we could ensure that the intersection of $[E]\times{\mathcal{A}}_{g-1}$ with $X$ were non-empty and transverse, then such property would hold for a Zariski open subset of $[E]\in{\mathcal{A}}_{1}$, i.e. for all but finitely many $[E]\in{\mathcal{A}}_{1}$. By taking the limit as $[E]$ approaches the boundary point $[i\infty]\in\partial{\mathcal{A}}_{1}^{*}$, this contradicts the compactness of $X$, thus proving that $\dim X\leq g-1$.

The recent advances in weakly special subvarieties show that this method essentially works for Hodge-generic subvarieties, up to relaxing the transversality condition to “the intersection having a component of the expected dimension”. An essential tool we will use for this is a simplified version of [KU23, Theorem 1.6(i)], whose proof relies on the Ax-Schanuel conjecture, proven for ${\mathcal{A}}_{g}$ by Bakker and the fourth author [BT19], and proven for an arbitrary Shimura variety by Mok, Pila and the fourth author [MPT19].

To deduce B, we first show that for any Hodge-generic compact subvariety $X^{\prime}$ of a non-compact Shimura variety ${\mathcal{S}}\subseteq{\mathcal{A}}_{g^{\prime}}$, the dimension of $X^{\prime}$ is bounded above as $\dim X^{\prime}\leq g^{\prime}-1$. This argument, applied to ${\mathcal{S}}={\mathcal{A}}_{g}$, itself gives a stronger version of A, for all Hodge-generic compact subvarieties $X\subsetneq{\mathcal{A}}_{g}$. We then show that, if a compact $X\subsetneq{\mathcal{A}}_{g}$ is not Hodge-generic, then it must either be contained in a compact Shimura variety, or must be contained (up to isogeny) in a product of Shimura varieties ${\mathcal{S}}^{\prime}\times{\mathcal{S}}^{\prime\prime}$, with ${\mathcal{S}}^{\prime}\subseteq{\mathcal{A}}_{g^{\prime}}$ non-compact, and ${\mathcal{S}}^{\prime\prime}\subseteq{\mathcal{A}}_{g^{\prime\prime}}$, with $g^{\prime}+g^{\prime\prime}=g$. Thus a maximal-dimensional non-Hodge-generic $X$ is either ${\mathcal{S}}^{\prime\prime}$ or $X^{\prime}\times{\mathcal{S}}^{\prime\prime}$, where $X^{\prime}$ is a $(g^{\prime}-1)$-dimensional subvariety of ${\mathcal{S}}^{\prime}\subseteq{\mathcal{A}}_{g^{\prime}}$. We thus complete the proof of B by examining the maximal possible dimension of compact Shimura subvarieties of ${\mathcal{A}}_{g}$, using the lists of symplectic representations provided by Milne [Mil11] and Lan [Lan17].

In Section 2 we recall basic facts on algebraic groups, symplectic representations and Shimura varieties. We also prove a structure theorem for Shimura subvarieties ${\mathcal{S}}\subset{\mathcal{A}}_{g}$ that are products of two Shimura varieties, one of which is non-compact (see 2.9).

In Section 3 we recall the statement of Ax-Schanuel for Shimura varieties, and we use it to prove 3.4, which is a stronger and more general version of A. This permits us to give a first estimate for the maximal dimension of a compact subvariety of ${\mathcal{A}}_{g}$ (see 3.7).

In Section 4 we investigate compact Shimura subvarieties of ${\mathcal{A}}_{g}$ that are induced by certain symplectic representations that we call “decoupled”, and determine which of them have dimension $d\geq g$ (see 4.14).

In Section 5 we investigate compact Shimura subvarieties of ${\mathcal{A}}_{g}$ that are induced by symplectic representations that are not decoupled. We show that such subvarieties have dimension either smaller than $g$, or smaller than some Shimura variety originating from a decoupled representation. We thus conclude the proof of B.

In Section 6 we discuss related problems for the moduli space of indecomposable abelian varieties, and we prove C for compact subvarieties of the moduli space ${\mathcal{M}}_{g}^{\rm ct}$ of curves of compact type.

The first author is grateful to Università Roma “La Sapienza” for hospitality in June and October 2023, when part of this work was done. We are grateful to Sorin Popescu who endowed the Stony Brook lectures in Algebraic Geometry, which allowed the first and fourth author to meet and think about these topics. We are very grateful to Salim Tayou and Nicolas Tholozan for enlightening discussions and pointers to the literature on these topics.

