# Effective atypical intersections and applications to orbit closures

Gregorio Baldi and David Urbanik

###### Abstract

We propose a unifying setting for dealing with monodromically atypical
intersections that goes beyond the usual Zilber-Pink conjecture. In particular
we obtain a new proof of finiteness of the maximal atypical orbit closures in
each stratum of translation surfaces $\Omega\mathcal{M}_{g}(\kappa)$, as given
by Eskin, Filip, and Wright. We also describe a concrete algorithm,
implementable in principle on a computer, which provably computes all maximal
orbit closures which are “atypical” in a sense described by Filip. The same
methods also give a general algorithm for computing atypical special loci
associated to systems of differential equations, and in particular give an
effective and o-minimal free proof of the geometric Zilber-Pink conjecture
variations of mixed Hodge structures.

###### Contents

  1. 1 Introduction
  2. 2 $\nabla$-weakly special subvarieties
  3. 3 Atypical weakly special subvarieties
  4. 4 Introduction to variations of mixed Hodge structures: period maps and Zilber-Pink
  5. 5 Recap on orbit closures
  6. 6 Finiteness of atypical orbit closures and density of the typical ones
  7. 7 Proof of Geometric Zilber-Pink for VMHS
  8. 8 Digression on Differentially-Defined Intersections
  9. 9 Algorithmic Setup
  10. 10 Algorithms

##  1 Introduction

In the last 15 years, moduli spaces of translation surfaces $(X,\omega)$ have
attracted a lot of attention, thanks to the work of Eskin, Filip, McMullen,
Mirzakhani, Mohammadi, Möller, Wright, and many others. A variety of tools
have been used to try to grasp them, mostly from dynamics, analysis, and
algebraic geometry. In particular, the work of the aforementioned authors
gives striking results on so-called _orbit closures_
$\mathcal{N}=\overline{\operatorname{GL}_{2}(\mathbb{R})\cdot(X,\omega)}$. The
aim of this paper is to push the Hodge-theoretic and algebro-geometric
understanding of such loci $\mathcal{N}$ one step further. Using tools from
differential geometry and differential algebra we will show that one can
understand the distribution of orbit closures just by knowing whether they are
_typical or atypical intersections_.

This new perspective will ultimately come from a new understanding of the
(geometric) Zilber–Pink philosophy in the following setting. We start with $S$
a smooth complex algebraic variety, $H$ an algebraic group (which is a
semidirect product of a unipotent and a semisimple group), and $\pi:P\to S$ an
algebraic $H$-principal bundle. The bundle $P$ will carry a $H$-principal flat
connection $\nabla$ with Galois group $\operatorname{Gal}(\nabla)=H$. We will
then study the behaviour of _atypical special subvarieties_ $W$ of $S$ (e.g.
in the sense of Hodge or Teichmüller theory) by pulling back to $P$ certain
algebraic data from an extended period domain. Our work will crucially use
recent breakthrough work of Blázquez-Sanz, Casale, Freitag, and Nagloo [21].
Thanks to an observation of Filip [44, §5] (partially motivated by recent
progress in Hodge theory regarding the distribution of the Hodge locus [15]),
such a setting recovers the atypical orbit closures. Indeed the main
motivation behind our work is 1.4, recalled below. Our result on the orbit
closures rests on the deep fact that orbit closures are algebraic and admit a
purely algebro-geometric characterization (see e.g. [44, Thm. 2]).

###  1.1 Special subvarieties in Teichmüller and Hodge theory, after Eskin-
Filip-Wright, and Baldi-Klingler-Ullmo

We start by recalling the main protagonists of the theory of moduli spaces of
translation surfaces, mostly following the recent notes [89, 44]. Let $X$ be
an irreducible smooth projective algebraic curve over $\mathbb{C}$, or simply
a compact _Riemann surface_. A _translation surface_ is a pair $(X,\omega)$,
where $X$ is a compact Riemann surface and $\omega$ is a non-zero global
section of the cotangent bundle of $X$ (also known as an _abelian
differential_). If $g\geq 1$ is the genus of $X$, then $\omega$ has $2g-2$
zeros, counted with multiplicities. Let $\mathcal{M}_{g}$ denote the moduli
space of genus $g$ compact Riemann surfaces and
$\Omega\mathcal{M}_{g}\to\mathcal{M}_{g}$ the bundle whose fibre over
$[X]\in\mathcal{M}_{g}$ is just the space of all holomorphic 1-forms on $X$.
It is a rank $g$ (algebraic) vector bundle on $\mathcal{M}_{g}$, naturally
equipped with a stratification by smooth (not necessarly irreducible)
algebraic subvarieties $\Omega\mathcal{M}_{g}(\kappa)$, where
$\kappa=(\kappa_{0},\dots,\kappa_{n})$ satisfies $\sum_{i}\kappa_{i}=2g-2$ and
denotes the multiplicities of zeros of $\omega$. We will also write
$n=\operatorname{length}(\kappa)-1$. (In fact, to avoid orbifold issues, we
will later work with finite covers of such moduli spaces).

A holomorphic 1-form $\omega$ on $X$ can be thought of as giving a collection
of charts on $X$ to $\mathbb{C}$, such that:

  * •

the transition maps are translations,

  * •

the charts can ramify at finitely many points corresponding to the zeros of
$\omega$,

  * •

they are locally given by $z\mapsto\int_{z_{0}}^{z}\omega$.

Such charts induce the so-called _period coordinates_ on
$\Omega\mathcal{M}_{g}(\kappa)$. We therefore obtain an action of
$\operatorname{GL}_{2}(\mathbb{R})$ on $\Omega\mathcal{M}_{g}(\kappa)$,
locally given by a diagonal action on a product of copies of
$\mathbb{C}\cong\mathbb{R}^{2}$. Such an action is a priori outside the realm
of algebraic geometry, but it turns out to be closely related to Hodge theory.

We will study the distribution of strict orbit closures
$\mathcal{N}:=\overline{\operatorname{GL}_{2}(\mathbb{R})\cdot(X,\omega)}\subset\Omega\mathcal{M}_{g}(\kappa)$,
where $\overline{(\cdot)}$ denotes the topological closure, also known in the
literature as _affine invariant submanifolds_. (For terminology and notations
we try to follow [44].)

We start by recalling a theorem of Eskin, Filip, and Wright (see §5.2 for
definitions):

######  Theorem 1.1 ([40, Thm. 1.5]).

In each (irreducible component of each) stratum
$\Omega\mathcal{M}_{g}(\kappa)$, all but finitely many orbit closures have
rank 1 and degree at most 2. In each genus there is a finite union of orbit
closures $\mathcal{N}$ of rank 2 and degree 1 such that all but finitely many
of the orbit closures of rank 1 and degree 2 are a codimension 2 subvariety of
one of these $\mathcal{N}$.

In fact, it will be better to work with an equivalent formulation of the
above. We will see that in the above result, the orbit closures which arise in
finite families are _atypical intersections_ , and the others are typical
ones. Already in the introduction of [40] a link with the André–Oort
conjecture is mentioned. But it’s only in the recent notes of Filip [44, §5]
where this is made more precise. The following, again due to Eskin, Filip, and
Wright [40, Thm. 1.3, 1.5, and 1.7], generalizes and gives a reformulation of
1.1 (see also [44, Thm. 3] and §5.3 for the definition of relatively
(a)typical).

######  Theorem 1.2.

Each orbit closure $\mathcal{M}$ contains only finitely many maximal atypical
suborbit closures. If an orbit closure $\mathcal{M}$ admits relatively typical
suborbit closures, then those are dense in $\mathcal{N}$.

