1 Introduction
If X is a complex manifold and
$f:Y \to X$
and
$g:Z \to X$
are two sufficiently nice maps from complex analytic spaces to X, then we say that an intersection between Y and Z is unlikely if
$\dim f(Y) + \dim g(Z) < \dim X$
Footnote
1
and that it is likely otherwise. Conjectures of Zilber-Pink type predict that in many cases of interest there are very few unlikely intersections. For example, the Zilber-Pink conjecture for Shimura varieties takes the following form. We let X be a Shimura variety and
$Y \subseteq X$
be a subvariety of X that is not contained in any proper weakly special subvariety (in the sense of Shimura varieties) of X. The conclusion of the conjecture in this case is that the union of all unlikely intersections between Y and special subvarieties of X is not Zariski dense in Y. In this paper, we address the complementary question of describing the likely intersections.
This is not the first time that the likely intersection problem has been addressed. In Section 4 we discuss several specializations of our main theorem and the relation between our results and those which appear in the literature. Three notable instances are the “all or nothing” theorem of Baldi, Klingler, and Ullmo [Reference Baldi, Klingler and Ullmo8] on the density of the typical Hodge locus, the results of Tayou and Tholozan [Reference Tayou and Tholozan33] describing the typical Hodge locus of polarized variations of Hodge structure over a smooth complex quasiprojective variety, and the work of Gao [Reference Gao12, Reference Gao14] (following on the work of André, Corvaja, and Zannier [Reference André, Corvaja and Zannier1]) on the generic rank of the Betti map, from which a sufficient condition for the density of torsion in subvarieties of abelian schemes is derived.
We formulate and prove our main theorem (Theorem 3.3) in terms of definable analytic maps from complex algebraic varieties to definable complex quotient spaces. See Section 2 for details. In brief, a definable complex quotient space S is a complex analytic space which may be presented as a double coset space
$\Gamma \backslash G / M$
where G is an open subgroup of the real points of an algebraic group,
$M \leq G$
is a suitable subgroup, and
$\Gamma \leq G$
is a discrete subgroup together with a choice of a definable (in a fixed o-minimal expansion of the real numbers) fundamental set
$\mathcal {F}$
. Examples of such definable complex quotient spaces include complex tori, Shimura varieties, Hopf manifolds, and (mixed) period spaces. We then define a special subvariety of a definable complex quotient space S to be the image of a map of definable complex quotient spaces
$S' \to S$
(or possibly taken only from some subcategory of definable complex quotient spaces) which is a definable map induced by an algebraic group homomorphism followed by translation. For example, the special subvarieties of complex tori would be translates by torsion points of subtori and the special subvarieties of period spaces would come from period subdomains. We must also consider a more general class of weakly special varieties, which are the fibers of maps of definable complex quotient spaces and the images of these fibers under other maps of definable complex quotient spaces. We identify a condition we call well-parameterization of weakly special subvarieties whereby all of the weakly special varieties in some given definable complex quotient space come from those appearing in countably many families of weakly special subvarieties. In our applications this condition is easy to verify as we restrict to subcategories of definable complex quotient spaces for which there are only countably many morphisms all told.
Consider
$S = \Gamma \backslash G / M = \Gamma \backslash D$
and
$D' \subseteq D$
a homogeneous space for a subgroup
$G' \leq G$
. We will write
$\pi _\Gamma :D \to S$
for the quotient map. It may happen that
$\pi _\Gamma (gD') \subseteq S$
is a special subvariety of S for many choices of
$g \in G$
, where “many” might mean that the set of such g is dense in G. For example, if
$D = \mathfrak {h}^2$
is the Cartesian square of the upper half plane,
$\Gamma = \operatorname {PGL}_2^2(\mathbb {Z})$
and
$D' = \{ (\tau ,\tau ) : \tau \in \mathfrak {h} \}$
is the diagonal, then all modular plane curves may be expressed as
$\pi _\Gamma (gD')$
as g ranges through
$\operatorname {PGL}_2^2(\mathbb {Q})$
acting via pairs of rational linear transformations. In such a situation we might expect that if X is a quasiprojective complex algebraic variety and
$f:X^{\text {an}} \to S$
is a definable complex analytic map from the analytification of X to S, then the set
$f^{-1} \bigcup _{g \in G, \pi _\Gamma (gD') \text { special }} \pi _\Gamma (g D')$
of special intersections is dense in X provided that the intersections are likely in the sense that
$\dim f(X) + \dim D' \geq \dim S$
. This is not quite right as the intersection may become unlikely after transformation through a special correspondence. We account for this complication with the notion of persistently likely intersections. Our main Theorem 3.3 asserts that if we know that the Ax-Schanuel theorem holds for our given category of definable complex quotient spaces (which satisfies some natural closure properties and the well-parameterization of weakly special subvarieties condition), then, in fact, when the intersection of
$f(X)$
with the family of sets
$\pi _\Gamma (gD')$
, where
$g\in G$
, is persistently likely, the set of special intersections is dense in X (even in the Euclidean topology).
When working in the setting of variations of Hodge structures, our result improves the “all or nothing” theorem of [Reference Baldi, Klingler and Ullmo8] first by giving a criterion for the typical Hodge locus to be dense (namely persistent likeliness with families), and secondly by obtaining a Euclidean dense set of likely intersections using a specific subcollection of the family of all special varieties. On the other hand, the general setting of definable quotients spaces does not allow for measure theoretic techniques, so we are unable to recover the equidistribution results appearing in, for example, [Reference Baldi, Klingler and Ullmo8] and [Reference Tayou and Tholozan33]. But our result does imply some of the applications of these equidistribution results. For example, in the setting of the moduli space of principally polarized abelian varieties of dimension g, given two subvarieties S and D of
$\mathcal {A}_g$
of complementary dimensions and so that
$\dim S\leq 2$
, our result implies the part of [Reference Tayou and Tholozan33, Theorem 1.22] stating that the set of points in S isogenous to a point in D is Euclidean dense in S. See also Section 4 for more applications. An advantage of the present work is that our results apply directly to the likely intersections problem in contexts not covered by [Reference Tayou and Tholozan33], such as for mixed Shimura varieties, or, more generally, for variations of mixed Hodge structures.
Unsurprisingly, Ax-Schanuel theorems play key roles in the existing proofs of the density of special intersections. What may be surprising is that our argument is not an abstraction of the proofs appearing, for example, in [Reference Baldi, Klingler and Ullmo8], [Reference Tayou and Tholozan33], or [Reference Gao12]. Instead, our arguments are inspired by the work of Aslanyan and Kirby [Reference Aslanyan and Kirby3], especially with the proof of their Theorem 3.1. The reader will recognize a resemblance between our notion of persistent likeliness and the J-broad and J-free conditions of [Reference Aslanyan and Kirby3], which themselves extend freeness and rotundity conditions from earlier works on existential closedness as a converse to Schanuel-type statements. While the contents of our arguments differ, the structure of many of the results of Daw and Ren in [Reference Daw and Ren11] inspired our approach.
Our own interest in the likely intersections problem was motivated by [Reference Pila and Scanlon26, Conjecture 3.13] in the second author’s work with Pila on effective versions of the Zilber-Pink conjecture.
This paper is organized as follows. In Section 2 we define definable quotient spaces and develop some of their basic theory. These definitions and results owe their form to the formalism of Bakker, Brunebarbe, Klingler, and Tsimerman [Reference Bakker, Klingler and Tsimerman5, Reference Bakker, Brunebarbe, Klingler and Tsimerman4] used to study arithmetic quotients and more generally mixed period spaces. In that section we express precisely what the Ax-Schanuel condition means and show that under the well-parameterization of weakly special subvarieties hypothesis, it implies a uniform version of itself. Section 3 is devoted to the statement and proof of our main theorem on dense special intersections. In Section 4 we detail several specializations of the main theorem, including to Hodge loci, intersections with modular varieties, and density of torsion in subvarieties of abelian schemes.
2 Complex quotient spaces and
$\mathsf {S}$
-special varieties
We express our theorems on likely intersections in terms of classes of definable complex quotient spaces. Our formalism is similar to what appears in [Reference Bakker, Klingler and Tsimerman5], though we explicitly include the fundamental domain giving the definable structure as part of our data.
Throughout we work in an appropriate o-minimal expansion of the real field
$\mathbb {R}$
(usually
$\mathbb {R}_{\mathrm {an},\exp }$
), and the word definable is meant with respect to this choice of o-minimal structure. We will also implicitly use a theorem of Pillay [Reference Pillay27, Proposition 2.5] that definable groups in o-minimal structures have a natural topology with respect to which they are topological groups, and even Lie groups in our context of expansions of the real field. Moreover, definable homomorphisms between definable groups are always continuous as in an o-minimal structure every definable function is continuous on a dense open set and for homomorphisms of topological groups it suffices to check continuity at one point. Furthermore, if the o-minimal expansion chosen has analytic cell decomposition (such as
$\mathbb {R}_{\mathrm {an},\exp }$
), then definable homomorphisms between definable groups are analytic maps.
Definition 2.1. A definable quotient space is given by the data of
-
• a definable group G,
-
• a definable compact subgroup
$M \leq G$
of G, -
• a discrete subgroup
$\Gamma \leq G$
of G, and -
•
$\mathcal {F} \subseteq D := G/M$
a definable open fundamental set for the action of
$\Gamma $
on D (that is,
$D = \bigcup _{\gamma \in \Gamma } \gamma \mathcal {F}$
and there is a finite subset
$\Gamma ' \subseteq \Gamma $
so that if
$x \in \mathcal {F}$
and
$\gamma x \in \mathcal {F}$
for some
$\gamma \in \Gamma $
, then
$\gamma \in \Gamma '$
) for which the closure
$\overline {\mathcal {F}}$
of
$\mathcal {F}$
is contained in
$\bigcup _{\gamma \in \Gamma "} \gamma \mathcal {F}$
for some finite subset
$\Gamma " \subseteq \Gamma $
.
We write
$S_{\Gamma ,G,M;\mathcal {F}}$
both for the quotient space
$\Gamma \backslash D = \Gamma \backslash G / M$
regarded as a definable, real analytic space where the definable structure comes from
$\mathcal {F}$
, and for the data
$(G,M,\Gamma ,\mathcal {F})$
giving this space. We denote the corresponding quotient map by
$\pi _{\Gamma }:D\rightarrow S_{\Gamma ,G,M;\mathcal {F}}$
. When the data are understood, we suppress them and write S for
$S_{\Gamma ,G,M;\mathcal {F}}$
and
$\pi :D \to S$
for the quotient map.
The class of definable quotient spaces forms a category
$\mathsf {D}\mathsf {Q}\mathsf {S}$
with the following notion of a morphism.
Definition 2.2. A morphism
$f:S_{\Gamma _1,G_1,M_1;\mathcal {F}_1} \to S_{\Gamma _2,G_2,M_2;\mathcal {F}_2}$
is given by a definable map of groups
$\varphi :G_1 \to G_2$
and an element
$a \in G_2$
for which
-
•
$\varphi (M_1) \leq M_2$
, -
•
$\varphi (\Gamma _1) \leq a^{-1} \Gamma _2 a$
, and -
• there is a finite set
$\Xi \subseteq \Gamma _2$
with
$a \overline {\varphi }(\mathcal {F}_1) \subseteq \bigcup _{\xi \in \Xi } \xi \mathcal {F}_2$
, where
$\overline {\varphi }:D_1 \to D_2$
is the map induced on the quotient spaces.
The induced map on the double quotient space is a definable real analytic map which we also denote by f.
