Likely intersections

If $X$ is an 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 $\dim f(Y)+\dim g(Z)<\dim X$ ¹¹1 We will be working with cases in which $f(Y)$ is automatically a finite Boolean combination of (at least) real analytic subvarieties of $X$. This happens, for example, when $f:Y\to X$ is proper or definable in an o-minimal expansion of the real field. In this case, we could take $\dim$ to mean the o-minimal dimension or the maximal dimension of a submanifold of $f(Y)$. 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 $Y\subseteq X$ a Hodge generic subvariety, meaning that there is no proper weakly special subvariety (in the sense of Shimura varieties) $T\subsetneq X$ with $Y\subseteq T$. 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 [8] on the density of the typical Hodge locus, the results of Tayou and Tholozan [29] describing the typical Hodge locus of polarized variations of Hodge structure over a smooth complex quasiprojective variety, and the work of Gao [12, 14] (following on the work of André, Corvaja, and Zannier [1]) 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 3.4) 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 the connected component 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 space $S^{\prime}\to S$ (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 $D^{\prime}\subseteq D$ a homogeneous space for a subgroup $G^{\prime}\leq G$. We will write $\pi_{\Gamma}:D\to S$ for the quotient map. It may happen that $\pi_{\Gamma}(gD^{\prime})\subseteq S$ 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 $D^{\prime}=\{(\tau,\tau):\tau\in\mathfrak{h}\}$ is the diagonal, then all modular plane curves may be expressed as $\pi_{\Gamma}(gD^{\prime})$ 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 $f^{-1}\bigcup_{g\in G,\pi_{\Gamma}(gD^{\prime})\text{ special }}\pi_{\Gamma}(gD^{\prime})$ of special intersections is dense in $X$ provided that the intersections are likely in the sense that $\dim f(X)+\dim D^{\prime}\geq\dim S$. 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 3.4 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 $\pi_{\Gamma}(D^{\prime})$ is persistently likely, the set of special intersections is dense in $X$.

When working in the setting of variations of Hodge structures, our result improves the “all or nothing” theorem of [8] first by giving a criterion for the typical Hodge locus to be dense (namely persistent likeliness), 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, [8] and [29]. 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 complimentary dimensions and so that $\dim S\leq 2$, our result implies the part of [29, 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 to contexts not covered by [29], 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 [8], [29] or [12]. Instead, our arguments are inspired by the work of Aslanyan and Kirby [3], 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 [3], 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 [11] inspired our approach.

Our own interest in the likely intersections problem was motivated by [24, 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 [5, 4] 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.

We express our theorems on likely intersections in terms of classes of definable complex quotient spaces. Our formalism is similar to what appears in [5], 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.

The class of definable quotient spaces forms a category $\mathsf{D}\mathsf{Q}\mathsf{S}$ with the following notion of a morphism.

The category $\mathsf{D}\mathsf{Q}\mathsf{S}$ has a terminal object and is closed under fiber products.

A useful observation is that morphisms of definable quotient spaces always factor as a surjective map followed by a map induced by an inclusion of subgroups.

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.

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 and closure under fiber products, 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 2.6 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 2.6 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. When we restrict to the case that the group $M$ is compact, then 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.

In Section 3 we will impose an additional restriction on $\mathsf{S}$.

In Definition 2.9 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 2.8, then $p:S_{3}\to S_{2}$ is a finite, and hence, proper, map of complex analytic spaces so that by Remmert’s proper mapping theorem its image $p(S_{3})=f(S_{1})$ is a complex analytic subvariety of $S_{2}$. Using this observation, we may modify Definition 2.9 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 finite.

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 it as an hypothesis and verify it in cases of interest.

Let us indicate now the key functional transcendence condition which may hold in a category $\mathsf{S}$ of definable complex quotient spaces.

Under the hypothesis that the weakly $\mathsf{S}$-special subvarieties are well-parameterized, the Ax-Schanuel condition implies a uniform version of itself.

Since our statement of this uniform version, expressed as Proposition 2.20 below, 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, then 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 2.20 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.

Note that in Definition 2.17, because we allow for the possibility that $f$ is not a finite map, it could happen that the dimension of $Y$ itself is greater than $d$. On the other hand, the intersection of $S^{\prime}$ with $f(X)$ may even be empty! Thus, the dimension of $Y$ could be less than $d$.

It follows from Lemma 2.16 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 $\mathsf{S}$ that we can recognize the pullbacks under $f$ of prespecials 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 $X$.

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 [27, 28] 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 [30] and appears as Theorem 1 of [22]. 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 [26]. 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 $x,y\in\check{D}(L)$,

where

is the common constant field of $L$. In particular, if this common constant field is $\mathbb{C}$, then we may express the quotient of $\check{D}$ by the $\mathbf{G}(\mathbb{C})$ 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 $\chi:X\to Z$ by the rule that for any meromorphic $\gamma:U\to X$ (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 [26] 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.)

A uniform version of the Ax-Schanuel condition follows from Lemma 2.16 using the compactness theorem.

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 2.8 and some further requirements. Let us specify with the following convention the properties we require.

With the following definition we specify what is meant by intersections being persistently likely.

Note that this definition does not actually require $f(X)\cap S^{\prime}$ to be non-empty for the intersection to be likely.

With this definition in place we may now state our main theorem.