With the above formulation, the result looks similar to what happens in Hodge
theory, as recently showed by Baldi, Klingler, and Ullmo [15] (see §4 for all
definitions, in a more general context). They consider a smooth quasi-
projective irreducible complex variety $S$ supporting a pure polarized
integral varietion of Hodge structures ($\mathbb{Z}$VHS from now on, and
usually simply denoted as a pair $(S,\mathbb{V})$) and study the (tensorial)
Hodge locus of $(S,\mathbb{V})$, which can be written a countable union of
algebraic subvarieties of $S$ (the so-called _special subvarieties_). In _op.
cit._ they propose a decomposition of the Hodge locus into the _typical_ and
_atypical_ part. Informally a special subvariety is atypical if it has less
than the expected codimension (i.e. the one suggested by a naïve dimension
count) and it is typical otherwise. They prove the following (see also the ICM
report [58] for more details and related discussions):

######  Theorem 1.3 ([15, Thm. 3.1 and 3.9]).

Let $(S,\mathbb{V})$ be a $\mathbb{Z}$VHS with $\mathbb{Q}$-simple algebraic
monodromy. We have:

  1. 1.

The positive period dimensional atypical Hodge locus lies in a strict
algebraic subvariety of $S$.

  2. 2.

The positive period dimensional typical Hodge locus is either empty or
analytically (therefore Zariski) dense in $S$;

One expects the same results to be true without the assumption _positive
period dimensional_ (in fact this would follow from the full power of the
Zilber-Pink conjecture). Nevertheless, the geometric part presents several
interesting incarnations (for example 1.2 is about positive dimensional
subvarieties). The statement 1. is a special case of the _geometric part of
Zilber-Pink_ (or simply geometric ZP) in the case where $\mathbb{V}$ has
$\mathbb{Q}$-simple monodromy.

The following appears in [44, pg.5], after discussing the two theorems
described above, as well as the main results of [4] (we will come back to this
point in §1.4).

######  Question 1.4 (S. Filip).

Can these methods and results be put into a common framework?

The goal of our paper is indeed to provide such an unifying framework, which
we believe can be of independent interest.

###  1.2 The abstract/unified setting

##### Special Manifolds and Intersections:

Many manifolds $M$ come naturally with a distinguished class of submanifolds
$\\{M_{i}\\}_{i\in I}$ exhibiting a _special_ behaviour. Examples include
abelian varieties with sub-abelian varieties and their torsion translates;
Shimura varieties and more general period domains with sub-Shimura varieties
and sub-period domains; and locally homogeneous spaces with totally geodesic
sub-spaces. Given a submanifold $S\subset M$, one then obtains from
$\\{M_{i}\\}_{i\in I}$ a distinguished class of _special intersections_ $S\cap
M_{i}$. Here the examples of interest are (mixed) Hodge loci, and the case of
particular interest to us: orbit closures within strata of abelian
differentials (c.f. (6.1) and the corresponding discussion). Zilber-Pink-type
results in such settings suggest distribution behaviour of the special
intersections can often be understood by simple dimension criteria. In
particular, one says that a component $Y$ of $S\cap M_{i}$ is _atypical_ if

$\operatorname{codim}_{M}(Y)<\operatorname{codim}_{M}(S)+\operatorname{codim}_{M}(M_{i}),$
(1.1)

and _typical_ otherwise. One then expects that:

  * (1)

$S$ contains only finitely many maximal atypical intersections.

  * (2)

The following are equivalent:

    * (i)

$S$ contains one typical intersection;

    * (ii)

the collection of typical intersections is dense in $S$; and

    * (iii)

there exists an $i\in I$ such that:

$\dim S-\operatorname{codim}_{M}(M_{i})\geq 0.$

Our goal is to discuss two concrete instances of such viewpoint. More
precisely we will describe a general strategy to tackle (1) in both the
setting of orbit closures and Hodge loci of variations of mixed Hodge
structures. The setting we work with might look indirect at first, so we
explain here the main protagonists, and how they relate to the above picture.

##### The Differential Viewpoint:

First of all, it will be crucial that all objects we work with are algebraic,
since the main tools come from algebraic geometry — in particular we will no
longer speak of manifolds $M$, but only of algebraic varieties $S$ and complex
analytic period domains $D$, and their quotients by opportune lattices
$\Gamma$. (With reference to the above viewpoint, one can think of $M$ as some
quotient of $\Gamma\backslash D$ into which $S$ embeds.) We consider arbitrary
(pointed) complex varieties $S$, supporting a local system $\mathbb{V}$ with
algebraic monodromy group $\mathbf{H}_{S}$ — for the applications,
$\mathbb{V}$ will underline a $\mathbb{Z}$-variation of mixed Hodge
structures. In §2.2 we introduce $P=P(s_{0})$, an algebraic principal
$\mathbf{H}_{S}$-subbundle of the algebraic bundle
$\operatorname{\mathcal{H\kern-1.0pto\kern-2.0ptm}}(\mathcal{H},\mathcal{O}_{S}\otimes\mathcal{H}_{s_{0}})$
associated to
$\operatorname{\mathcal{H\kern-1.0pto\kern-2.0ptm}}(\mathbb{V},\mathbb{V}_{s_{0}})$
(via Riemann-Hilbert), where $\mathcal{H}$ is the algebraic bundle associated
to $\mathbb{V}$. The bundle $P$ has an equivariant algebraic connection
induced from the connection on $\mathbb{V}$, which locally foliates $P$ by
flat leaves $\mathcal{L}$. Through each point $x\in P$, there is a unique leaf
$\mathcal{L}_{x}$.

The gist of our approach is the following. In mixed Hodge theory, the special
intersections in $S$ are particular cases of _weakly special subvarieties_ ,
i.e. the maximal subvarieties of $S$ with a given monodromy group. A recent
spectacular result of Blázquez-Sanz, Casale, Freitag, and Nagloo [21], known
as _Ax-Schanuel_ asserts that, if an algebraic subvariety $V\subset P$ has an
_atypical intersection_ $U\subset V\cap\mathcal{L}$ with some leaf
$\mathcal{L}$ then the projection of $U$ to the base lies in a strict weakly
special subvariety. Our strategy now proceeds as follows:

  1. I.

Relate the atypicality inequality (1.1) to an analogous one appearing in the
torsor $P\to S$ constructed in §2.2.

  2. II.

Over-parametrize by an algebraic parameter space $\mathcal{Y}$ all
subvarieties of $P$ that can give rise to atypical intersections with flat
leaves in $P$ above $S$;

  3. III.

Use the Ax-Schanuel Theorem _in families_ (3.12) to show that, all atypical
intersections with leaves parameterized by $\mathcal{Y}$ project into the
fibres of finitely many families of weakly special subvarieties of $S$ (not
necessarly atypical themselves); and

  4. IV.

Conclude by either using some rigidity or maximality property.

To implement the above strategy, Hodge theory (essentially through the _fixed
part theorem_) is needed at two steps. It is used to guarantee, a priori, that
the atypical intersections we are interested in can be over-parametrized by
algebraic families (of subvarieties of $P$), as well as to say that the weakly
special subvarieties one obtains after applying Ax-Schanuel fibre-by-fibre can
be assembled in countably many families (of subvarieties of $S$).

Arguments of this kind are not new, see for example [15, §3.6] for a history
of successful applications of Ax-Schanuel-type theorems to Zilber-Pink. Also,
in the case of subvarieties of Shimura varieties, a “differential-geometric”
approach (rather than the o-minimal one of Baldi-Klingler-Ullmo) to the
geometric Zilber-Pink theorem can already found in [20], and the
generalization to arbitrary VMHS doesn’t pose particular difficulties. There
is however a key difference between earlier approaches and ours: namely, we
use (an opportune) Ax-Schanuel in $P$ rather than the Hodge-theoretic one
associated to some period domain $D$. At first sight, this might look like a
small and technical difference but, in reality, it makes the scope of the
Zilber-Pink theory much broader. The reason is that the Hodge-theoretic Ax-
Schanuel considers only atypical intersections inside $S\times D$, which
typically has much smaller dimension than $P$. This means that there can be
more atypical intersections in $P$, since there is more freedom to impose
algebraic conditions on leaves $\mathcal{L}\subset P$ then on the graph of
some period map. This difference is crucial for treating the case of orbit
closures: the space $\Omega\mathcal{M}_{g}(\kappa)$ supports a
$\mathbb{Z}$VMHS, but the atypicality of the orbit closure $\mathcal{N}$ can
be detected only in an opprtune automorphic bundle above the mixed Shimura
variety associated to the $\mathbb{Z}$VMHS. Said in more plain terms: the
conditions defining orbit closures include algebraic conditions on periods of
the one-form $\omega$ which do not come from conditions on Hodge structures,
but can be imposed inside $P$. In particular our main results are evidence in
favour of a deeper Zilber-Pink philosophy that enriches already the case of
varieties mapping to mixed Shimura varieties, but also includes mixed Hodge
theory.