Remark 2.3. If
$f:S_{\Gamma _1,G_1,M_1;\mathcal {F}_1} \to S_{\Gamma _2,G_2,M_2;\mathcal {F}_2}$
is a morphism of definable quotient spaces as in Definition 2.2, then it is definable in the sense that its graph on the specified fundamental domains
$$\begin{align*}\operatorname{Graph}(f) := \{ (x,y) \in \mathcal{F}_1 \times \mathcal{F}_2 : f(\pi_{\Gamma_1}(x)) = \pi_{\Gamma_2}(y) \} \end{align*}$$
is definable. Indeed, it is clear that the induced map
$\overline {\varphi }:D_1 \to D_2$
is definable and for
$(x,y) \in \mathcal {F}_1 \times \mathcal {F}_2$
we have that
$(x,y) \in \operatorname {Graph}(f)$
if and only if
$\bigvee _{\xi \in \Xi } \xi \cdot y = a \overline {\varphi } (x)$
.
The category
$\mathsf {D}\mathsf {Q}\mathsf {S}$
has a terminal object.
Proposition 2.4. There is a terminal object in
$\mathsf {D}\mathsf {Q}\mathsf {S}$
.
Proof. Take
$G = M = \Gamma = \{ 1 \}$
to be the trivial group and then let
$\mathcal {F} := G/M$
, which is a singleton. The one-point space
$\{ \ast \} = S_{G,\Gamma ,M;\mathcal {F}}$
is a definable quotient space and for any definable quotient space
$S = S_{G',\Gamma ',M';\mathcal {F}'}$
, the unique set theoretic map
$S \to \{ \ast \}$
is induced by the unique map of groups
$\varphi :G' \to \{ 1 \}$
and
$a = 1 \in \{ 1 \}$
.
A useful observation is that a morphism of definable quotient spaces always factors as a surjective map followed by a map induced by an inclusion of subgroups.
Proposition 2.5. Every map
$f:S_1 \to S_2$
of definable quotient spaces fits into a commutative diagram
where
$q:S_1 \to S_3$
is surjective and the map
$p:S_3 \to S_2$
is induced by an inclusion of subgroups. Moreover, p is closed and has compact fibers.
Proof. Take
$S_i = S_{G_i,\Gamma _i,M_i;\mathcal {F}_i}$
for
$1 \leq i \leq 2$
and let
$\varphi :G_1 \to G_2$
be a definable homomorphism and
$a \in G_2$
so that
$f:S_1 \to S_2$
is given by
$\Gamma _1 x M_1 \mapsto \Gamma _2 a \varphi (x) M_2$
.
Define
$G_3 := \varphi (G_1) \leq G_2$
, a definable group. Set
$\Gamma _3 := \varphi (\Gamma _1)$
,
$M_3 := \varphi (M_1)$
, and
$\mathcal {F}_3 := \overline {\varphi }(\mathcal {F}_1)$
where
$\overline {\varphi }:G_1/M_1 \to G_3/M_3$
is the induced map. We make the following observations:
-
(a) Since
$\Gamma _2$
is discrete and
$\Gamma _3$
is a subgroup of
$a^{-1}\Gamma _2a$
, then
$\Gamma _3$
is discrete. -
(b) The map
$\overline {\varphi }$
is clearly definable. A general fact about continuous group actions on topological spaces is that the projection map under a group action is an open map. Therefore the maps
$G_1\xrightarrow {\varphi } G_3\cong G_1/\ker \varphi $
,
$G_1\xrightarrow {\pi _1} G_1/M_1$
, and
$G_3\xrightarrow {\pi _3} G_3/M_3$
are continuous open maps. So given an open subset
$U\subseteq G_1/M_1$
, we have that
$\overline {\varphi }(U) = \pi _3\circ \varphi \circ \pi _1^{-1}(U)$
is open in
$G_3/M_3$
. This gives that
$\overline {\varphi }$
is open. -
(c) Since
$\varphi $
is continuous and
$M_1$
is compact,
$M_3$
is also compact. -
(d) Since
$\overline {\varphi }$
is definable and open, then so is
$\mathcal {F}_3$
. We also get that
$$\begin{align*}\bigcup_{\gamma\in\Gamma_3}\gamma\mathcal{F}_3 = \bigcup_{\gamma\in\Gamma_1}\varphi(\gamma)\overline{\varphi}(\mathcal{F}_1) = \overline{\varphi}(G_1/M_1) = G_3/M_3.\end{align*}$$
-
(e) We will now show that
$\mathcal {F}_3$
is a fundamental set for the action of
$\Gamma _3$
on
$G_3/M_3$
. For this, let
$\Xi $
be a finite subset of
$\Gamma _2$
such that
$a\mathcal {F}_3\subseteq \bigcup _{\xi \in \Xi }\xi \mathcal {F}_2$
, and let
$\Xi ^{-1}:=\{\xi ^{-1} : \xi \in \Xi \}$
. Since
$\mathcal {F}_2$
is a fundamental set for the action of
$\Gamma _2$
on
$G_2/M_2$
, there is a finite subset
$\Gamma _2'\subset \Gamma _2$
such that whenever
$\gamma y\in \mathcal {F}_2$
for some
$y\in \mathcal {F}_2$
and some
$\gamma \in \Gamma _2$
, then
$\gamma \in \Gamma _2'$
.Define
$\Gamma _3':= (a^{-1}\Xi \Gamma _2'\Xi ^{-1}a) \cap \Gamma _3$
, which is clearly a finite subset of
$\Gamma _3$
. Suppose now that
$x\in \mathcal {F}_3$
and
$\gamma \in \Gamma _3$
are such that
$\gamma x\in \mathcal {F}_3$
. Choose
$\xi _1,\xi _2\in \Xi $
such that
$ax\in \xi _1\mathcal {F}_2$
and
$a\gamma x\in \xi _2\mathcal {F}_2$
. Choosing
$\tilde {\gamma }\in \Gamma _2$
such that
$\gamma = a^{-1}\tilde {\gamma }a$
, we then get
$\tilde {\gamma }ax\in \xi _2\mathcal {F}_2$
. In other words,
$\xi _1^{-1}ax, \xi _2^{-1}\tilde {\gamma }ax\in \mathcal {F}_2$
. Since
$\xi _2^{-1}\tilde {\gamma }ax = (\xi _2^{-1}\tilde {\gamma }\xi _1)\xi _1^{-1}ax$
and
$\xi _2^{-1}\tilde {\gamma }\xi _1\in \Gamma _2$
, we conclude that
$\xi _2^{-1}\tilde {\gamma }\xi _1\in \Gamma _2'$
. Therefore,
$\tilde {\gamma }\in \Xi \Gamma _2'\Xi ^{-1}$
, which gives that
$\gamma \in \Gamma _3'$
. -
(f) We will now show that the closure of
$\mathcal {F}_3$
is contained in finitely many
$\Gamma _3$
-translates of
$\mathcal {F}_3$
. Let
$\Gamma _2"$
be a finite subset of
$\Gamma _2$
such that
$\overline {\mathcal {F}_2}\subseteq \bigcup _{\gamma \in \Gamma _2"}\gamma \mathcal {F}_2$
. Then, keeping the notation of
$\Xi $
and
$\Gamma _2'$
used above, we get
$$\begin{align*}a\overline{\mathcal{F}_3}\subseteq\bigcup_{\xi\in\Xi}\xi\overline{\mathcal{F}_2}\subseteq\bigcup_{\xi\in\Xi}\xi\left(\bigcup_{\gamma\in\Gamma_2"}\gamma\mathcal{F}_2\right).\end{align*}$$
Choose
$x\in \overline {\mathcal {F}_3}$
. Let
$\gamma _0\in \Gamma _3$
be such that
$\gamma _0 x\in \mathcal {F}_3$
. Let
$\xi _1,\xi _2\in \Xi $
and
$\gamma \in \Gamma _2"$
be such that
$a\gamma _0x\in \xi _1\mathcal {F}_2$
and
$ax\in \xi _2\gamma \mathcal {F}_2$
. Let
$\tilde {\gamma _0}\in \Gamma _2$
be such that
$\gamma _0 = a^{-1}\tilde {\gamma _0}a$
. We then get that
$\xi _1^{-1}\tilde {\gamma _0}ax,\gamma ^{-1}\xi _2^{-1}ax\in \mathcal {F}_2$
, and since
$\xi _1^{-1}\tilde {\gamma _0}ax = \left (\xi _1^{-1}\tilde {\gamma _0}\xi _2\gamma \right )\gamma ^{-1}\xi _2^{-1}ax$
, we conclude that
$\xi _1^{-1}\tilde {\gamma _0}\xi _2\gamma \in \Gamma _2'$
. Therefore We can therefore use the finite set
$$\begin{align*}\gamma_0 \in a^{-1}\Xi\Gamma_2'\left(\Gamma_2"\right)^{-1}\Xi^{-1}a.\end{align*}$$
$\Gamma _3":= a^{-1}\Xi \Gamma _2'\left (\Gamma _2"\right )^{-1}\Xi ^{-1}a\cap \Gamma _3$
to obtain that
$\overline {\mathcal {F}}_3\subseteq \bigcup _{\gamma \in \Gamma _3"}\gamma \mathcal {F}_3$
.
The observations above show that the data
$(G_3,M_3,\Gamma _3,\mathcal {F}_3)$
yield a definable quotient space. Set
$S_3 := S_{G_3,\Gamma_3,M_3; {\mathcal{F}}_3}$
and let
$q:S_1 \to S_3$
be the map
$\Gamma _1 x M_1 \mapsto \Gamma _3 \varphi (x) M_3$
. The map
$p:S_3 \to S_2$
is then given by
$\Gamma _3 x M_3 \mapsto \Gamma _2 a x M_2$
.
We now show that the fibers of p are compact. Given
$x\in G_3$
, the fiber over
$p(\Gamma _3 xM_3)$
is given by
$\left (a^{-1}\Gamma _2axM_2\right )\cap G_3$
. Observe that if
$\gamma _1,\gamma _2\in \Gamma _2$
are such that
$a^{-1}\gamma _1axM_2\cap a^{-1}\gamma _2 axM_2\neq \emptyset $
, then
$a^{-1}\gamma _1axM_2 = a^{-1}\gamma _2 axM_2$
. On the other hand, since p is a definable map, the fibers of p have finitely many connected components. Since
$M_2$
is compact, disjoint translates of
$M_2$
can be separated by Euclidean open sets. So there is a finite subset
$\Delta \subset \Gamma _2$
such that
$$\begin{align*}\left(a^{-1}\Gamma_2axM_2\right)\cap G_3 = \bigcup_{\gamma\in\Gamma_2}\left(a^{-1}\gamma axM_2\right)\cap G_3 = \bigcup_{\gamma\in\Delta}\left(a^{-1}\gamma axM_2\right)\cap G_3.\end{align*}$$
Thus the fibers of p are compact and in fact consist of a bounded number of homogeneous spaces for the compact group
$M_2 \cap G_3$
.
Lastly, we show that p is a closed map. Choose a closed subset
$K\subseteq S_3$
and let
$K'$
be the preimage of K in
$G_3$
, then
$p(K) = \Gamma _2aK'M_2$
. By the definition of the topology of the quotient space,
$p(K)$
is closed in
$S_2$
if and only if the product set
$K_2:=\Gamma _2aK'M_2$
is a closed subset of
$G_2$
. Let
$\{\gamma _iak_im_i\}_{i\in \mathbb {N}}$
denote a sequence of elements of
$K_2$
which converges to an element
$z\in G_2$
, where
$\gamma _i\in \Gamma _2$
,
$k_i\in K'$
and
$m_i\in M_2$
. To complete the proof we will show that
$z\in K_2$
. For this we first observe the following reductions.