Let $G$ be an algebraic group over a number field ${\mathbb{F}}$; $G$ is called simple (resp. almost-simple) if every proper closed normal subgroup of $G$ is trivial (resp. finite).

The group $G$ is called geometrically simple (resp. geometrically almost-simple) if $G_{\mathbb{C}}$ is simple (resp. almost-simple), and it is semisimple if $G_{\mathbb{C}}$ is, namely if all connected normal closed subgroups of $G_{\mathbb{C}}$ are trivial.

If $G$ is almost-simple, then $Z(G)$ is finite and $G^{\mathrm{ad}}:=G/Z(G)$ is simple; hence, $G$ cannot be isogenous to $G_{1}\times G_{2}$ for any positive-dimensional algebraic groups $G_{1}$ and $G_{2}$.

An isogeny is a surjective homomorphism of algebraic groups $G^{\prime}\rightarrow G$ with finite kernel; the kernel is then necessarily contained in $Z(G^{\prime})$. An algebraic group $G$ is simply connected if every isogeny $G^{\prime}\rightarrow G$ is an isomorphism.

If $G$ is semisimple and $G_{1},\dots,G_{r}$ are all the almost-simple closed normal subgroups of $G$, then the product map $G_{1}\times\dots\times G_{r}\rightarrow G$ is an isogeny (see [Mil17, Theorem 22.121]). The Galois group $\operatorname{Gal}({\mathbb{C}}/{\mathbb{F}})$ fixes ${\mathbb{F}}$, and thus it acts on $G_{\mathbb{C}}$ by permuting its almost-simple factors and by fixing $G_{\mathbb{F}}$. As a consequence, the stabilizer of the factors of $G_{\mathbb{C}}$ has a number field ${\mathbb{F}}^{\prime}\supseteq{\mathbb{F}}$ as fixed field, and so $G_{{\mathbb{F}}^{\prime}}$ splits as a product of geometrically almost-simple factors.

If $H_{\mathbb{F}}$ is a geometrically almost-simple group over a totally real number field ${\mathbb{F}}$ and $G=H_{{\mathbb{F}}/{\mathbb{Q}}}$ is the ${\mathbb{Q}}$-group obtained from $H_{\mathbb{F}}$ by restriction of the scalars, then $\operatorname{Prod}_{\sigma}H_{\sigma,{\mathbb{R}}}\rightarrow G_{\mathbb{R}}$ is an isogeny, where $\sigma$ ranges over all embeddings ${\mathbb{F}}\hookrightarrow{\mathbb{R}}$ and $H_{\sigma,{\mathbb{R}}}:=\sigma(H_{{\mathbb{F}}})\otimes_{\sigma({\mathbb{F}})}{\mathbb{R}}$.

A split torus $T$ is a product of a number of copies of the multiplicative group $\mathbb{G}_{m}$. An algebraic group is called anisotropic if it contains no positive-dimensional split tori. A semisimple group $G$ over a number field ${\mathbb{F}}$ is anisotropic if and only if $G_{\mathbb{F}}$ contains no nontrivial unipotent elements.

In this paper we will only deal with connected Shimura varieties. So, when we mention a Shimura variety we mean that it is connected.

In this section we recall definitions and basic properties, see also [Mil05] and [Mil11].

We will denote by $(G^{\mathrm{ad}}_{{\mathbb{R}}})^{+}$ the connected component of $G^{\mathrm{ad}}_{\mathbb{R}}$ that contains the identity.

The group $G_{\mathbb{R}}$ splits (up to isogeny) into a product of geometrically almost-simple factors: denote by $G_{c}$ the product of all its compact factors, and by $G_{nc}=G_{1}\times\dots\times G_{k}$ the product of all its non-compact factors. Now, $\mathcal{D}^{+}$ is a connected Hermitian symmetric domain, and $(G^{\mathrm{ad}}_{\mathbb{R}})^{+}$ acts on $\mathcal{D}^{+}$ transitively, isometrically, and holomorphically. The group $S^{1}$ acts on $\mathcal{D}^{+}$ via $\operatorname{Ad}\circ u$ by fixing the point $u$ and by complex multiplication on $T_{u}\mathcal{D}^{+}$. The compact subgroup $G^{\theta}\cap(G^{\mathrm{ad}}_{\mathbb{R}})^{+}$ is the connected component of the identity of the stabilizer of $u\in\mathcal{D}^{+}$.