###  1.3 Applications of the main result to Teichmüller and Hodge theory

####  1.3.1 What do we mean by effective?

When we use the term “effective” to describe a proof or a result, we will mean
in the sense that the proof describes an explicit algorithm, implementable in
principle on a computer, to compute the object whose existence is described by
the proof; in particular, if we say that the proof of the finiteness of a set
of orbit closures is effective, we mean that the proof gives an algorithm to
compute this set. Note that our use of the term “algorithm” is to be always
understood in the sense of “algorithm that is provably guaranteed to terminate
with the correct output” — we generally do not speak of “non-terminating
algorithms”, nor does the word “algorithm” refer to a kind of heuristic
procedure or method.

To make such statements truly precise in an algebro-geometric context, it is
important to specify computational models. More precisely, one should specify
precisely in what terms the data involved in the proof — whether it be an
algebraic variety, a connection, or an algebraic map — is being presented,
both when viewed as “input data” and when viewed as “output data”. To do this
carefully, we will largely follow the conventions of [83, §2]. We give
additional details in §9.

####  1.3.2 Teichmüller–new proofs and new results

The first result is a new and effective proof of the first part of 1.1 (which
is a special case of 1.2), which appears in §6.

######  Theorem 1.5 (Effective finiteness for atypical intersections).

Each stratum $\Omega\mathcal{M}_{g}(\kappa)$ contains only finitely many
maximal atypical suborbit closures. There is moreover an algorithm that
computes them.

In particular the above gives a positive answer to the first part of the
following, see also related discussions from [65].

######  Question 1.6 ([44, Qtn. 5.1.5. (see also §6.2 and 6.3)]).

Can one make the bound on the number of [maximal] atypical orbit closures
effective. More generally, can one effectively bound their numerical
invariants (rank, torsion corank) and arithmetic invariants (discriminant of
order of number field, index of lattice)?

More precisely, we will actually obtain a new approach to 1.2 in its entirety.
Namely we will give, a priori different, definitions of _(a)typical suborbit
closures_ $\mathcal{N}$ (relatively to some
$\mathcal{M}\subset\Omega\mathcal{M}_{g}(\kappa)$) and prove both the
finiteness and density results. At least in the case where $\mathcal{M}$
agrees with some stratum our definitions will agree with the ones given by
Eskin, Filip, and Wright. See indeed 6.3 for all details.

Another related finiteness is the one appearing in the work of Wright,
recalled below (we added the word _maximal_ in the statement below, since we
believe it is necessary).

######  Theorem 1.7 ([90, Thm. 1.2]).

There are only finitely many (maximal) totally geodesic submanifolds (with
respect to the Kobayashi metric) of $\mathcal{M}_{g,n}$ of dimension greater
than $1$.

We recall how the above theorem is proven, and the relationship between
totally geodesic submanifolds and orbit closures, simply following the end of
[90, §2]. By Theorem 1.3 in _op. cit._ , if $N$ is a totally geodesic
submanifold of dimension at least 2, then $\Omega N$ has rank at least 2. Here
we denote by $\Omega N$ the locus of square roots of quadratic differentials
in the largest dimensional stratum of the cotangent bundle to $N$. Since
$\Omega N$ determines $N$, and there are a finite list of strata that may
contain $\Omega N$ for $N$ a totally geodesic submanifold of
$\mathcal{M}_{g,n}$, the result in turn follows from 1.1. In particular 1.5
gives also an effective version of the above finiteness (since there is an
algorithm to compute the finitely many orbit closures of rank at least 2).

####  1.3.3 Hodge–new proofs and new results

Let $S$ be a smooth quasi-projective variety, and $\mathbb{V}\to S$ an
admissible graded-polarizable integral VMHS.

######  Definition 1.8.

By a _family_ $f:Z\to Y$ of subvarieties of a variety $V$ we mean a pair
$(f,\pi)$, where $f$ is a surjective map of varieties, and $\pi:Z\to V$ is an
algebraic map restricting to a closed embedding on each fibre of $f$.

######  Theorem 1.9 (Geometric mixed ZP).

There are only finitely many families of maximal monodromically atypical
subvarieties of $(S,\mathbb{V})$.

######  Remark 1.10.

One may interpret the statement of 1.9 as simply saying that all such weakly
special subvarieties fit into a common family over a (possibly disconnected)
base.

Our proof of 1.9 is effective in the sense that it gives an algorithm to
compute the monodromically atypical families given the data of the algebraic
vector bundle, algebraic connection, and algebraic Hodge filtration underlying
$(S,\mathbb{V})$ as input. One can also obtain from the algorithm a priori
bounds on the degree of the polynomials defining the fibres of the finitely
many monodromically atypical families. (In fact, this is also true in many
cases in the orbit closure setting as well, though we will not dwell on it;
the key point is explained in §3.2.1.)

The mixed version was not known before, not even for mixed Shimura varieties.
(The prior work of Gao [46] resolved the so-called geometric mixed André-Oort
conjecture, and later the geometric Zilber-Pink conjecture [47, Thm. 8.2] for
the case of mixed Shimura varieties of “Kuga type”.) The first instance of
such can be found already for subvarieties of abelian varieties in [51, §9].
We remark here that the approach of [15] to 1.3 is not, a priori, effective:
in general the use of o-minimality is highly non-constructive and makes
describing an effective procedure to determine the finitely many families
described in 1.9 difficult. Instead our approach is more similar to previous
work on effective atypical intersections [31, 83], and in particular [20]. Our
approach can be used to give an effective version of the following corollary
of 1.9.

######  Corollary 1.11.

The images of the finitely many families of maximal atypical subvarieties in
$S$ are not Zariski dense when $\mathbb{V}$ has $\mathbb{Q}$-simple algebraic
monodromy group.

Finally we hope that an effective geometric Zilber-Pink could have interesting
number theoretic applications, once plugged in the main result of [14], as
well as some applications to the geometry of the _Noether-Lefschetz locus_
(cf. [13] and for example Qtn. 8 in _op. cit._).

####  1.3.4 Effective characterization of complex totally geodesic
subvarieties in non-arithmetic ball quotients

Finally, we record another interesting application of our effective proof of
the geometric Zilber-Pink conjecture (this time for pure $\mathbb{Z}$VHS).
Recall indeed the following question of Bader, Fisher, Miller and Stover (that
could be compared with 1.6, see indeed §1.4 for a deeper comparison between
the two worlds). See §10.4 for more details.

######  Question 1.12 ([5, Qtn. 9.5]).

Can one classify the totally geodesic curves on nonarithmetic Deligne–Mostow
[68, 37] orbifolds?

Once our new approach to the pure geometric Zilber-Pink is plugged in the main
result of [17] we at least get an algorithm that solves their question. The
following is an effective version of the finiteness theorem from [17, 5]:

######  Corollary 1.13.

Let $\Gamma\subset\operatorname{PU}(1,n)$ be a Deligne-Mostow non-arithmetic
lattice. Then there exist an algorithm which computes the finitely many
maximal complex totally geodesic subvarieties of
$S_{\Gamma}:=\Gamma\backslash\mathbb{B}_{\mathbb{C}}^{n}$.