-
(i) We may assume that each
$m_i$
is the identity. Indeed, since
$M_2$
is compact, the sequence
$\{m_i\}_{i\in \mathbb {N}}$
has a convergent subsequence, so by passing to this subsequence we may assume that
$m_i\to m$
, for some
$m\in M_2$
. Since
$K_2$
is invariant under multiplication on the right by
$M_2$
, we then have
$\gamma _iak_i\in K_2$
and
$$\begin{align*}\lim_{i\to\infty}\gamma_iak_i = \lim_{i\to\infty}\gamma_iak_im_im_i^{-1} = zm^{-1}.\end{align*}$$
-
(ii) We may assume that
$zM_2\in \mathcal {F}_2$
. Indeed, there exists
$\gamma \in \Gamma _2$
such that
$zM_2\in \gamma \mathcal {F}_2$
, so using that
$K_2$
is closed under multiplication on the left by
$\Gamma _2$
we obtain that
$\gamma ^{-1}\gamma _iak_i\in K_2$
. On the other hand and
$$\begin{align*}\lim_{i\to\infty}\gamma^{-1}\gamma_iak_i = \gamma^{-1}z,\end{align*}$$
$\gamma ^{-1}zM_2\in \mathcal {F}_2$
.
-
(iii) We may assume that
$k_iM_3\in \mathcal {F}_3$
for all
$i\in \mathbb {N}$
. To see this, for every
$i\in \mathbb {N}$
choose
$\delta _i\in \Gamma _3$
and
$x_i\in G_3$
such that
$k_i = \delta _i x_i$
and
$x_iM_3\in \mathcal {F}_3$
. Then where
$$\begin{align*}\gamma_iak_i = \gamma_ia\delta_ix_i=\gamma_ia\delta_ia^{-1}ax_i = \gamma_i'ax_i,\end{align*}$$
$\gamma _i' := \gamma _i a \delta _ia^{-1}$
which belongs to Γ2 as aδ
i
a
−1 ∈ aΓ3
a
−1 ≤Γ2 and γ
i
∈Γ2.
Let
$U\subseteq \mathcal {F}_2$
be an open neighborhood of
$zM_2$
and let
$N\in \mathbb {N}$
be such that
$\gamma _iak_iM_2\in U$
for all
$i>N$
. As we saw in item (f) above, there exists a finite set
$\Pi _0\subseteq \Gamma _2$
such that
$$\begin{align*}a\overline{\mathcal{F}_3}\subseteq\bigcup_{\xi\in\Pi_0}\xi\mathcal{F}_2.\end{align*}$$
Since we have reduced to the case that k i M 3 ∈F3 for all i, restricting to a subsequence, we may suppose that there is one ξ 0 ∈Π0 so that the image ak i M 2 of ak i M 3 lies in ξ 0F2 for all i.
For every i > N we have that γ
i
ak
i
M
2 ∈ U ⊆F2 and ξ
0
−1
ak
i
M
2 ∈F2. In particular, γ
i
ξ
0F2 ∩F2≠∅. By the definition of fundamental domain,
$ \mathcal {F}_2$
has nonempty intersection with only finitely many of its
$\Gamma _2$
-translates. Thus, γ
i
takes only finitely many values, and restricting to a subsequence again, we may assume that the whole sequence is constant, say
$\gamma _i=\gamma _0$
. This then gives that the sequence
$\gamma _0ak_i$
converges to z. Since K is closed in
$S_3$
, then
$K'$
is closed in
$G_3$
and we conclude that
$z\in \gamma _0a K'\subseteq K_2$
.
For the problems we consider in this paper, we require that our definable quotient spaces come equipped with a complex structure and for the domain D to arise as a subset of an algebraic variety.
Definition 2.6. A definable complex quotient space is a definable quotient space
$S_{\Gamma ,G,M;\mathcal {F}}$
together with the data of a complex algebraic group
$\mathbf {G}$
and an algebraic subgroup
$\mathbf {B} \leq \mathbf {G}$
of
$\mathbf {G}$
for which G is a definable subgroup of
$\mathbf {G}(\mathbb {C})$
,
$M = \mathbf {B}(\mathbb {C}) \cap G$
, and
$D = G/M \subseteq (\mathbf {G}/\mathbf {B})(\mathbb {C}) =: \check {D}(\mathbb {C})$
is an open domain in the complex points of the algebraic variety
$\check {D}$
.
A morphism
$f:S_{\Gamma _1,G_1,M_1;\mathcal {F}_1} \to S_{\Gamma _2,G_2,M_2;\mathcal {F}_2}$
of definable complex quotient spaces is a morphism of definable quotient spaces for which the definable map of groups is given by a map of algebraic groups
$\varphi :\mathbf {G}_1 \to \mathbf {G}_2$
for which
$\varphi (\mathbf {B}_1) \leq \mathbf {B}_2$
.
The class of definable complex quotient spaces with this notion of morphism forms a category
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
.
We leave it to the reader to check that the proofs of the basic closure properties for the category
$\mathsf {D}\mathsf {Q}\mathsf {S}$
, such as the existence of a terminal object, go through for the category
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
. In practice, the morphisms in
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
we consider satisfy a stronger conclusion than what Proposition 2.5 gives. That is, in practice, a map
$f:S_{G_1,\Gamma _1,M_1;\mathcal {F}_1} \to S_{G_2,\Gamma _2,M_2;\mathcal {F}_2}$
in
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
is given by a map of algebraic groups
$\varphi :\mathbf {G}_1 \to \mathbf {G}_2$
(and an element
$a \in G_2$
) for which
$M_1$
is a finite index subgroup of
$\varphi ^{-1} (M_2)$
. It then follows from the proof of Proposition 2.5 that f factors as
$f = p \circ q$
where
$q:S_1 \to S_3$
is a surjective
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
morphism and
$p:S_3 \to S_2$
is a
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
morphism with finite fibers.
As we have defined definable complex quotient spaces, such a space
$S_{G,\Gamma ,M;\mathcal {F}}$
may have singularities. Since M is compact, the singularities are at worst locally isomorphic to those coming from a quotient by a finite group. In our applications, we will consider only cases where these quotients may be desingularized by passing to a finite cover by another definable complex quotient space.
For some purposes we may wish to restrict to an even smaller category
$\mathsf {S}$
. We always assume about our given category
$\mathsf {S}$
of definable complex quotient spaces that it satisfies some basic closure properties. Let us specify these with the following convention.
Convention 2.7. The category
$\mathsf {S}$
is a subcategory of
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
satisfying the following conditions.
-
• The one-point space
$\{ \ast \}$
is a terminal object of
$\mathsf {S}$
. -
• Every
$\mathsf {S}$
-morphism
$f:S_1 \to S_2$
factors as
$f = p \circ q$
where
$q:S_1 \to S_3$
is a surjective
$\mathsf {S}$
-morphism and
$p:S_3 \to S_2$
is a proper
$\mathsf {S}$
-morphism with finite fibers.
In Section 3 we will impose an additional restriction on
$\mathsf {S}$
.
Definition 2.8. If
$f:S_1 \to S_2$
is an
$\mathsf {S}$
-morphism, then the image
$f(S_1)$
is called an
$\mathsf {S}$
-special subvariety of
$S_2$
.
In Definition 2.8 we refer to
$f(S_1)$
as a special subvariety. It is, in fact, always a complex analytic subvariety of
$S_2$
. Indeed, if we factor
$f = p \circ q$
as given by Convention 2.7, then by Remmert’s proper mapping theorem, the image
$p(S_3) = f(S_1)$
is a complex analytic subvariety of
$S_2$
.
With Convention 2.7 in place, we may modify Definition 2.8 to require the morphism
$f:S_1 \to S_2$
witnessing that
$f(S_1)$
is an
$\mathsf {S}$
-special subvariety of
$S_2$
to be a proper S-morphism with finite fibers.
Definition 2.9. An
$\mathsf {S}$
-family of weakly special subvarieties of
$S \in \mathsf {S}$
is given by a pair of
$\mathsf {S}$
-morphisms
for which
$\zeta $
is a map with finite fibers over its image and
$\xi $
is surjective. For each
$b \in S_2$
, the image
$\zeta \xi ^{-1} \{ b \}$
is a weakly
$\mathsf {S}$
-special subvariety of S.
Remark 2.10. An
$\mathsf {S}$
-special variety is weakly
$\mathsf {S}$
-special as if
$f:S_1 \to S$
expresses
$f(S_1)$
as an
$\mathsf {S}$
-special subvariety of S with f finite, then we can take
$\zeta = f$
,
$S_2 = \{ \ast \}$
,
$\xi :S_1 \to S_2$
the unique map to
$\{ \ast \}$
. If we take
$\mathsf {S}$
to be the category
$\mathsf {D}\mathbb {C}\mathsf {Q}\mathsf {S}$
, then the converse that every weakly
$\mathsf {S}$
-special variety is actually
$\mathsf {S}$
-special holds.
In another extreme, every singleton in
$S \in \mathsf {S}$
is a weakly special variety witnessed by
$S=S_1 = S_2$
and
$\xi = \zeta = \operatorname {id}_S$
.
For our results it will be important that all
$\mathsf {S}$
-weakly special varieties come from countably many
$\mathsf {S}$
-families of weakly special subvarieties. We isolate this as an hypothesis and verify this condition in cases of interest.
Definition 2.11. We say that the weakly
$\mathsf {S}$
-special subvarieties of
$S \in \mathsf {S}$
are well-parameterized if there are countably many
$\mathsf {S}$
-families of weakly special subvarieties of S,
for
$i \in \mathbb {N}$
, so that for every weakly
$\mathsf {S}$
-special subvariety
$S' \subseteq S$
of S there is some
$i \in \mathbb {N}$
and
$c \in S_{2,i}$
so that
$S' = \zeta _{i} \xi _{i}^{-1} \{ c \}$
. More generally, we say that the weakly
$\mathsf {S}$
-special subvarieties are well-parameterized if for every
$S \in \mathsf {S}$
the weakly
$\mathsf {S}$
-special subvarieties of S are well-parameterized.
Remark 2.12. When
$\mathsf {S}$
is itself countable, by which we mean that there are only countably many objects in
$\mathsf {S}$
and the set of
$\mathsf {S}$
-morphisms between any two such objects is itself countable, then the weakly
$\mathsf {S}$
-special subvarieties are well-parameterized.
Remark 2.13. The well-parameterization condition may fail in some cases. From the theory of Douady spaces, we know that all complex analytic subvarieties of a given compact definable complex quotient space S are parameterized by countably many complex analytic families of analytic spaces. However, when parameterizing weakly special varieties, the Douady universal family need not arise as a weakly special family as defined in Definition 2.9. Moreover, there need not be natural parameterizations of the weakly special varieties in the cases that S is noncompact. The Hopf manifold construction may be instructive here. For example, consider p and q two multiplicatively independent complex numbers of modulus less than one,
$X := \mathbb {C}^2 \smallsetminus \{ (0,0) \}$
,
$\Gamma := \langle (p,q) \rangle $
the subgroup of
$\operatorname {Aut}(X)$
generated by the map
$(x,y) \mapsto (px, qy)$
, and
$M := \Gamma \backslash X$
. The Hopf manifold M may be realized as a definable complex quotient space and is a compact complex manifold with a noncompact, connected group of automorphisms. One by one, these automorphisms define special subvarieties of
$M \times M$
, but they cannot be parameterized by a family of weakly special varieties.
Let us indicate now the key functional transcendence condition which may hold in a category
$\mathsf {S}$
of definable complex quotient spaces.