Note that $\mathcal{D}^{+}=\mathcal{D}_{1}^{+}\times\dots\times\mathcal{D}^{+}_{k}$ with $\mathcal{D}^{+}_{i}=G^{\mathrm{ad}}_{i}/K_{i}$, where $K_{i}=G^{\theta}\cap G^{\mathrm{ad}}_{i}$ is a compact inner form of $G^{\mathrm{ad}}_{i}$, i.e. $G^{\mathrm{ad}}_{i}$ has an inner automorphism which is a Cartan involution. We claim that $G_{i}$ is geometrically almost-simple (see the proof of [Mil11, Theorem 3.13]). Indeed, if $G_{i}$ were not geometrically almost-simple, then $G_{i,{\mathbb{C}}}$ would be isomorphic to the real group $H_{{\mathbb{C}}/{\mathbb{R}}}$ obtained from a complex group $H_{\mathbb{C}}$ by the restriction of scalars. But then such $G_{i}$ has no compact inner form by the following:

Recall that a subgroup $\Gamma\subset G_{nc}$ is called arithmetic if there exists a simply connected rational algebraic group $G^{\prime}$ and a surjective homomorphism $\varphi:G^{\prime}_{\mathbb{R}}\rightarrow G_{nc}$ with compact kernel such that $\Gamma$ is commensurable to $\varphi(G^{\prime}_{\mathbb{Z}})$. An arithmetic subgroup is automatically a lattice. Moreover, denoting $\Gamma_{i}:=\Gamma\cap G_{i}(F)$, the product $\Gamma_{1}\times\dots\times\Gamma_{k}$ is then a finite index subgroup of $\Gamma$, and so the map $\prod_{i}(\Gamma_{i}\backslash\mathcal{D}^{+}_{i})\rightarrow\Gamma\backslash\mathcal{D}^{+}$ is a finite étale cover.

As every arithmetic subgroup $\Gamma$ of $G_{nc}$ contains a torsion-free normal subgroup $\Gamma^{\prime}$ of finite index, the connected Shimura variety ${\mathcal{S}}=\Gamma\backslash\mathcal{D}^{+}$ is a quotient of ${\mathcal{S}}^{\prime}=\Gamma^{\prime}\backslash\mathcal{D}^{+}$ by the finite group $\Gamma/\Gamma^{\prime}$ (see [Mil11, Theorem 3.6]). Moreover, ${\mathcal{S}}^{\prime}$ is a smooth quasi-projective variety over ${\mathbb{C}}$ (see [Mil11, Theorems 4.2-4.3]) and so ${\mathcal{S}}$ can be seen as an irreducible quasi-projective variety too, but also as an irreducible, smooth, Deligne-Mumford stack.

We say that a connected Shimura variety associated to the Shimura datum $(G,\mathcal{D}^{+})$ is indecomposable if $G$ is almost-simple. As for the compactness, we recall the following criterion.

If ${\mathcal{S}}_{j}=\Gamma_{j}\backslash\mathcal{D}_{j}^{+}$ is a Shimura variety associated to the Shimura datum $(G_{j},\mathcal{D}^{+}_{j})$ for $j=1,2$, a morphism of Shimura varieties ${\mathcal{S}}_{1}\rightarrow{\mathcal{S}}_{2}$ is a map induced by a homomorphism $\rho:G_{1}\rightarrow G_{2}$ of algebraic groups such that $\rho(\mathcal{D}_{1}^{+})\subseteq\mathcal{D}_{2}^{+}$ and $\rho(\Gamma_{1})\subseteq\Gamma_{2}$.

A point of the Shimura variety ${\mathcal{S}}$ corresponding to the homomorphism $u:S^{1}\rightarrow(G^{\mathrm{ad}}_{\mathbb{R}})^{+}$ is called Hodge-generic if the smallest algebraic ${\mathbb{Q}}$-subgroup $G^{\prime}$ of $G^{\mathrm{ad}}$ such that $G^{\prime}_{\mathbb{R}}$ contains the image of $u$ is the whole $G^{\mathrm{ad}}$. A subvariety $X\subseteq{\mathcal{S}}$ is called Hodge-generic (within ${\mathcal{S}}$) if its general point is. This is equivalent to requiring $X$ to contain a point that is Hodge-generic in ${\mathcal{S}}$.

A special subvariety of ${\mathcal{S}}$ is a subvariety induced by a Shimura sub-datum of $(G,\mathcal{D}^{+})$. A weakly special subvariety of ${\mathcal{S}}$ is either a point, or a special subvariety, or a subvariety ${\mathcal{S}}_{H}\times\{p\}$ of a special subvariety ${\mathcal{S}}_{H}\times{\mathcal{S}}_{K}$ of ${\mathcal{S}}$, where $p$ is a Hodge-generic point in ${\mathcal{S}}_{K}$, up to isogeny.