We will explain in §10.4 how we model the varieties $S_{\Gamma}$; in
particular, one can always find a representative of the commensurability class
of $\Gamma$ such that $S_{\Gamma}$ and the natural $\mathbb{Z}$VHS over it
comes from geometry.

It seems very hard to make the strategy from [4, 5] effective, since it relies
on dynamical methods (as it was for 1.1). More generally, the proof of the
above corollary works for any non-arithmetic lattice given as the monodromy
group of an “explicit” family of varieties.

###  1.4 Final comments: dynamics vs functional transcendence

To put 1.4 into context, we quickly survey some other results partially
motivated by 1.1. On the dynamical side, this result inspired [4] (and the
later [5]), where it was shown that the arithmeticity of a finite-volume
hyperbolic manifold is determined by its maximal totally geodesic
submanifolds. Both results inspired the main result of [17], which already
obtained finiteness results thanks to some functional transcendence. From
here, the functional transcendence/Zilber-Pink viewpoint was repeatedly
advanced further. The last mentioned result, and the insightful notes [57],
were the basic foundation for [15] (later generalised even further, in a
different direction, in [82]), which finally hinted to the right level of
generality of the theory that we develop here.

By drawing a diagram where the arrow “$\rightsquigarrow$” is understood as
“inspired”

[40, Thm. 1.5] $\rightsquigarrow$ [4, Thm. 1.1] and [5, Thm. 1.1]
$\rightsquigarrow$ [17, Thm. 1.2.1] $\rightsquigarrow$ [15, Thm. 3.1]

we see that the approach of our paper _closes the circle_ , by offering a
common setting where one can deal with all of the above. The only one notable
exception is [4, Thm. 1.1], since it is outside complex algebraic geometry.
See also the ICM report [45] for an overview of such results as well as [16]
for more about the second arrow.

We remark here that in dynamics one often studies the frame flow and the frame
bundle (e.g. [66, page 1053], to name one). Likewise for our approach it is
essential to work with a frame bundle, as discussed in §2.2. It would be
exciting to explore this similarity, and in particular, to find, loosely
speaking, atypical intersection/functional transcendence phenomena in the
theory of finite-volume hyperbolic manifolds. To conclude, we also mention
that the very first results towards the geometric André-Oort conjecture (a
very special case of 1.3) were built on dynamical tools (Ratner theory to be
preicese), see indeed [29].

###  1.5 Related works

Our paper is not the first one to use ideas from functional transcendence to
study orbit closures, although the previous ones had different motivations. To
conclude the introduction, we mention here the history, to the best of our
knowledge, of results on translation surfaces/abelian differentials linked to
the Zilber-Pink viewpoint. A first link between Zilber-Pink in
$\mathbb{G}_{m}^{n}\times\mathbb{G}_{a}^{n}$ and orbit closures was observed
by Bainbridge, Habegger, and Möller [6]. Klingler and Lerer [59] started
analysing strata of abelian differentials, through the lens of _bi-algebraic
geometry_ (however the $\mathrm{GL}_{2}(\mathbb{R})$-action does not enter in
the picture). In [12], Bakker and Tsimerman proved [59, Conj. 2.15], giving an
_Ax–Lindemann_ type of statement for abelian differentials. See also the
related [38], where the authors construct a non-linear bi-algebraic curve,
resp. surface, of translation surfaces of genus 7, resp. 10. Even more
recently, Benirschke [18] investigated some special cases of Ax-Schanuel for
translation surfaces. In a different but related direction, we also mention
the work of Pila and Tsimerman, regarding an Ax-Schanuel for complex curves
and differentials [75, Thm. 3.2]. We hope the framework we consider will have
more applications to this emerging viewpoint.

On the technical side, as we discussed above, the novelty of our approach is
to work in the period torsor $P$. Its construction and the link between
monodromy and differential Galois group is certainly not new and was noticed
by several people in the nineties (but certainly known even before, since it
is a generalization of Picard-Vessiot theory), and inspired by the work of
Deligne [33]. However we weren’t able to find a reference that accommodates
our construction. For example O. Gabber pointed out to us [25, §5] (in
particular point b) of §5.4) and various papers and preprints of B. Malgrange
(like [63], where a more complicated setting is studied). The oldest reference
we are aware of is due to Schlessinger. The role of the Gauss-Manin connection
(and related foliated bundles) for the study of some Hodge loci is also
highlighted in the work of Movasati, see for example [69]. With the aim of
obtaining effective versions of the geometric Zilber-Pink for Shimura
varieties (first obtained in [32]), Binyamini and Daw [20] also noticed the
advantage of the differential algebraic geometry viewpoint for atypical
intersection problems (regarding the arithmetic side of the story, we refer
also to Binyamini’s work [19]). The framework considered by Binyamini and
Binyamini-Daw, together with the very general Ax-Schanuel of Blázquez-Sanz,
Casale, Freitag, and Nagloo discussed above (and its Hodge theoretic
incarnation from [12]) were the main source of inspiration for our work.
Finally we mention also the work of Pila and Scanlon [74], who used some
interesting constructibility and model theoretic arguments related to
algebraic differential equations as well as [60] (especially Thm. 6.1 in _op.
cit._).

###  1.6 Outline of paper

In §2, we recall the general formalism of $H$-bundles and principal
connections (beyond the Hodge theoretic setting) and recall some basic facts
on the Riemann-Hilbert correspondence and weakly special subvarieties. The
main new result of §3 is the _Ax-Schanuel in families_ , that will behind the
next applications. From §4 on we crucially use Hodge theory. We first recall
some of the key facts of the theory of $\mathbb{Z}$VMHS, and then, in §6,
apply such ideas to explore atypical intersections in an enhanced Hodge-
theoretic setting — incorporating both the data of a VMHS and the periods of
the natural holomorphic one form $\omega$ — to establish Eskin-Filip-Wright
finiteness (1.2). In §7 we present a similar argument to obtain the geometric
mixed Zilber-Pink, confirming that our (foliated) approach answers 1.4. In the
rest of the paper, we explain how our results are effective and discuss
algorithmic aspects. In particular in §10, we provide the effective
refinements of the finiteness statements obtained in previous sections.

###  1.7 Acknowledgements

The first author thanks the organizers of the workshop ‘Finiteness results for
special subvarieties: Hodge theory, o-minimality, dynamics’ at the Lake Como
School of Advanced Studies (partially supported by the Grant ERC-2020-ADG
n.101020009 - TameHodge) where various discussions on Zilber-Pink and orbit
closures happened (in particular with S. Filip, D. Fisher, Z. Gao, B.
Klingler, L. Lerer, N. Miller, M. Stover, and E. Ullmo). In particular we
thank S.Filip, whose notes [44] helped us clarify the theory of orbit closures
and the link with atypical intersections. We thank also O. Gabber for
discussions on foliated $H$-bundles.

The authors were partially supported by the NSF grant DMS-1928930, while they
were in residence at the MSRI in Berkeley, during the Spring 2023 semester.
Key parts of the paper were done also while the authors were hosted by the
I.H.E.S. (France) which we thank for the excellent working conditions.

Finally we hope it will be clear to the reader that our applications to
Teichmüller theory own a great debt to the breakthrough work of Eskin, Filip
and Wright.

##  2 $\nabla$-weakly special subvarieties

Before starting our discussion on weakly special subvarieties and the Riemann-
Hilbert correspondence, we recall a well-known theorem of Chevalley (see e.g.
Prop. 3.1 in Deligne’s ariticle in the collection [36]).

Let $G$ be an affine algebraic group, acting faithfully on a vector space $V$.
This gives an inclusion $G\subset\operatorname{GL}(V)$.