Definition 2.14. Fix a category
$\mathsf {S}$
of definable complex quotient spaces. We say that
$f:X^{\text {an}} \to S' \subseteq S_{\Gamma ,G,M;\mathcal {F}} \in \mathsf {S}$
, a definable complex analytic map from the analytification of a complex algebraic variety X to a weakly
$\mathsf {S}$
-special variety
$S' \subseteq S$
satisfies the Ax-Schanuel condition relative to
$\mathsf {S}$
if whenever
$k \in \mathbb {Z}_+$
is a positive integer, and
$(\gamma ,\widetilde {\gamma }):\Delta ^k \to X^{\text {an}} \times D$
is a complex analytic map, where
$$\begin{align*}\Delta = \{ z \in \mathbb{C} : \| z \| < 1 \} \text{ ,}\end{align*}$$
with
$\pi _\Gamma \circ \widetilde {\gamma }= f \circ \gamma $
, then either
$$\begin{align*}\operatorname{tr.deg}_{\mathbb{C}}(\mathbb{C}(\gamma,\widetilde{\gamma})) \geq \dim S' + \operatorname{rk}(d\widetilde{\gamma}) \end{align*}$$
or
$f(\gamma (\Delta ^k))$
is contained in a proper weakly
$\mathsf {S}$
-special subvariety of
$S'$
.
Under the hypothesis that the weakly
$\mathsf {S}$
-special subvarieties are well-parameterized, the Ax-Schanuel condition implies a uniform version of itself, expressed as Proposition 2.19.
Since our statement of this uniform version is a bit dense, we take this opportunity to explain it with a few words. Basically, what it says is that if we are given an
$\mathsf {S}$
-family of weakly special varieties and a family of algebraic varieties which might witness the failure of the transcendence degree lower bound in the Ax-Schanuel property, the weakly special variety in the alternative provided by the Ax-Schanuel property may be chosen from one of finitely many preassigned
$\mathsf {S}$
-families of weakly special varieties.
Before proving Proposition 2.19 we require two lemmas. The first describes families of weakly special subvarieties algebraically. The second allows us to recast Ax-Schanuel in differential algebraic terms.
Lemma 2.15. Let
$f:X^{\text {an}} \to S_{\Gamma ,G,M;\mathcal {F}} =: S \in \mathsf {S}$
be a definable complex analytic map from a complex algebraic variety to a definable complex quotient space in
$\mathsf {S}$
. Let
be an
$\mathsf {S}$
-family of weakly special subvarieties. Then there are algebraically constructible sets B and
$T \subseteq X \times B$
so that the set of fibers
$\{ T_b : b \in B(\mathbb {C}) \}$
is equal to
$\{ f^{-1} \zeta \xi ^{-1} \{ c \} : c \in S_2 \}$
.
Proof. The fiber product
$X^{\text {an}} \times _S S_1$
is a definable complex analytic variety and the map
$\zeta ':X^{\text {an}} \times _S S_1 \to X^{\text {an}}$
is a definable, finite, holomorphic map. As such, the ramification locus
$D \subseteq X$
of
$\zeta '$
is a definable, complex analytic subvariety of X. By the Peterzil-Starchenko definable Chow theorem [Reference Peterzil and Starchenko25], D is an algebraic subvariety of X. By Noetherian induction, the family of fibers
$\{ f\vert _{D}^{-1} \zeta \xi ^{-1} \{ c \} : c \in S_2 \}$
is represented by a constructible family. Thus, it suffices to replace X by
$X \smallsetminus D$
so that we may assume that
$\zeta ':X^{\text {an}} \times _S S_1 \to X^{\text {an}}$
is unramified and has finite fibers.
By the Riemann Existence Theorem [Reference Raynaud29, Théorème 5.1], there is an algebraic variety
$X'$
, a regular map of algebraic varieties
$\zeta ':X' \to X$
and an analytic map
$f':(X')^{\text {an}} \to S_1$
realizing
$(X')^{\text {an}}$
as the fiber product
$X^{\text {an}} \times _S S_1$
. The fiber equivalence relation
$$\begin{align*}E_{\xi \circ f'} := \{ (x,y) \in X' \times X' : \xi (f'(x)) = \xi(f'(y)) \} \end{align*}$$
is analytic and definable, and hence algebraic by the definable Chow theorem. The quotient
$B := X'/E_{\xi \circ f'}$
may be realized within the category of constructible sets as a constructible set [Reference Poizat28, Lemme 2 and Théorème 7]. Let us write
$\nu :X' \to B$
for the quotient map. We may then take
$$\begin{align*}T := \{ (x,b) \in X \times B : (\exists x' \in X') \zeta'(x') = x \text{ and } \nu(x') = b \}.\\[-37pt] \end{align*}$$
Definition 2.16. Let
$f:X^{\text {an}} \to S \in \mathsf {S}$
be a definable complex analytic map from the complex algebraic variety X to the definable complex quotient space S in
$\mathsf {S}$
. We say that a subvariety
$Y \subseteq X$
is relatively weakly
$\mathsf {S}$
-special of relative dimension at most d if there is a weakly
$\mathsf {S}$
-special
$S' \subseteq S$
of dimension at most d for which
$Y = f^{-1} S'$
.
Note that in Definition 2.16, because we allow for the possibility that the map f does not have finite fibers, it could happen that the dimension of Y itself is greater than d. On the other hand, the intersection of
$S'$
with
$f(X)$
may even be empty! Thus, the dimension of Y could be less than d.
It follows from Lemma 2.15 that if the weakly
$\mathsf {S}$
-special subvarieties are well-parameterized, then for any definable complex analytic map
$f:X^{\text {an}} \to S \in \mathsf {S}$
from a complex algebraic variety X to some definable complex quotient space S in
$\mathsf {S}$
we can recognize the pullbacks under f of weakly special varieties, in the sense that for each number d the collection of relatively weakly
$\mathsf {S}$
-special subvarieties of dimension at most d comprise a countable collection of algebraic families of subvarieties of X.
Definition 2.17. Let
$f:X^{\text {an}} \to S \in \mathsf {S}$
be a definable complex analytic map from the complex algebraic variety X to a definable complex quotient space S in
$\mathsf {S}$
. Given any field M over which X and a countable collection of families of relatively weakly
$\mathsf {S}$
-special subvarieties of X including all such relatively weakly
$\mathsf {S}$
-special subvarieties are defined, by an M-relatively weakly
$\mathsf {S}$
-special variety of dimension at most d we mean an M-variety of the form
$Y_b$
where
$Y \subseteq X \times B$
is an algebraic family of relatively weakly
$\mathsf {S}$
-special subvarieties of dimension at most d and
$b \in B(M)$
.
Using the Seidenberg embedding theorem we may reformulate the Ax-Schanuel property in differential algebraic terms. To be completely honest, the embedding theorem as stated and proven by Seidenberg in [Reference Seidenberg31, Reference Seidenberg and Seidenberg32] is not quite sufficient in that he starts with a finitely generated differential subfield
$K \subseteq \mathcal {M}(U)$
of a differential field of meromorphic functions on the open domain
$U \subseteq \mathbb {C}^n$
and then shows that for any finitely generated differential field extension L of K at the cost of shrinking U to some open subdomain
$V \subseteq U$
we may embed L into
$\mathcal {M}(V)$
over the embedding of K. For our purposes, we will need to start with a possibly countably generated differential field
$K \subseteq \mathcal {M}(U)$
. The necessary extension of embeddings theorem is a consequence of the the Cauchy-Kovalevskaya theorem [Reference von Kowalevsky34] and appears as Theorem 1 of [Reference Pavlov, Pogudin and Razmyslov24]. Iterating this construction countably many steps, we see that if
$K \subseteq \mathcal {M}(U)$
is a countable differential subfield of the meromorphic functions on some open domain in
$\mathbb {C}^n$
and L is a countably generated differential field extension of K, then L embeds into the differential field of germs of meromorphic functions at some point
$x \in U$
over the embedding of K.
We recall the generalized Schwartzian and generalized logarithmic derivative constructions from [Reference Scanlon30]. Consider
$S = S_{G,\Gamma ,M;\mathcal {F}}$
a definable complex quotient space and fix an integer k. From the action
$\mathbf {G} \curvearrowright \check {D}$
of the algebraic group
$\mathbf {G}$
on the quasiprojective algebraic variety
$\check {D}$
and a positive integer k, there is a differentially constructible map
$\widetilde {\chi }:\check {D} \to Z$
from
$\check {D}$
to some algebraic variety Z so that for any differential field
$(L,\partial _1,\ldots ,\partial _k)$
extending
$\mathbb {C}$
(where the derivations
$\partial _i$
commute and vanish on
$\mathbb {C})$
we have that for
$x, y \in \check {D}(L)$
,
$$\begin{align*}\widetilde{\chi}(x) = \widetilde{\chi}(y) \Longleftrightarrow (\exists g \in \mathbf{G}(C) ) g x = y \end{align*}$$
where
$$\begin{align*}C = \{ a \in L : \partial_i(a) = 0 \text{ for } 1 \leq i \leq k \} \end{align*}$$
is the common constant field of L. In particular, if this common constant field is
$\mathbb {C}$
, then we may express the quotient of
$\check {D}(L)$
by
$\mathbf {G}(\mathbb {C})$
as the image of
$\widetilde {\chi }$
. When
$f:X^{\text {an}} \to S$
is a definable, complex analytic map from the analytification of a quasiprojective algebraic variety X to S, then we may define a differentially analytically constructible function
$\chi :X^{\text {an}} \to Z$
by the rule that for any meromorphic
$\gamma :U \to X^{\text {an}}$
(where
$U \subseteq \mathbb {C}^k$
is an open domain in
$\mathbb {C}^k$
),
$\chi (\gamma ) := \widetilde {\chi } (\pi _\Gamma ^{-1} (f(\gamma ))$
where
$\pi _\Gamma ^{-1}$
is any branch of the inverse of
$\pi _\Gamma $
. Theorem 3.12 of [Reference Scanlon30] shows that
$\chi $
is actually differentially constructible. (That theorem is stated in the case where
$X^{\text {an}} = S$
and
$f = \operatorname {id}_S$
, but the proof goes through in the more general case.)
Lemma 2.18. Fix
$\mathsf {S}$
a category of definable complex quotient spaces. Suppose that the weakly
$\mathsf {S}$
-special subvarieties are well-parameterized and that
$f:X^{\text {an}} \to S' \subseteq S_{\Gamma ,G,M;\mathcal {F}} \in \mathsf {S}$
is a definable complex analytic map from the analytification of a complex algebraic variety X to an
$\mathsf {S}$
-weakly special subvariety of a definable complex quotient space in
$\mathsf {S}$
, satisfying the Ax-Schanuel condition. Let K be a countable subfield of
$\mathbb {C}$
over which X and a complete collection of algebraic families of relatively weakly
$\mathsf {S}$
-special varieties are defined. Let k and d be two positive integers.
Then for any differential field
$(L,\delta _1, \ldots , \delta _k)$
with k-commuting derivations for which K is a subfield of the constants C of L and C is algebraically closed and any C-relatively weakly
$\mathsf {S}$
-special Y of dimension at most d, if
$(\gamma ,\widetilde {\gamma }) \in Y(L) \times \check {D}(L)$
satisfies
$\chi (\gamma ) = \widetilde {\chi }(\widetilde {\gamma })$
,
$\operatorname {rk}( \left ( \delta _i \gamma \right )_{1 \leq i \leq k}) = k$
, and
$\operatorname {tr.deg}_C C(\gamma ,\widetilde {\gamma }) < d + k$
, then there is a C-relatively weakly
$\mathsf {S}$
-special
$Z \subsetneq Y$
for which
$\gamma \in Z(L)$
.