Let $G$ be a connected semisimple algebraic group over ${\mathbb{Q}}$ and let $(V,\omega)$ be a rational symplectic representation of $G$ (i.e. a homomorphism $G\to\operatorname{Sp}(V,\omega)$, for $V$ a vector space over ${\mathbb{Q}}$ with a symplectic form $\omega$). A symplectic subspace of $V$ is a vector subspace on which $\omega$ restricts to a (non-degenerate) symplectic form.

We will often invoke the following useful result.

Using the results recalled above, we now study Shimura subvarieties of ${\mathcal{A}}_{g}$ that are a product of a non-compact Shimura variety and another Shimura variety, showing that then each factor embeds into a suitable ${\mathcal{A}}_{k}$, and that, up to isogeny, the embedding of the product is the product of the embeddings.

Let $(G,\mathcal{D}^{+})$ be a connected Shimura datum and let $\Gamma$ be a lattice in $G_{\mathbb{Z}}$. Denote by ${\mathcal{S}}$ the Shimura variety $\Gamma\backslash\mathcal{D}^{+}$ and by $\pi:\mathcal{D}^{+}\rightarrow{\mathcal{S}}$ the natural projection, and let $\check{\mathcal{D}}^{+}$ be the compact Hermitian domain dual to $\mathcal{D}^{+}$, which is a projective variety.

This will be usually applied when $W$ is the image inside $\mathcal{D}^{+}$ of the map induced by a morphism of Shimura data.

As a consequence of 3.1, one obtains 3.2 below, which is essentially a simplified version of [KU23, Theorem 1.6(i)], whose proof we include for completeness.

Recall that if $X\subseteq{\mathcal{S}}$ is an irreducible subvariety and $X_{\mathrm{sm}}$ is its (connected) smooth locus, then the algebraic monodromy of $X$ at a point $x\in X_{\mathrm{sm}}$ is the Zariski closure of the image inside $G_{\mathbb{Q}}$ of the induced homomorphism $\pi_{1}(X_{\mathrm{sm}},x)\rightarrow\Gamma$.

Moreover, recall that for a subvariety $Y\subseteq{\mathcal{S}}$, denoting $\widetilde{Y}:=\pi^{-1}(Y)\subseteq\mathcal{D}^{+}$, for any $\gamma\in G_{\mathbb{Q}}$ the image $\pi(\gamma\cdot\widetilde{Y})$ is a subvariety of ${\mathcal{S}}$ called the $\gamma$-translate of $Y$.

The following application will be useful for us.

Given an irreducible subvariety $X\subset{\mathcal{A}}_{g}$, by definition $X$ is Hodge-generic within the smallest Shimura subvariety $S$ of ${\mathcal{A}}_{g}$ containing $X$. In this section we prove the following.

The first main result of the present paper is an immediate consequence.

Note that we of course get a stronger version of A, for compact subvarieties through any Hodge-generic point of ${\mathcal{A}}_{g}$, not just through an abstract very general point of ${\mathcal{A}}_{g}$. Note also that the proof of 3.4 in this special case ${\mathcal{S}}={\mathcal{A}}_{g}$ does not require either technical 3.5 or 3.6, and implements the idea discussed in the introduction, making use of 3.3 to see that the intersections are indeed of expected dimension.

The idea of the proof of 3.4, for arbitrary non-compact ${\mathcal{S}}$, is to first produce a morphism of Shimura varieties $\Phi:{\mathcal{S}}^{\prime}_{1}\times{\mathcal{S}}^{\prime}_{2}\rightarrow{\mathcal{A}}_{g}$ with ${\mathcal{S}}^{\prime}_{1}={\mathcal{A}}_{1}(N)$ and ${\mathcal{S}}^{\prime}_{2}={\mathcal{A}}_{g-1}(N)$ such that $\Phi(\{y\}\times{\mathcal{S}}^{\prime}_{2})$ intersects ${\mathcal{S}}$ for almost every $y\in{\mathcal{S}}^{\prime}_{1}$ (where ${\mathcal{A}}_{g}(N)$ denotes the moduli of ppav with a full level $N$ structure). By 3.3(ii), the compactness of $X$ then forces $\dim X\leq g-1$.

The construction of $\Phi$ uses the following preliminary lemma, in which we construct a modular curve mapping to ${\mathcal{S}}$.

We can now construct the desired morphism $\Phi$.