######  Theorem 2.1 (Chevalley’s Theorem).

For any algebraic subgroup $H\subset G$, there exists a representation $S(V)$
constructed from $V$ by standard tensor operations (i.e. taking direct sums,
duals, or tensor products, in any order and any finite number of times), and a
line $L\subset S(V)$, such that $H$ is the stabilizer of $L$ in $G$.

If moreover $H$ is reductive, then there is a vector $v\in S(V)$, such that
$H$ is the stabilizer of $v$ in $G$.

###  2.1 Preliminary definitions

We start with a pointed complex irreducible smooth variety $(S,s_{0})$ and a
local system $\mathbb{V}$ on $S$, corresponding to a representation
$\rho:\pi_{1}(S,s_{0})\to\operatorname{GL}(\mathbb{V}_{s_{0}})$.

######  Definition 2.2.

Given an irreducible (not necessary smooth) subvariety
$(Y,s_{0})\subset(S,s_{0})$, we define its _algebraic monodromy group_
$\mathbf{H}_{Y}=\mathbf{H}_{Y,\mathbb{V}}=\mathbf{H}_{Y,\mathbb{V},s_{0}}$ as
the identity component of the Zariski closure of the induced monodromy
representation
$\pi_{1}(Y^{\textrm{nor}},s_{0})\to\operatorname{GL}(\mathbb{V}_{s_{0}})$,
where $Y^{\textrm{nor}}\to Y$ denotes the normalization. When $\mathbb{V}$ and
the point $s_{0}$ are understood, we will just write $\mathbf{H}_{Y}$.

Note that algebraic monodromy groups are unchanged by passing to an open
subvariety of $Y$, or replacing $Y$ with a smooth resolution — this is because
if $X$ is an irreducible complex normal variety and $U\subset X$ is an open
subvariety, $\pi_{1}(U)\to\pi_{1}(X)$ is surjective.

######  Definition 2.3.

A geometrically irreducible variety $Y\subset S$ is said to be _weakly
special_ if it is maximal under inclusion among geometrically irreducible
subvarieties of $S$ for the dimension of $\mathbf{H}_{Y}$. In other words, any
$Y^{\prime}$ strictly containing $Y$ satisfies
$\mathbf{H}_{Y}\varsubsetneq\mathbf{H}_{Y^{\prime}}$.

Directly from the definition $S$ is a weakly special subvariety, and in cases
of interest — for instance when $\mathbb{V}$ comes from a family of cohomology
groups of algebraic varieties inducing a quasi-finite period map — every
closed point $s\in S$ will also be weakly special. We will primarily interest
ourselves in other weakly specials, not equal to either $S$ or a closed point.
The former we call _strict_ weakly special subvarieties.

###  2.2 A recap of Riemann-Hilbert

Given a smooth irreducible algebraic variety $S$ and complex local system
$\mathbb{V}\to S$, there is the _Riemann-Hilbert correspondence_ functor

$\displaystyle\tau:\textrm{LocSys}(S^{\textrm{an}})$
$\displaystyle\to\textrm{MIC}_{\textrm{reg}}(S)$ $\displaystyle\mathbb{V}$
$\displaystyle\mapsto\tau(\mathbb{V})=(\mathcal{H},\nabla).$

Here $\textrm{MIC}_{\textrm{reg}}$ denotes the category of locally free
modules $\mathcal{H}$ on $S$ with regular-singular integrable connection
$\nabla$. The functor has the property that the space of
$\nabla^{\textrm{an}}$-flat sections of the locally free sheaf
$\mathcal{H}^{\textrm{an}}$ may be identified with $\mathbb{V}$ up to natural
isomorphism. Moreover, the analytic vector bundle
$\mathbb{V}\otimes_{\mathbb{C}}\mathcal{O}^{\textrm{an}}$ with its natural
flat connection has a unique algebraic structure $(\mathcal{H},\nabla)$ with
regular singularities. The functor $\tau$ is an equivalence of (Tannakian)
categories [33].

Given a fixed such $(\mathcal{H},\nabla)$, the tensor powers
$\mathcal{H}^{a,b}:=\mathcal{H}^{\otimes a}\otimes(\mathcal{H}^{*})^{\otimes
b}$ also carry natural algebraic connections, where we use $(-)^{*}$ to denote
the dual vector bundle. Correspondingly, write
$\mathbb{V}^{a,b}=\mathbb{V}^{\otimes a}\otimes(\mathbb{V}^{*})^{\otimes b}$.
Now let $\mathbf{H}_{S}$ be the algebraic monodromy group of $\mathbb{V}$.
Using Chevellay’s 2.1 on invariants, we may understand the group
$\mathbf{H}_{S}$ as the identity component of the stabilizer of a one-
dimensional subsystem $\mathbb{L}\subset\bigoplus_{a,b}\mathbb{V}^{a,b}$, and
hence it admits an algebraic realization $\mathbf{H}_{S,s}$ in each fibre
$\operatorname{GL}(\mathcal{H}_{s})$ as the identity component of the
stabilizer of $\tau(\mathbb{L})_{s}$.

Associated to this data we may construct a certain algebraic subbundle
$P(s_{0})$ of the principal left
$\operatorname{GL}(\mathcal{H}_{s_{0}})$-bundle
$\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H}):=\operatorname{\mathcal{I\kern-1.0pts\kern-2.0pto}}(\mathcal{H},\mathcal{O}_{S}\otimes_{\mathbb{C}}\mathcal{H}_{s_{0}})$;
here the fibres
$\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H})_{s}$ above
points $s\in S$ are identified with the set of isomorphisms
$\textrm{Iso}(\mathcal{H}_{s},\mathcal{H}_{s_{0}})$. We write points of
$\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H})$ as pairs
$(\beta,s)$, where $\beta:\mathcal{H}_{s}\to\mathcal{H}_{s_{0}}$. To construct
$P(s_{0})$ we define

$P(s_{0}):=\\{(\beta,s)\in\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H}):\beta(\tau(\mathbb{L})_{s})=\tau(\mathbb{L})_{s_{0}}\\}$
(2.1)

where $\mathbb{L}\subset\bigoplus_{a,b}\mathbb{V}^{a,b}$ is as above. This
defines an algebraic subbundle of
$\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H})$, and using
flat frames we may observe that each $P(s_{0})$ is naturally a left torsor for
the Zariski closure $\mathbf{H}^{\textrm{full}}_{S}$ of the monodromy
representation at $s_{0}$.

######  Proposition 2.4.

Locally in the étale topology on $S$, the torsor $P(s_{0})$ is trivial. More
precisely around any point $s\in S$ there exists an étale map $\iota:U\to S$
whose image contains $s$ such that
$\iota^{*}P(s_{0})\simeq\bigsqcup_{i=1}^{k}\mathbf{H}_{S}\times U$ as an
algebraic bundle, where $k$ is the number of connected components of
$\mathbf{H}^{\textrm{full}}_{S}$.

###### Proof.

We may observe the map $P(s_{0})\to S$ is of finite presentation and flat; the
latter condition may be checked in the analytic topology, and is implied by
the fact that $P(s_{0})$ is analytically a fibration. This implies that
$P(s_{0})$ is an fppf torsor: we have a trivialization
$P(s_{0})\times\mathbf{H}^{\textrm{full}}_{S}\xrightarrow{\sim}P(s_{0})\times_{S}P(s_{0})$
given on points by $((\beta,s),g)\mapsto((\beta,s),(g\circ\beta,s))$. Then use
the fact that every algebraic fppf torsor is an étale torsor; c.f. [54]. ∎

We may interpret the statement of the proposition as saying that each
$P(s_{0})$ is an étale torsor for the Zariski closure of the monodromy of
$\mathbb{V}$.

We will usually work instead with the component $P$ of $P(s_{0})$ which
contains the point $(\textrm{id}_{s_{0}},s_{0})$, where
$\textrm{id}_{s_{0}}:\mathcal{H}_{s_{0}}\to\mathcal{H}_{s_{0}}$ is the
identity map. Then $P$ is an étale torsor for $\mathbf{H}_{S}$.