Proof. Consider Y and
$(\gamma ,\widetilde {\gamma }) \in Y(L)$
as in the statement of the lemma. Let M be a countable differential subfield of L with an algebraically closed field of constants
$C'$
containing K and over which Y and the point
$(\gamma ,\widetilde {\gamma })$
are defined. By the embedding theorem, we may realize M as a differential field of germs of meromorphic functions. Let
$S' \subseteq S$
be the
$\mathsf {S}$
-weakly special variety of dimension at most d for which
$Y = f^{-1} S'$
. By Ax-Schanuel applied to
$\gamma $
and
$\widetilde {\gamma }$
regarded as meromorphic functions, there is a proper weakly
$\mathsf {S}$
-special subvariety
$S" \subsetneq S'$
with the image of
$f \circ \gamma $
contained in
$S"$
. The algebraic variety
$Z := f^{-1} S"$
is then relatively
$\mathbb {C}$
-weakly special. Let
$M' := M^{\text {dc}}$
be the differential closure of M and
$M" := M(\mathbb {C})^{\text {dc}}$
be the differential closure of the differential field generated over M by
$\mathbb {C}$
. In
$M"$
,
$\gamma $
satisfies the condition that it belongs to a
$\mathsf {C}$
-relatively weakly special variety of relative dimension strictly less than d where
$\mathsf {C}$
is the constant field. As
$M"$
is an elementary extension of
$M'$
, the same is true in
$M'$
[Reference Marker20, page 53]. Since the constant field of the differential closure is the algebraic closure of the constant field of the initial field, we see that
$\gamma $
belongs to a C-relatively weakly special variety of relative dimension strictly less than d.
A uniform version of the Ax-Schanuel condition follows from Lemma 2.15 using the compactness theorem.
Proposition 2.19. Let
$f:X^{\text {an}} \to S' \subseteq S_{\Gamma ,G,M;\mathcal {F}} \in \mathsf {S}$
be a definable complex analytic map from the analytification of a complex algebraic variety X to a definable complex quotient space satisfying the same hypotheses as in Lemma 2.18.
Given an
$\mathsf {S}$
-family
of weakly
$\mathsf {S}$
-special subvarieties of S, a positive integer
$k \in \mathbb {Z}_+$
, and a family
$Y \subseteq (X \times \check {D}) \times B$
of subvarieties of
$X \times \check {D}$
, then there are finitely many
$\mathsf {S}$
-families of weakly special subvarieties
for
$1 \leq i \leq n$
so that for any pair of parameters
$b \in B$
and
$c \in S_2$
and analytic map
$(\gamma ,\widetilde {\gamma }):\Delta ^k \to Y_b \subseteq X \times D$
with
$\gamma (\Delta ^k) \subseteq \zeta \xi ^{-1} \{ c \} =: S^{\prime }_c$
,
$f \circ \gamma = \pi _\Gamma \circ \widetilde {\gamma }$
,
$\operatorname {rk}(d \widetilde {\gamma }) = k$
, and
$\dim Y_b < k + \dim S_c'$
, there is some
$i \leq n$
and
$d \in S_{2,i}$
for which
$f \circ \gamma (\Delta ^k) \subseteq \zeta _i \xi _i^{-1} \{ d \} \subsetneq S^{\prime }_c$
.
Proof. Apply the compactness theorem to Lemma 2.15. See the proofs of [Reference Aslanyan2, Theorem 3.5] or [Reference Kirby18, Theorem 4.3] for details on how to formalize the compactness argument.
Remark 2.20. Proposition 2.19 will play a crucial role in the proof of our main Theorem 3.3. When applied, we will be considering instances of the failure of the Ax-Schanuel dimension lower bound in families of weakly special subvarieties. In order to carry out our inductive argument, we need to know that the proper weakly special varieties which account for a failure of the lower bound on the dimension of intersections predicted by Ax-Schanuel vary uniformly.
3 Density of special intersections
In this section we state and prove our general theorem that, when persistently likely, intersections with special varieties are dense.
Throughout this section
$\mathsf {S}$
is a category of definable complex quotient spaces satisfying our usual hypotheses from Convention 2.7 and some further requirements. Let us specify with the following convention the properties we require.
Convention 3.1. The category
$\mathsf {S}$
of definable complex quotient spaces satisfies the following conditions.
-
• Each weakly
$\mathsf {S}$
-special variety is definably isomorphic to a finite union of spaces in
$\mathsf {S}$
via maps of definable quotient spaces. -
• The terminal definable complex quotient (one point) space
$\{ \ast \}$
belongs to
$\mathsf {S}$
as do the unique maps
$S \to \{ \ast \}$
for
$S \in \mathsf {S}$
. -
• If
$f:S_1 \to S_2$
is an
$\mathsf {S}$
-morphism, then there are
$\mathsf {S}$
-morphisms
$q:S_1 \to S_3$
and
$p:S_3 \to S_2$
so that
$f = p \circ q$
, q is surjective, and p has finite fibers. -
• For every
$S \in \mathsf {S}$
there is some smooth
$S' \in \mathsf {S}$
and a finite surjective
$\mathsf {S}$
-morphism
$S' \to S$
. -
• The
$\mathsf {S}$
-weakly special varieties are well-parameterized. -
• Every definable analytic map
$f:X^{\text {an}} \to S \in \mathsf {S}$
from the analytification of an algebraic variety to a definable complex quotient space in
$\mathsf {S}$
considered in this section satisfies the Ax-Schanuel condition relative to
$\mathsf {S}$
.
With the following definition we specify what is meant by intersections being persistently likely.
Definition 3.2. Let
$S = S_{G,\Gamma ,M;\mathcal {F}} \in \mathsf {S}$
and let
$D' \subseteq D = G/M$
be a homogeneous space for some definable subgroup
$G' \leq G$
. Let
$S' := \pi (D')$
. Let
$f:X^{\text {an}} \to S$
be a complex analytic map from the analytification of an algebraic variety X to S. We say that X has likely intersection with
$S'$
if
$\dim f(X) + \dim S' \geq \dim S$
, where here, the dimension is the local o-minimal dimension. That is,
$S'$
is a possibly countable union of definable sets and
$\dim S'$
is the max of the dimensions of the definable subsets of
$S'$
. We say the intersection is persistently likely if whenever
$\zeta :S_1 \to S$
and
$\xi :S_1 \to S_2$
are surjective
$\mathsf {S}$
-morphisms with
$\zeta $
a map with finite fibers, then
$\dim \xi (\zeta ^{-1} f(X)) + \dim \xi (\zeta ^{-1} S') \geq \dim S_2$
. We say that X has likely, respectively, persistently likely, intersection with the family
$\{ \pi (gD') \}_{g \in G}$
if there is a definable subset
$Q\subset G$
(possibly empty)Footnote
2
with
$\dim Q < \dim G$
such that for every
$g \in G\setminus Q$
the variety X has likely, respectively, persistently likely, intersection with
$\pi (gD')$
.
Note first that the definition of likely intersection does not require
$f(X)\cap S'$
to be nonempty for X and S to have likely intersection. We also note that while the intersection of X and
$S'$
may be persistently likely, this does not imply that X has persistently likely intersection with
$\{ \pi (gD') \}_{g \in G}$
. However, one can check that in many cases of interest, such as when S is an abelian variety or a power of the modular curve, the implication holds.
With these definitions in place we may now state our main theorem.
Theorem 3.3. Let
$S = S_{G,\Gamma ,M;\mathcal {F}} \in \mathsf {S}$
and let
$f:X^{\text {an}} \to S$
be a definable complex analytic map from the irreducible quasi-projective complex algebraic variety X to S. Let
$D' \subseteq D = G/M$
be a homogeneous space for some definable subgroup
$G' \leq G$
.
Suppose that the intersection of X with the family
$\{ \pi (gD') \}_{g \in G}$
is persistently likely. Let
$U \subseteq X(\mathbb {C})$
be an open subset of the complex points of X. Then the set
$B := \{ g \in G : f(U) \cap \pi _\Gamma (gD' \cap \mathcal {F}) \neq \varnothing \}$
has nontrivial interior.
In particular, by unwinding the meaning of density, one sees that if the set
$\{ g \in G : \pi _\Gamma (gD') \text {is a special subvariety of } S \}$
is Euclidean dense in G, then the set of special intersections,
$$\begin{align*}f^{-1} \bigcup_{g \in G, \pi_\Gamma(g D') \text{ is special}} \pi_\Gamma(g D') \end{align*}$$
is dense in X for the Euclidean topology.
Proof. We break the proof of Theorem 3.3 into several claims. The claims at the beginning of the proof are really just reductions permitting us to consider a simpler situation. The main steps of the proof begin with Claim 3.3.9 in which we compute the dimension of the incidence correspondence R. We then use this computation to show that B and G have the same o-minimal dimension, so that B has nontrivial interior in G.
Claim 3.3.1. We may assume that
$f:X^{\text {an}} \to S$
is an embedding.
Proof of Claim:
The equivalence relation
$E_f := \{ (x,y) \in X \times X : f(x) = f(y) \}$
is a definable and complex analytic subset of the quasi-projective algebraic variety
$X \times X$
. Hence, by the definable Chow theorem,
$E_f$
is itself algebraic. Let Y be a nonempty Zariski open subset of
$X/E_f$
, considered as a constructible set. Then f induces an embedding
$\overline {f}:Y \hookrightarrow S$
whose image is dense in
$f(X)$
. Shrinking U, we may assume that
$f(U) \subseteq \overline {f}(Y)$
and then replacing U by
$U' := \overline {f}^{-1} f(U) \subseteq Y(\mathbb {C})$
, we see that if the theorem holds for
$\overline {f}:Y^{\text {an}} \to S$
and
$U'$
, then it also holds for
$f:X^{\text {an}} \to S$
and U.
With Claim 3.3.1 in place, from now on we will regard X as a locally closed subvariety of S. With the next claim we record the simple observations that it suffices to prove the theorem for any given open subset of U in place of U and that we may take U to be definable.
Claim 3.3.2. If Theorem 3.3 holds for some nonempty open
$V \subseteq U$
in place of U, then it holds as stated. Moreover, we may assume that U is definable.
Proof of Claim:
The set
$\{ g \in G : V \cap \pi (g D' \cap \mathcal {F}) \neq \varnothing \}$
is a subset of
$\{ g \in G : U \cap \pi (g D' \cap \mathcal {F}) \neq \varnothing \}$
. Hence, if the former set has nonempty interior, so does the latter. For the “moreover” clause apply the main body of the claim to the case that
$V \subseteq U$
is a nonempty open ball.
From now on we will take U to be definable and will continue to refer to the open subset of X under consideration as U even after taking various steps to shrink it.
Another basic reduction we shall employ is that it suffices to prove the theorem for a finite cover of S.
Claim 3.3.3. If
$\rho :\widetilde {S} \to S$
is a surjective
$\mathsf {S}$
-morphism with finite fibers, then we may find an instance of the statement of Theorem 3.3 with
$\widetilde {S}$
in place of S so that the truth of Theorem 3.3 for
$\widetilde {S}$
implies the result of S.
Proof of Claim:
Filling the Cartesian square
we obtain a complex analytic space
$Y := X^{\text {an}} \times _S \widetilde {S}$
. Since
$\overline {\rho }:Y \to X^{\text {an}}$
is finite, Y is itself the analytification of an algebraic variety. Let
$\widetilde {X}$
be an irreducible component of this algebraic variety and then let
$\widetilde {U} := \overline {\rho }^{-1} U$
.