###  2.3 Principal Bundles Preliminaries

In this section we first introduce the formalism of $H$-bundles and principal
connections. The reader may refer for example to [21, §2] for more details,
although we will try to be as self contained as possible. Throughout we work
over the complex numbers.

Let $S$ be an (irreducible) smooth variety, and $H$ a linear algebraic group.
Let $\pi:P\to S$ be an $H$-_principal_ bundle, that is $H$ acts on $P$ (on the
right, with the action simply denoted by $\cdot$) and such that the action
induces an isomorphism

$P\times H\simeq P\times_{S}P,\ (p,g)\mapsto(p,p\cdot g).$

In particular the fibres of $\pi$ are principal homogeneous spaces, and the
choice of a point $p\in P_{s}:=\pi^{-1}(s)$ induces an isomorphism $H\cong
P_{s}$, given by $g\mapsto p\cdot g$. We have the following exact sequence of
$S$-bundles

$0\to\textrm{ker}\,d\pi\to TP\to TS\times_{S}P\to 0,$ (2.2)

where $\textrm{ker}\,d\pi$ is the so called _vertical bundle_. A _connection_
$\nabla$ is simply a section of (2.2). We say that $\nabla$ is $H$-_principal_
if it is $H$-equivariant. We additionally say that $\nabla$ is _flat_ if it
lifts to a morphism of Lie algebras: the equation
$\nabla_{[v,w]}=[\nabla_{v},\nabla_{w}]$ holds on the level of maps of
$\mathbb{C}$-schemes. From now on all connections are $H$-principal and flat.

######  Definition 2.5.

An _integral submanifold_ of $P$ is a locally closed complex submanifold whose
tangent subbundle lies entirely in the image of $\nabla$. A _leaf_ is a
maximal integral submanifold.

######  Example 2.6.

The component $P$ containing $(\textrm{id}_{s_{0}},s_{0})$ of the bundle
$P(s_{0})$ introduced in §2.2 is a principal $\mathbf{H}_{S}$-bundle that
naturally comes with an algebraic flat connection. To translate between the
notion of connection in §2.2 and the notion of connection given here, we
consider the algebraic connection

$\delta:\operatorname{\mathcal{H\kern-1.0pto\kern-2.0ptm}}(\mathcal{H},\mathcal{O}_{S}\otimes_{\mathbb{C}}\mathcal{H}_{s_{0}})\to\Omega^{1}_{S}\otimes\operatorname{\mathcal{H\kern-1.0pto\kern-2.0ptm}}(\mathcal{H},\mathcal{O}_{S}\otimes_{\mathbb{C}}\mathcal{H}_{s_{0}}).$

Using this map we construct a map of algebraic bundles

$\nabla:TS\times_{S}\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H})\to
T\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H}).$

This map sends a pair $(v,(\beta,s))$ to the differential operator
$\nabla(v,(\beta,s))$ at $(\beta,s)$ which computes the derivative of a
function on $\operatorname{\mathcal{P\kern-1.0pte\kern-2.0ptr}}(\mathcal{H})$
in the flat direction along $v$. By using $\delta$ to write out an explicit
system of algebraic differential equations, one can see $\nabla$ is an
algebraic map. Then $\nabla$ restricts to a map $TS\times_{S}P\to TP$, which
gives the desired section of (2.2).

######  Remark 2.7.

We remark here that similar considerations appeared (at least in the VHS
setting) in [28, Thm. 2.5] and [12, Lem. 2.8]. In these references, compared
to our account, the authors use _o-minimality_ to produce an algebraic bundle
$P$.

##  3 Atypical weakly special subvarieties

###  3.1 Ax-Schanuel after Blázquez-Sanz, Casale, Freitag, and Nagloo

######  Definition 3.1 ([21, Def. 3.4]).

An algebraic group $H$ is _sparse_ if for any strict Lie subalgebra
$\mathfrak{l}\subsetneq\mathfrak{h}=\operatorname{Lie}(H)$, the algebraic
envelope $\mathfrak{l}^{a}$ of $\mathfrak{l}$ is again a strict algebraic Lie
subalgebra of $\mathfrak{h}$ (i.e. there exists a strict algebraic subgroup
$L=L_{\mathfrak{l^{a}}}\subset H$ such that
$\mathfrak{l}\subset\operatorname{Lie}(L)$).

We remark that a subgroup of a sparse group does not have to be sparse. The
main applications of the paper will concern the case where $H$ is a semidirect
product of a unipotent group and a semisimple one; such groups are sparse. For
example, for the main results on the orbit closures in
$\Omega\mathcal{M}_{g}(\kappa)$, $H$ will actually be the complex group
$\operatorname{Sp}_{2g}\ltimes\mathbb{G}_{\operatorname{a}}^{2g\cdot n}$
(where $n+1$ is the length of $\kappa$).

Fix a principal $H$-bundle $\pi:P\to S$ with flat $H$-equivariant connection
$\nabla$, as in §2.3. Assume that $H$ is sparse. See [21, §2.2 and 2.3] for
more details on the next definition:

######  Definition 3.2.

An irreducible subvariety $V\subset P$ is a _minimal $\nabla$-invariant
variety_ if it is the Zariski closure of a $\nabla$-horizontal leaf.

######  Definition 3.3 ([21, Lem. 2.3]).

Let $V$ be a minimal $\nabla$-invariant variety. Then the _Galois group of
$S$_

$H_{V}:=\operatorname{Gal}(\nabla)=\operatorname{Gal}(S)=\\{g\in H:V\cdot
g=V\\},$

is an algebraic subgroup of $H$ and $\pi_{|V}$ is an $H_{V}$-principal bundle.

In our setting, without loss of generality, from now on we assume that
$H=\operatorname{Gal}(\nabla)=\operatorname{Gal}(S)$. This will always be true
for us as a consequence of 3.8 below.

######  Definition 3.4.

A Zariski closed irreducible subvariety $Y\subset S$ is _$\nabla$ -special_ if
its Galois group $\operatorname{Gal}(Y)=\operatorname{Gal}(\nabla_{|Y})$ is
strictly contained in $\operatorname{Gal}(S)$, and it is maximal for this
property.

The following is the main theorem of Blázquez-Sanz, Casale, Freitag, and
Nagloo. (For an effective version of the below see also [39, Thm. 4].)

######  Theorem 3.5 ([21, Thm. 3.6]).

Let $H$ be a sparse group, and let $\nabla$ be an $H$-principal connection on
$\pi:P\to S$ with Galois group $H$. Let $V$ be a subvariety of $P$, $x\in V$,
and let $\mathcal{L}\subset P$ be a horizontal leaf through $x$. Let $U$ be an
analytic irreducible component of $V\cap\mathcal{L}$.

If

$\operatorname{codim}_{P}U<\operatorname{codim}_{P}V+\operatorname{codim}_{P}\mathcal{L},$

(or equivalently $\dim V<\dim U+\dim H$), then the projection of $U$ in $S$ is
contained in a $\nabla$-special subvariety (in the sense of 3.4).

######  Remark 3.6.

We refer to [12, Ex. 3.5] for a discussion about how Ax-Schanuel fails when
$H$ doesn’t have sparse monodromy/differential Galois group.

We will apply this theorem through the following corollary:

######  Corollary 3.7.

Let $S$ be a smooth algebraic variety and complex local system $\mathbb{V}\to
S$. Let $(P,\nabla)$ the associated bundle of 2.6. Assume that the monodromy
group $H=\mathbf{H}_{S}$ of $\mathbb{V}$ is sparse. Let $V$ be a subvariety of
$P$, $x\in V$, and let $\mathcal{L}\subset P$ be a formal horizontal leaf
through $x$. Let $U$ be an analytic irreducible component of
$V\cap\mathcal{L}$

If

$\operatorname{codim}_{P}U<\operatorname{codim}_{P}V+\operatorname{codim}_{P}\mathcal{L},$

then the projection of $U$ in $S$ is contained in a weakly special subvariety
(in the sense of 2.3).

To prove 3.7 we check that $\nabla$-special subvarieties (in the sense of 3.4)
are indeed weakly special (as in 2.3). We note that both the definition of
algebraic monodromy group and differential Galois group of a variety $Y\subset
S$ are stable under replacing $Y$ with either a smooth resolution or
normalization. Moreover the construction of the bundle in 2.6 is compatible
with restriction and pullback. It is therefore enough to prove the following.

######  Lemma 3.8.

Let $P$ be constructed as in 2.6. Then
$\mathbf{H}_{S}=\operatorname{Gal}(\nabla)$.

###### Proof.

The construction of $P$ implies that
$\operatorname{Gal}(\nabla)\subset\mathbf{H}_{S}$, so it suffices to observe
that any $V$ as in 3.2 necessarily contains a monodromy orbit in each of its
fibres above points $s\in S(\mathbb{C})$. But any point $p\in V(\mathbb{C})$
mapping to $s\in S(\mathbb{C})$ contains a leaf $\mathcal{L}$ passing through
$p$, and hence also contains any analytic continuation of $\mathcal{L}$ which
passes through the fibre of $P\to S$ above $s$. The points of all such
analytic continuations above $s$ are in the image of a monodromy orbit at $s$
coming from the monodromy representation associated to $\mathbb{V}$. Allowing
$s$ to vary one sees that $V=P$. ∎

######  Example 3.9.

We stress that 3.8 uses crucially that the algebraic structure on $P$ arises
from a connection with regular singularities, and that one has
$\mathbf{H}_{S}\subsetneq\operatorname{Gal}(\nabla)$ in general. For instance,
the connections given by $\nabla=d-df$ on $\mathbb{A}^{1}\setminus\\{0\\}$ and
with $f$ a function on $\mathbb{A}^{1}\setminus\\{0\\}$ all have an underlying
constant local system, and hence $\mathbf{H}_{S}=\\{\textrm{id}\\}$, but can
have non-trivial Galois groups.

###  3.2 A constructibility fact

A useful fact which we will use repeatedly is the following:

######  Proposition 3.10.

Let $(P,\nabla)$ be an algebraic $H$-principal bundle with connection on an
algebraic variety $S$, and let $f:\mathcal{Z}\to\mathcal{Y}$ be a family of
subvarieties of $P$ (in the sense of 1.8). Then the locus

$\mathcal{Y}(f,e):=\\{(x,y)\in
P\times\mathcal{Y}:\dim_{x}(\mathcal{L}_{x}\cap\mathcal{Z}_{y})\geq e\\}$
(3.1)

is algebraically constructible.

Above and in the rest of the text, by $\dim_{x}$ we denote the local analytic
dimension at the point $x$.

###### Proof.

By stratifying the base $\mathcal{Y}$, we may assume $\mathcal{Y}$ is smooth.

Consider the bundle $(Q,\delta)$ obtained from $(P,\nabla)$ by pulling back
along the natural projection $\pi:S\times\mathcal{Y}\to S$. From the
definition of the fibre product one obtains an embedding $\mathcal{Z}\subset
Q$. Then $\mathcal{Z}\subset Q$ is foliated by the intersections
$\mathcal{L}_{z}\cap f^{-1}(y)$ where $\mathcal{L}_{z}\subset Q$ is an
analytic leaf passing through $z\in\mathcal{Z}$. The locus

$\mathcal{Z}(f,e):=\\{z\in\mathcal{Z}:\dim_{z}\mathcal{L}_{z}\cap
f^{-1}(y)\geq e\\}$

is closed analytic. Note that if $x=\pi(z)$ is the image of $z$ in $P$, then
the germ of the intersection $\mathcal{L}_{z}\cap f^{-1}(y)$ at $z$ is
identified with the germ of the intersection $\mathcal{L}_{x}\cap f^{-1}(y)$
at $x$, with $\mathcal{L}_{x}$ a leaf of $P$ passing through $x$.

In fact we claim that $\mathcal{Z}(f,e)$ is also algebraic, and describe an
explicit way to construct it. We may assume that $\mathcal{Z}$ is irreducible.
We denote points of $\mathcal{Z}(f,e)\subset\mathcal{Z}$ by $(x,y)$, in
accordance with the embedding into the product $P\times\mathcal{Y}$. For each
$e$, we define

$\mathcal{Z}(f,e)^{1}:=\\{(x,y)\in\mathcal{Z}:\dim T_{x}\mathcal{L}_{x}\cap
T_{x}f^{-1}(y)=e\\}.$

Because $\nabla$ is algebraic, so is the bundle $\bigcup_{\mathcal{L}\subset
P}T\mathcal{L}$ above $P$, hence the locus $\mathcal{Z}(f,e)^{1}$ is defined
by algebraically constructible conditions.

Now let $e_{f}$ be the dimension of the smallest component of $\mathcal{L}\cap
f^{-1}(y)$ as $\mathcal{L}$ and $y$ vary over all possible choices. Then for
$e\leq e_{f}$ we have $\mathcal{Z}(f,e)=\mathcal{Z}(f,e_{f})=\mathcal{Z}$, so
it suffices to show that $\mathcal{Z}(f,e)$ is algebraic for $e>e_{f}$. Now
for $e=e_{f}+1$ the locus $\mathcal{Z}(f,e)^{1}$ is not dense in
$\mathcal{Z}$: the intersections of the fibres $f^{-1}(y)$ with the leaves
$\mathcal{L}$ must be smooth at a generic point, so have tangent space
dimension at most $e_{f}$ at a generic point. Letting
$\mathcal{Z}^{\prime}\subset\mathcal{Z}$ denote the closure of
$\mathcal{Z}(f,e)^{1}$, we may construct the family
$f^{\prime}:\mathcal{Z}^{\prime}\to\mathcal{Y}$ and reduce to showing the
algebraicity of $\mathcal{Z}(f^{\prime},e)$ for all $e$. The proof is
therefore complete by Noetherian induction.

The locus $\mathcal{Y}(f,e)$ is just the image of $\mathcal{Z}(f,e)$, hence is
algebraically constructible. ∎

Once we have an algorithmic way of working with principal bundles and their
connections, the above proof gives an algorithm for computing the loci
$\mathcal{Z}(f,e)$ and $\mathcal{Y}(f,e)$. This will be the essential input in
making our results effective in §9.

####  3.2.1 A priori degree bounds

Although it is not our main focus, let us explain how one can use the above
proof to obtain an a priori bound on the degree of the closure of
$\mathcal{Y}(f,e)$, and therefore also the closure of its image in $S$. Since
in applications one will use such $\mathcal{Y}(f,e)$ to construct atypical
(weakly) special loci, including atypical orbit closures, we believe this can
be of interest in situations where computing the orbit closure itself is
computationally difficult but computing the degree estimates is more
tractable.