The map
$\rho :\widetilde {S} \to S$
comes from a homomorphism of algebraic groups
$\varphi :\widetilde {\mathbf {G}} \to \mathbf {G}$
and some element
$a \in G$
where
$\widetilde {S} = S_{\widetilde {G},\widetilde {\Gamma },\widetilde {M};\widetilde {\mathcal {F}}}$
. This map induces a map
$\widehat {\rho }:\widetilde {G}/\widetilde {M} = : \widetilde {D} \to D$
. Let
$\widetilde {D}'$
be a component of
$\widehat {\rho }^{-1} D'$
. If we succeed in showing that
$\{ g \in G' : \overline {f}(\widetilde {U}) \cap \pi _{\widetilde {\Gamma }} (g \widetilde {D}' \cap \widetilde {\mathcal {F}}) \}$
contains some nonempty open set V, then
$a \varphi (V)$
would be a nonempty open subset of
$\{ g \in G : f(U) \cap \pi _\Gamma (gD' \cap \mathcal {F}) \}$
, as required.
Let us record a useful consequence of Claim 3.3.3.
Claim 3.3.4. We may assume that S is smooth.
Proof of Claim:
By Convention 3.1, we may find a surjective
$\mathsf {S}$
-morphism
$\widetilde {S} \to S$
with finite fibers with
$\widetilde {S}$
smooth. By Claim 3.3.3, if we know the theorem for
$\widetilde {S}$
, then we may deduce it for S.
Another useful consequence of Claim 3.3.3 is that we may assume that
$f(X)$
is not contained in any proper weakly special subvariety of S.
Claim 3.3.5. We may assume that there is no proper weakly special variety
$S" \subsetneq S$
with
$f(X) \subseteq S"$
.
Proof of Claim:
Let us prove Theorem 3.3 by induction on the dimension of S. If
$f(X) \subseteq S" \subsetneq S$
where
$S"$
is a weakly special variety, then we could find an
$\mathsf {S}$
-morphism
$\zeta :S_1 \to S$
with finite fibers, a surjective
$\mathsf {S}$
-morphism
$\xi :S_1 \to S_2$
, and a point
$b \in S_2$
so that
$f(X) \subseteq \zeta (\xi ^{-1} \{ b \})$
. By Claim 3.3.3, we may assume that
$S_1 = S$
and
$\zeta = \operatorname {id}_S$
. That is,
$f(X) \subseteq \xi ^{-1} \{ b \} =: S_b \subsetneq S$
. By Convention 3.1, the weakly special variety
$S_b$
is definably isomorphic to a space in
$\mathsf {S}$
. By induction on dimension, Theorem 3.3 already holds for
$S_b$
.
By our hypothesis that the intersection between X and
$S'$
is persistently likely, it is, in particular, likely. Since we have reduced to the case that
$f:X^{\text {an}} \to S$
is an embedding by Claim 3.3.1, we may express the likeliness of this intersection by an equation
$$\begin{align*}\dim_{\mathbb{C}} (X) + \dim_{\mathbb{C}} (S') = \dim_{\mathbb{C}}(S) + k \end{align*}$$
for some nonnegative integer k. Notice that we have expressed this equality with dimensions as complex analytic spaces. Later, when we write “
$\dim $
” without qualification we mean the o-minimal dimension, for which we would have
$$\begin{align*}\dim X + \dim S' = \dim S + 2k . \end{align*}$$
By the uniform Ax-Schanuel condition, which holds in
$\mathsf {S}$
by Convention 3.1 and Proposition 2.19, there is a finite list of families of weakly special varieties
for
$1 \leq i \leq n$
where
$\zeta _i:S_{1,i} \to S$
is an
$\mathsf {S}$
-morphism with finite fibers and
$\xi _i:S_{1,i} \to S_{2,i}$
is a surjective
$\mathsf {S}$
-morphism and if
$\ell $
is a natural number with
$(\gamma ,\widetilde {\gamma }):\Delta ^{\ell } \to U \times D$
complex analytic with
$\pi _\Gamma \circ \widetilde {\gamma } = g \circ \gamma $
,
$\operatorname {rk}(d \gamma ) = \ell> k$
, and
$\widetilde {\gamma }(\Delta ^\ell ) \subseteq g D'$
for some
$g \in G$
, then for some
$i \leq n$
and
$b \in S_{2,i}$
we have
$\widetilde {\gamma }(\Delta ^\ell ) \subseteq \zeta _i (\xi _i^{-1} \{ b \} ) \subsetneq S$
.
Claim 3.3.6. We may assume that
$\zeta _i:S_{1,i} \to S$
is surjective for each
$i \leq n$
.
Proof of Claim:
By Claim 3.3.5, we have reduced to the case that
$f(X)$
is not contained in any proper weakly special subvariety of S. For any
$i \leq n$
with
$\zeta _i(S_{1,i}) \neq S$
, we would thus have that
$\zeta _i(S_{1,i}) \cap f(X)$
is a proper complex analytic subvariety of
$f(X)$
. Thus, we may shrink U so that for such an i we have
$\zeta _i (S_{1,i}) \cap U = \varnothing $
. We will thus never encounter weakly special varieties of the form
$\zeta _i(\xi ^{-1} \{ b \})$
with
$\gamma (\Delta ^\ell ) \subseteq \zeta _i(\xi ^{-1} \{ b \}) \cap U$
. Thus, we may omit these families of weakly special varieties from our list.
We may adjust our family of weakly special varieties to remember only the maps
$\xi _i:S_{1,i} \to S_{2,i}$
.
Claim 3.3.7. We may assume that
$S_{1,i} = S$
and
$\zeta _i:S_{1,i} \to S$
is the identity map
$\operatorname {id}_S:S \to S$
.
Proof of Claim:
Work by induction on n. In the inductive case of
$n+1$
, apply Claim 3.3.3 to replace S by
$S_{1,n+1}$
. We then need to replace
$S_{1,i}$
for
$i \leq n$
by
$S_{1,i} \times _S S_{1,n+1}$
. Conclude by induction.
We shrink U once again to ensure that all of the fibers of
$\xi _i$
have the same dimension when restricted to U.
Claim 3.3.8. We may shrink U to a smaller nonempty open set so that for all
$i \leq n$
there is some number
$d_i \in \mathbb {N}$
so that for all
$u \in U$
we have
$\dim (\xi _i^{-1} \{ \xi _i(u)\} \cap f(U) ) = d_i$
.
Proof of Claim:
For each
$i \leq n$
and each natural number
$j \leq \dim U$
, let
$$\begin{align*}F_{i,j} := \{ u \in U : \dim_u (f^{-1} \xi_i^{-1} \{ \xi_i(f(u)) \}) = j\} .\end{align*}$$
Here
$\dim _u ( ~ )$
refers to the o-minimal dimension at u.
The definable set U is the finite disjoint union of the definable sets
$$\begin{align*}\bigcap_{i=1}^n F_{i,d_i} \end{align*}$$
as
$(d_1, \ldots , d_n)$
ranges through
$[0,\dim (U)]^n$
. We may cell decompose U subjacent to these definable sets. Let V be an open cell in this cell decomposition. Then for some sequence
$(d_1, \ldots , d_n)$
we have
$V \subseteq \bigcap _{i=1}^n F_{i,d_i}$
. Because V is an open cell, for each
$u \in V$
, we have
$$ \begin{align*} \dim_u (f^{-1} \xi_i^{-1} \{ \xi(f(u)) \}) &= \dim_u (V \cap f^{-1} \xi_i^{-1} \{ \xi(f(u)) \})\\ &= \dim (V \cap f^{-1} \xi_i^{-1} \{ \xi_i (f(u)) \} ) . \end{align*} $$
Apply Claim 3.3.2 to conclude.
Consider now the following incidence correspondence.
$$\begin{align*}R := \{ (u,g) \in U \times G : f(u) \in \pi_\Gamma(gD' \cap \mathcal{F} ) \}\end{align*}$$
Note that R is definable.
Claim 3.3.9. We have
$$\begin{align*}\dim(R) = \dim(G) + 2k .\end{align*}$$
Proof of Claim:
Fix the base point
$\ast \in D' = G'/M'$
corresponding to
$M' = M \cap G'$
in the coset space. For
$u \in U$
, let
$\widetilde {u} \in \mathcal {F}$
with
$\pi _\Gamma (\widetilde {u}) = u$
. Let
$g_0 \in G$
with
$g_0 \ast = \widetilde {u}$
. We will check that
$R_u := \{ g \in G : (u,g) \in R \}$
is a homogenous space for
$M \times G'$
with fibers isomorphic to
$G' \cap M$
. Indeed, if
$h \in G'$
and
$m \in M$
, we have
$\widetilde {u} = g_0 m h h^{-1} \ast $
, demonstrating that
$f(u) \in \pi _{\Gamma }((g_0 m h) D' \cap \mathcal {F})$
. That is,
$g_0 m h \in R_u$
. On the other hand, if
$g \in R_u$
, then we can find some h so that
$g_0 \ast = \widetilde {u} = gh^{-1} \ast $
. That is,
$m := g_0^{-1} gh^{-1} \in M$
, the stabilizer of
$\ast $
in G. That is,
$g = g_0 m h \in g_0 M G'$
. We compute that for
$g_1, g_2 \in G'$
and
$m_1, m_2 \in M$
, we have
$g_0 g_1 m_1 = g_0 g_2 m_2$
only if
$g_2^{-1} g_1 = m_2 m_1^{-1} =: h \in G' \cap M = M'$
.
Using the fiber dimension theorem, since all fibers over U have the same dimension,
$\dim (M \times G') - \dim (M \cap G')$
, we now compute that
$$ \begin{align*} \dim R & = \dim f(U) + \dim R_u \text{ for any } u \in U \\ & = \dim U + \dim(M \times G') - \dim (M \cap G') \\ & = \dim X + \dim M + (\dim G' - \dim (M \cap G')) \\ & = \dim X + \dim M + \dim S' \\ & = \dim X + (\dim G - \dim S) + \dim S' \\ & = \dim G + 2k .\\[-37pt] \end{align*} $$
Abusing notation somewhat, for
$g \in G$
we will also write
$R_g$
for the fiber
$\{ u \in U : (u,g) \in R \}$
. Note that
$R_g$
is definably, complex analytically isomorphic to
$f(U) \cap \pi _\Gamma (g D' \cap \mathcal {F}')$
which is a locally closed complex analytic subset of S. It follows that the o-minimal dimension of
$R_g$
is always even.
For each
$i \leq \dim X$
, let us define
$$\begin{align*}B_i := \{ g \in G : \dim R_g = i \} .\end{align*}$$
Claim 3.3.10. For
$i < 2k$
, we have
$B_i = \varnothing $
.
Proof of Claim:
We have reduced through Claim 3.3.4 to the case that S is smooth. Hence, each component of
$f(U) \cap \pi _\Gamma (g D' \cap \mathcal {F}')$
has complex dimension at least
$\dim U + \dim D' - \dim S = k$
.
The set B of the statement of the theorem may be expressed as
$$\begin{align*}B = \bigcup_{i=0}^{\dim U} B_i .\end{align*}$$
By Claim 3.3.10, we actually have
$$\begin{align*}B = \bigcup_{i=2k}^{\dim U} B_i .\end{align*}$$
With the next claim we show that (again by shrinking U) we may arrange that
$B = B_{2k}$
.
Claim 3.3.11. Possibly after shrinking U, we have
$B_i = \varnothing $
for
$i> 2k$
.
Proof of Claim:
Suppose that
$\ell> k$
and
$g \in B_{2\ell }$
. Shrinking U if necessary, we may assume that
$g\in G\setminus Q$
, where Q is given by the definition of X having persistently likely intersection with the family
$\{\pi (gD')\}_{g\in G}$
. Then the complex analytic set
$f(U) \cap \pi _\Gamma (gD' \cap \mathcal {F})$
has a component L of complex dimension
$\ell $
. Let
$(\gamma ,\widetilde {\gamma }):\Delta ^\ell \to U \times g D'$
be a complex analytic map with
$\operatorname {rk}(d \gamma ) = \ell $
and
$\pi _\Gamma \circ \widetilde {\gamma } = f \circ \gamma $
. By our choice of the witnesses to the Ax-Schanuel property for
$\mathsf {S}$
, for some
$i \leq n$
and
$b \in S_{2,i}$
we have
$$\begin{align*}f \circ \gamma (\Delta^\ell) \subseteq \xi_i^{-1} \{ b \} =: S_b \subsetneq S .\end{align*}$$
By our reduction from Claim 3.3.8 and the fiber dimension theorem, we have
$$\begin{align*}\dim X = \dim U = \dim f(U) = \dim \xi_i f(U) + \dim (f(U) \cap S_b) .\end{align*}$$
Moreover, by the homogeneity of
$\pi _\gamma (gD')$
, we also have
$$\begin{align*}\dim \pi_\gamma(gD') = \dim \xi_i\pi_\gamma(gD') + \dim (\pi_\gamma(gD') \cap S_b) \end{align*}$$
and, of course,
$$\begin{align*}\dim S = \dim S_{2,i} + \dim S_b .\end{align*}$$
By our hypothesis of persistently likely intersections, we have that X has a persistently likely intersection with
$\pi (gD')$
, hence
$$\begin{align*}\dim_{\mathbb{C}} \xi_i f(U) + \dim_{\mathbb{C}} \xi_i \pi_\gamma(gD') = \dim_{\mathbb{C}} S_{2,i} + k' \end{align*}$$
for some
$k' \geq 0$
.
Written in terms of o-minimal dimension this says
$$\begin{align*}\dim \xi_i f(U) + \dim \xi_i \pi_\gamma(gD') = \dim S_{2,i} + 2 k' .\end{align*}$$
Combining these equalities, we compute that
$$\begin{align*}\dim (f(U) \cap S_b) + \dim (\pi_\gamma(gD') \cap S_b) = \dim S_b + 2k - 2k' . \end{align*}$$
Since,
$k' \geq 0$
, this means that the expected (complex) dimension of a component of
$f(U) \cap \pi _\gamma (gD') \cap S_b$
is at most k, but L is such a component of complex dimension greater than k. That is, L is an atypical component of the intersection inside
$S_b \subsetneq S$
. Applying uniform Ax-Schanuel again, we may extend the family of weakly special varieties
for
$n+1 \leq i \leq n_2$
so that each such atypical component will satisfy
$L \subseteq \zeta _i (\xi _i^{-1} \{ b_2 \} ) \subsetneq S_b \subsetneq S$
for some
$i \leq n_2$
and
$b_2 \in S_{2,i}$
.
Repeating the reductions of the earlier claims and this extension of the list of weakly special witnesses to Ax-Schanuel
$\dim S + 1$
times, we reach a contradiction to the hypothesis that
$B_i$
is nonempty for some
$i> 2k$
.
Thus,
$B = B_{2k}$
. So we have
$$\begin{align*}\dim G + 2k = \dim R = \dim B_{2k} + 2k = \dim B + 2k .\end{align*}$$
Subtracting
$2k$
from both sides, we conclude that
$\dim B = \dim G$
. Hence, by cell decomposition, B contains an open subset of G.
4 Applications
In this section we illustrate Theorem 3.3 by considering various situations in which it applies.
4.1 Arithmetic quotients
Our formalism is derived from that of Bakker, Klingler, and Tsimerman in [Reference Bakker, Klingler and Tsimerman5] for the study of arithmetic quotients. They consider definable complex quotient spaces
$S_{G,\Gamma ,M;\mathcal {F}}$
in which the algebraic group
$\mathbf {G}$
is a semisimple
$\mathbb {Q}$
-algebraic group,
$\Gamma $
is arithmetic (so commensurable with
$\mathbf {G}(\mathbb {Z})$
for some / any choice of an integral model for
$\mathbf {G}$
), and M is compact. They often require
$\Gamma $
to be neat; we will return to that issue in a moment. The definable fundamental domain
$\mathcal {F}$
is not chosen sufficiently carefully in [Reference Bakker, Klingler and Tsimerman5], an issue that was then addressed and fixed in [Reference Bakker, Bakker, Klingler and Tsimerman6]. A similar issue is addressed in [Reference Orr and Schnell23] in that one needs to take
$\mathcal {F}$
to be constructed from a Siegel set associated to a maximal compact subgroup of G containing M.
If we drop the neatness requirement on
$\Gamma $
, then an arithmetic quotient need not be smooth, but because every arithmetic group has a neat subgroup of finite index, for any arithmetic quotient S we may find a smooth arithmetic quotient
$\widetilde {S}$
and a surjective map of arithmetic quotients
$\widetilde {S} \to S$
with finite fibers.
The one-point space is clearly a terminal object in the category of arithmetic quotients. That the weakly special varieties are isomorphic to arithmetic quotients is essentially [Reference Moonen22, Theorem 4.3]. We say “essentially” because Moonen works with the additional hypothesis that the ambient variety is a Shimura variety though his arguments for the characterization of the totally geodesic varieties are algebraic group theoretic and apply to general arithmetic quotients. Since there are only countably many arithmetic quotients all told and at most countably many maps of algebraic groups between algebraic groups defined over the rational numbers, it follows that the weakly special varieties are well-parameterized within the category of arithmetic quotients.
The main theorem of [Reference Bakker and Tsimerman7] is that period mappings associated to polarized variations of integral Hodge structures satisfy the Ax-Schanuel condition with respect to
$\mathsf {S}$
. It is an interesting open question whether every definable analytic map
$f:X^{\text {an}} \to S$
where S is an arithmetic definable complex quotient space necessarily satisfies the Ax-Schanuel condition with respect to
$\mathsf {S}$
.
Our last observation in verifying Convention 3.1 and the hypotheses of Theorem 3.3 for arithmetic quotients is that if
$D' \subseteq D$
is a homogeneous space for which
$\pi _\Gamma (D') \subseteq S$
is a special variety, then for every
$g \in \mathbf {G}(\mathbb {Q}) \cap G$
,
$\pi _\Gamma (g D')$
is also special. Thus the set of
$g \in G$
for which
$\pi _\Gamma (g D')$
is special is dense in G for the Euclidean topology.
Returning to the case where we know
$f:X^{\text {an}} \to S$
to be a period mapping and
$f(X)$
is not contained in any proper weakly special subvariety, the union of
$f^{-1} S'$
ranging over all proper special subvarieties
$S' \subsetneq S$
is called the Hodge locus. In [Reference Klingler and Otwinowska19], a dichotomy theorem is proven for a modified form of the Hodge locus which they call the Hodge locus of positive period dimension: either this locus is Zariski dense in X or it is itself a proper algebraic subvariety of X. In [Reference Khelifa and Urbanik17], tight conditions for the density of the Hodge loci are established.
Because the special subvarieties of S come from
$\mathbb {Q}$
-semisimple algebraic subgroups of
$\mathbf {G}$
and there are only finitely many such subgroups up to
$G = \mathbf {G}(\mathbb {R})^+$
-conjugacy, all special subvarieties of S come from finitely many families of homogeneous spaces in the sense of Theorem 3.3. That is, we can find finitely many homogeneous spaces
$D_1, \ldots , D_n \subseteq D$
so that for any special subvariety
$S' \subseteq S$
there is some
$g \in G$
and
$i \leq n$
with
$S' = \pi _\Gamma (g D_i)$
. Thus, if the intersection of X with special subvarieties of the form
$\pi (gD_i')$
is persistently likely, then the Hodge locus is Euclidean dense in X. In fact, the subset of the Hodge locus of the form
$f^{-1} \bigcup _{h \in G, \pi _\Gamma (h D_i) \text { special }} \pi _\Gamma (h D_i)$
is Euclidean dense in X. In [Reference Baldi, Klingler and Ullmo8] a theorem of a similar flavor is proven. They show that if the typical Hodge locus is nonempty, then it is analytically dense in X. Here the typical Hodge locus is the union of all components of
$f^{-1} S'$
of expected dimension as
$S'$
ranges through the special subvarieties of S. The proof in [Reference Baldi, Klingler and Ullmo8] uses some elements in common with ours. Notably, Ax-Schanuel plays a central role in both proofs. To pass from a nonempty typical locus to one which is dense, they argue through an analysis of Lie algebras to find enough special varieties. Such a technique is not available to us in general as we must postulate the existence of special varieties of a given shape. On the other hand, such an argument does not immediately lend itself to a study of intersections with a restricted class of special varieties.
Definability of the period mappings associated to admissible, graded polarized, variations of mixed Hodge structures has been established by Bakker, Brunebarbe, Klingler, and Tsimerman in [Reference Bakker, Brunebarbe, Klingler and Tsimerman4] and then Ax-Schanuel for these maps was proven independently by Chiu [Reference Chiu10] and Gao and Klingler [Reference Gao and Klingler15]. Indeed, Chiu has established a stronger Ax-Schanuel theorem with derivatives for such period maps associated to variations of mixed Hodge structures [Reference Chiu9]. These results give the necessary ingredients to extend our result on the density of Hodge loci to variations of mixed Hodge structures. We will return to the special case of universal abelian schemes over moduli spaces in Section 4.3 to draw a conclusion from the combination of our Theorem 3.3 and Ax-Schanuel in the context of mixed Shimura varieties.
4.2 Modular varieties
For the sake of illustration, let us consider a very special case of Theorem 3.3. We take
$S = \mathbb {A}^n = Y_0(1)^n$
. That is, S is affine n-space (for some positive integer n) regarded as the coarse moduli space of n-tuples of elliptic curves. We may see S as an arithmetic quotient space, taking
$\mathbf {G} = \operatorname {PGL}_2^n$
,
$\Gamma = \operatorname {PGL}_2^n(\mathbb {Z})$
, and the homogeneous space D may be identified with
$\mathfrak {h}^n$
where
$\mathfrak {h} = \{ z \in \mathbb {C} : \operatorname {Im}(z)> 0 \}$
is the upper half plane.
If
$\sigma = \langle \sigma _1, \ldots , \sigma _m \rangle $
is a finite sequence taking values in
$\{ 1, \ldots , n \}$
we may define
$\pi _\sigma :S \to \mathbb {A}^m$
by
$(x_1,\ldots ,x_n) \mapsto (x_{\sigma _1},\ldots ,x_{\sigma _m})$
. For
$J \subseteq \{1, \ldots , n \}$
we list the elements of J in order as
$J = \{ j_1 < j_2 < \ldots < j_m \}$
and write
$\pi _J$
for
$\pi _{\langle j_1, \ldots , j_m \rangle }$
. For a singleton
$J = \{ j \}$
, we just write
$\pi _j$
for
$\pi _J$
.
For
$S' \subseteq S$
a special subvariety of S provided that for each
$i \leq n$
the projection map
$\pi _i:S' \to \mathbb {A}^1$
is dominant, then
$S'$
defines a partition
$\Pi (S')$
of
$\{ 1, \ldots , n \}$
by the rule that i and j lie in a common element of the partition if and only if
$\dim \pi _{\langle i, j \rangle } S' = 1$
. Given a partition
$\Pi $
of
$\{ 1, \ldots , n \}$
we say that
$S'$
is a special variety of type
$\Pi $
if
$\Pi (S') = \Pi $
. Let us observe that a special variety of type
$\Pi $
has dimension equal to
$\# \Pi $
.
Fix a partition
$\Pi $
with
$\# \Pi = m$
. Let
$D_\Pi ' \subseteq D$
be the homogeneous subspace of D defined by
$\tau _i = \tau _j$
if and only there is some
$\nu \in \Pi $
with
$\{ i, j \} \subseteq \nu $
. Then
$\pi _\Gamma (D_\Pi ') =: S'$
is the corresponding multidiagonal subvariety of S and is a special variety of type
$\Pi $
. Indeed, the special varieties of the form
$\pi _\Gamma (gD_\Pi ')$
as g ranges through
$\operatorname {PGL}_2^n(\mathbb {Q}) \cap G$
are exactly the special varieties of type
$\Pi $
.
For a partition
$\Pi $
of
$\{ 1, \ldots , n \}$
and subset
$J \subseteq \{1, \ldots , n\}$
of
$\{1, \ldots , n\}$
, we define
$$\begin{align*}\Pi \upharpoonright J := \{ \nu \cap J : \nu \in \Pi, \nu \cap J \neq \varnothing \} \end{align*}$$
to be the restriction of the partition
$\Pi $
to J.
It is easy to check that for any special variety
$S' \subseteq S$
and subset
$J \subseteq \{ 1, \ldots , n \}$
, the projection
$\pi _J(S')$
is a special variety and
$\Pi ( \pi _J(S')) = \Pi \upharpoonright J$
. With these combinatorial preliminaries in place, we may state the specialization of Theorem 3.3 to the case of
$Y_0(1)^n$
.
Proposition 4.1. Let
$n \geq 1$
be a positive integer and
$\Pi $
a partition of
$\{ 1, \ldots , n \}$
. If
$X \subseteq \mathbb {A}^n$
is an irreducible complex algebraic subvariety of affine n-space, regarded as the coarse moduli space of n-tuples of elliptic curves, and for every
$J \subseteq \{ 1, \ldots , n \}$
we have
$\# \Pi \upharpoonright J + \dim \pi _J(X) \geq \#J$
, then
$$\begin{align*}X \cap \bigcup_{S' \subseteq \mathbb{A}^n \text{ special of type } \Pi } S' \end{align*}$$
is dense in X for the Euclidean topology.
Proof. Let us check that the intersection between X and the family
$\{\pi (h D_\Pi )\}_{h \in G}$
is persistently likely. Let
be a pair of surjective maps of arithmetic quotients with
$\zeta $
finite. The arithmetic quotients
$S_1$
and
$S_2$
will take the form
$S_1 = S_{\operatorname {PGL}_2^n,\Gamma _1,M_1; \mathcal {F}_1}$
and
$S_2 = S_{\operatorname {PGL}_2^k,\Gamma _2,M_2; \mathcal {F}_2}$
with
$k \leq n$
where
$\Gamma _j$
is an arithmetic group for
$j = 1$
and
$2$
and the corresponding homogeneous spaces are
$\mathfrak {h}^n$
and
$\mathfrak {h}^k$
, respectively. Since each
$\operatorname {PGL}_2$
factor is simple, the maps of algebraic groups corresponding to
$\zeta $
and
$\xi $
are given by coordinate projections followed by an inner automorphism defined over
$\mathbb {Q}$
. That is, the map of groups corresponding to
$\xi $
is given by
$\langle g_1, \ldots , g_n \rangle \mapsto \langle g_{j_1}, \dots , g_{j_k} \rangle $
followed by an inner automorphism of
$\operatorname {PGL}_2^k$
defined over
$\mathbb {Q}$
for some collection of k distinct numbers
$j_1, \ldots , j_k$
between
$1$
and n, and likewise for
$\zeta $
. Let
$J = \{ j_1, \ldots , j_k \}$
, then permuting coordinates we see that this family of weakly special varieties fits into the commuting square
where
$\widetilde {\zeta }:S_2 \to \mathbb {A}^k$
is finite. We then have
$$ \begin{align*} \dim \xi \zeta^{-1} X + \xi \zeta^{-1} \pi(h D_\Pi) & = \dim \pi_J X + \dim \pi_J (\pi(h D_\Pi)) \\ & = \dim \pi_J X + \# \Pi \upharpoonright J \\ & \geq k \\ & = \dim S_2 \end{align*} $$
Since
$\operatorname {PGL}_2^n(\mathbb {Q})^+$
is the commensurator of
$\operatorname {PGL}_2^n(\mathbb {Z})$
and is dense in
$G = \operatorname {PGL}_2^n(\mathbb {R})^+$
, the concluding “in particular” clause of Theorem 3.3 applies and we find that the intersections of X with special varieties of type
$\Pi $
is dense in X in the Euclidean topology.
Instances of Proposition 4.1 appear in the literature. Habegger shows in [Reference Habegger16, Theorem 1.2] that if
${X \subseteq \mathbb {A}^2}$
is a curve defined over the algebraic numbers, then there is a constants
$c = c(X)> 0$
and
$p_0(X)> 0$
so that for every prime number
$p> p_0(X)$
there is an algebraic point
$P \in X(\mathbb {Q}^{\text {alg}}) \cap Y_0(p) (\mathbb {Q}^{\text {alg}})$
with logarithmic height
$h(P) \geq c \log (p)$
where here
$Y_0(p)$
is the modular curve parametrizing the isomorphism classes of pairs of elliptic curves
$\langle E, E' \rangle $
for which there is an isogeny
$E \to E'$
of degree p. Habegger’s result implies in particular that for
$n = 2$
and
$\Pi = \{ \{ 1, 2 \} \}$
, if
$X \subseteq \mathbb {A}^2_{\mathbb {Q}^{\text {alg}}}$
is an affine plane curve defined over the algebraic numbers, then the intersection of X with the special varieties of type
$\Pi $
is Zariski dense in X. Using equidistribution results, this Zariski density could be upgraded to Euclidean density.
In the discussion after Remark 3.4.5 in [Reference Zannier35], Zannier sketches an argument showing that if
$X \subseteq \mathbb {A}^2$
is a rational affine plane curve, then the intersections of X with special curves of type
$\Pi $
, as in the previous paragraph, are dense in X in the Euclidean topology.
4.3 Torsion in families of abelian varieties
If
$\pi :A \to B$
is an abelian scheme of relative dimension g over the irreducible quasiprojective complex algebraic variety B and
$X \subseteq A$
is a quasi-section of
$\pi $
, by which we mean that
$\pi $
restricts to a generically finite map on X, then under some mild nondegeneracy conditions, we expect that the set
$$\begin{align*}\pi(X \cap A_{\text{tor}}) = \{ b \in B(\mathbb{C}) : (\exists n \in \mathbb{Z}_+) X_b \cap A_b(\mathbb{C})[n] \neq \varnothing \} \end{align*}$$
of points on the base over which X meets the torsion subgroup of the fiber is dense in B if and only if
$\dim B \geq g$
. Masser-Zannier prove in [Reference Masser and Zannier21] that when
$B = \mathbb {P}^1 \smallsetminus \{0, 1, \infty \}$
and
$\pi :A \to B$
is the square of the Legendre family of elliptic curves defined in affine coordinates by
$y_1^2 = x_1(x_1-1)(x_1-\lambda )$
and
$y_2^2 = x_2(x_2 - 1)(x_2 - \lambda )$
where
$\lambda $
ranges over B, and X is the curve defined by
$x_1 = 2$
and
$x_2 = 3$
, then the set
$\pi (X \cap A_{\text {tor}})$
is finite. This theorem sparked much work on torsion in families of abelian varieties culminating in a result announced by Gao and Habegger that, at least for such abelian schemes
$\pi :A \to B$
defined over
$\mathbb {Q}^{\text {alg}}$
, if
$X \subseteq A$
is an algebraic variety, also defined over
$\mathbb {Q}^{\text {alg}}$
so that the group generated by X is Zariski dense in A and
$\pi (X \cap A_{\text {tor}})$
is Zariski dense in B, then
$\dim X \geq g$
.
In the opposite direction, André, Corvaja, and Zannier study in [Reference André, Corvaja and Zannier1] the problem of density of torsion through an analysis of the rank of the Betti map. In an appendix to that paper written by Gao, it is shown that if
$\pi :A \to B$
is a principally polarized abelian scheme of relative dimension g which has no nontrivial endomorphism (on any finite covering), and for which the image of S in the moduli space
$\mathcal {A}_g$
of abelian varieties of dimension g itself has dimension at least g and
$X \subseteq A$
is the image of a section of
$\pi $
, then
$\pi (X \cap A_{\text {tor}})$
is dense in B in the Euclidean topology. The proof of this result made use of the Ax-Schanuel theorem for pure Shimura varieties and was subsequently upgraded. See in particular Gao’s work on the Ax-Schanuel theorem for the universal abelian variety [Reference Gao13] and on the Betti map in [Reference Gao12, Reference Gao14].
Gao’s main theorem, Theorem 1.1, in [Reference Gao12] may be seen as a geometric elaboration of what Theorem 3.3 means for the density of torsion. Gao considers an abelian scheme
$\pi :A \to B$
of relative dimension g over a quasiprojective complex algebraic variety B and a closed irreducible subvariety
$X \subseteq A$
and then establishes the conditions under which the generic rank of the Betti map restricted to X may be smaller than expected. It is noted with [Reference André, Corvaja and Zannier1, Proposition 2.2.1] that density of the torsion in X follows from the Betti map, generically, having rank
$2g$
on X. Thus, the converse of Gao’s condition gives a criterion for when the torsion is dense.
In more detail, taking finite covers if necessary, one may pass from the problem of density of torsion in X as a subvariety of A, to the density of torsion in
$\widetilde {\iota }(X)$
in
$\mathfrak {A}$
where
$\mathfrak {A} \to \mathcal {A}$
is a universal abelian variety over a moduli space
$\mathcal {A}$
of abelian varieties of some fixed polarization type with some fixed level structure and the Cartesian square
expresses
$A \to B$
as coming from this universal family. To ease notation, we replace B by
$\iota (B)$
and X by
$\widetilde {X}$
. Shrinking the moduli space, possibly taking covers, and moving to an abelian subscheme of
$\mathfrak {A}_B$
, we may arrange that X is not contained in any proper weakly special varieties. At this point, Theorem 3.3 says that
$X \cap \mathfrak {A}_{\text {tor}}$
is dense in X in the Euclidean topology if the intersection of X with the zero section is persistently likely. Gao’s criterion expresses geometrically what persistent likelihood means here: for any abelian subscheme
$\mathfrak {A}'$
of
$\mathfrak {A}_B$
, if
$p:\mathfrak {A}_B \to \mathfrak {A}_B / \mathfrak {A}'$
is the quotient map, then
$\dim p(X)$
is at least the relative dimension of
$\mathfrak {A}_B / \mathfrak {A}'$
over B.
Acknowledgments
During the writing of this paper S.E. was partially supported through the NSF grant RTG DMS-1646385 and the EPSRC fellowship EP/T018461/1, and T.S. was partially supported by NSF grants DMS-1800492, FRG DMS-1760414, and DMS-22010405. The authors have no competing interests, financial or otherwise. S.E. thanks E. Ullmo and G. Baldi for hosting him at IHÉS, for sharing an early version of [Reference Baldi, Klingler and Ullmo8], and for discussing problems around the density of special intersections. T.S. thanks Z. Gao, M. Orr, and U. Zannier for offering detailed accounts of the state of the art on likely intersection problems for modular curves and torsion in abelian schemes. Both authors thank the referees for their close reading, for their insightful questions, and for suggesting several improvements.
Competing interest
None.
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