The basic input is the following elementary fact (see for example [53]):

######  Lemma 3.11.

Let $f:\mathbb{A}^{m}\to\mathbb{A}^{m}$ be a polynomial map. Suppose
$f=(f_{1},\ldots,f_{m})$, and let $\kappa=\operatorname{max}_{i}\\{\deg
f_{i}\\}$. Then for an algebraic subvariety $W\subset\mathbb{A}^{m}$ of
dimension $r$, one has $\deg\overline{f(W)}^{\operatorname{Zar}}\leq[\deg
V]\kappa^{r}$. ∎

If one then runs the argument of 3.10, then this fact can be used to bound the
degree of $\overline{\mathcal{Y}(f,e)}^{\textrm{Zar}}$ and its image in $S$
after choosing explicit coordinates for $S,P,\mathcal{Z}$, $\mathcal{Y}$, etc,
and that we have an explicit presentation for $\nabla$ as a section of (2.2).
Indeed, since $\mathcal{Y}(f,e)$ is the image of $\mathcal{Z}(f,e)$, this
reduces to bounding the degree of $\mathcal{Z}(f,e)$. Then $\mathcal{Z}(f,e)$
is constructed as the closure of loci of the form
$\mathcal{Z}(f^{\prime},e^{\prime})^{1}$ for appropriate
$(f^{\prime},e^{\prime})$ determined by $(f,e)$. These loci, in turn, are
projections of loci in the tangent bundle $TP$ defined by explicit equations,
so we may bound their degrees as well. The number of such loci needed in the
proof can be bounded by bounding the number of steps needed in the Noetherian
induction, which can be done by using the degree bound on each
$\mathcal{Z}^{\prime}$ appearing in the proof to bound the number of its
components as well.

###  3.3 Ax-Schanuel in Families

Given a family $h$ of subvarieties of $S$, we refine the set
$\mathcal{Y}(f,e)$ introduced in the previous section (3.1), as follows:

$\displaystyle\mathcal{Y}(f,e,h):=\left\\{(x,y)\in
P\times\mathcal{Y}:\begin{array}[]{c}\textrm{ exists an embedded analytic germ
}\\\ (x,\mathcal{D})\subset(x,\mathcal{Z}_{y}\cap\mathcal{L}_{x})\textrm{ of
dimension }\\\ \textrm{ at least }e\textrm{ mapping into a fibre of
}h\end{array}\right\\}$

The following complements 3.7. It says that if one applies the Ax-Schanuel
theorem to a family of subvarieties then the resulting intersections belong to
a family of weakly special subvarieties. (For a family version of Ax-Schanuel
in a special case see [71, Prop. 2.1].)

######  Theorem 3.12.

Fix some family of subvarieties of $P$, $f:\mathcal{Z}\to\mathcal{Y}$, and an
integer $e\geq 1$, such that each fibre of $f$ has dimension $<\dim H+e$.

Suppose that there exists a countable collection of families
$\\{h_{i}:C_{i}\to B_{i}\\}_{i=1}^{\infty}$ of subvarieties of $S$, all of
whose fibres are weakly special, and that all weakly special subvarieties of
$S$ are among the fibres of these families. Then the set $\mathcal{Y}(f,e)$
introduced in 3.10 is contained in $\bigcup_{i=1}^{m}\mathcal{Y}(f,e,h_{i})$
for some index $m$ after reordering the $i$’s.

This will be deduced from:

######  Theorem 3.13.

Consider a constructible subset $\mathcal{K}\subset P\times\mathcal{Y}$.
Instead of the dimension condition $<\dim H+e$ in 3.12, suppose one knows that
each germ $(x,\mathcal{D})$ of dimension $\geq e$ and corresponding to a point
$\mathcal{Y}(f,e)\setminus\mathcal{K}$ maps into a fibre of some family in the
collection $\\{h_{i}:C_{i}\to B_{i}\\}_{i\in I}$, where $I\subset\mathbb{N}$
is some subset. Then in fact this is true for some finite subset of $I$.

###### Proof.

We first notice that 3.12 reduces to 3.13 as a consequence of 3.7.

We define $f\cap h_{i}$ to be the family of subvarieties of $P$ whose fibres
are the intersections of fibres of $f$ with the subbundles
${\left.\kern-1.2ptP\vphantom{\big{|}}\right|_{f^{-1}(y)}}\subset P$. Then the
base of the family $f\cap h_{i}$ is a space $\mathcal{Y}_{i}$ with a natural
map $\mathcal{Y}_{i}\to\mathcal{Y}$. Then we have that
$\textrm{im}[\mathcal{Y}(f\cap h_{i},e)\to
P\times\mathcal{Y}]=\mathcal{Y}(f,e,h_{i})$, so $\mathcal{Y}(f,e,h_{i})$ is
constructible by 3.10. By assumption we have that
$\mathcal{Y}(f,e)\setminus\mathcal{K}$ is contained in the countable union
$\bigcup_{i=1}^{\infty}\mathcal{Y}(f,e,h_{i})$, but since everything is
constructible, it is in fact contained in finitely many of these sets by 3.14
below. ∎

######  Lemma 3.14.

Let $Y$ be a complex variety and $\mathcal{I}\subset Y(\mathbb{C})$ be a
constructible subset, and $\\{U_{r}\\}_{r=1}^{\infty}$ be constructible
subsets of $Y(\mathbb{C})$. If
$\mathcal{I}\subset\bigcup_{r=1}^{\infty}U_{r}$, then in fact it lies in the
union of finitely many of the $U_{r}$.

###### Proof.

This follows from [83, Lem. 4.32], where one takes the sequence of
intersections in [83, Lem. 4.32] to be trivial (all equal to $\mathcal{I}$)
and the countable union to be
$\mathcal{I}=\bigcup_{r=1}^{\infty}(\mathcal{I}\cap U_{r})$. ∎

##  4 Introduction to variations of mixed Hodge structures: period maps and
Zilber-Pink

In this section we briefly recall the Zilber–Pink conjecture for admissible
variations of polarizable integral mixed Hodge structures and explain more
carefully 1.3 (and its variants in the mixed setting). There is a first
version of the Zilber–Pink conjecture appearing [57], we explain here the
_strong version_ , following [15] (in _op. cit._ the authors state only the
pure version, but, as they notice, it’s easy to obtain the general mixed
formulation).

###  4.1 Notations

First of all some general notations/conventions. In this section an algebraic
variety $S$ is a reduced scheme of finite type over the field of complex
numbers, not necessarily irreducible. If $S$ is an algebraic (resp. analytic)
variety, by a subvariety $Y\subset S$ we always mean a _closed_ algebraic
(resp. analytic) subvariety. A pure $\mathbb{Q}$-Hodge structure of weight $n$
on a finite dimensional $\mathbb{Q}$-vector space $V_{\mathbb{Q}}$ is a
decreasing filtration $F^{\bullet}$ on the complexification $V_{\mathbb{C}}$
such that
$V_{\mathbb{C}}=\oplus_{p\in\mathbb{Z}}F^{p}\oplus\overline{F^{n-p}}$. The
category of pure $\mathbb{Q}$-Hodge-structures is Tannakian, and the category
of polarizable pure $\mathbb{Q}$-Hodge-structures is semisimple (all pure
Hodge structures appearing in this paper will be polarizable). Recall that the
Mumford–Tate group $\mathbf{MT}(V)\subset\operatorname{GL}(V)$ of a
$\mathbb{Q}$-Hodge structure $V$ is the Tannakian group of the Tannakian
subcategory $\langle V\rangle^{\otimes}$ of $\mathbb{Q}$-Hodge structures
generated by $V$. Equivalently, it is the smallest $\mathbb{Q}$-algebraic
subgroup of $\operatorname{GL}(V)$ whose base-change to $\mathbb{R}$ contains
the image of $h:\mathbb{S}\to\operatorname{GL}(V_{\mathbb{R}})$ (here
$\mathbb{S}$ denotes the _Deligne torus_ , i.e. the Weil restriction to
$\mathbb{R}$ of the complex multiplicative group). It is also the fixator in
$\operatorname{GL}(V)$ of the Hodge tensors for $V$. As $V$ is polarised, this
is a reductive group. See for example [15] for more Hodge-theoretic details.
We will recall below in more detail the case of VMHS, but assuming some
familiarity with the pure setting. As a reference for Tannakian categories we
refer to the article of Deligne and Milne from [36] and for general facts in
Hodge theory to [34]. For basic knowledge on variations of _mixed_ Hodge
structures and _mixed_ Mumford–Tate domains, we refer for example to [48,
§2-5] and also to [76].

###  4.2 The Tannakian category of $\mathbb{Q}$VMHS, first definitions

A mixed $\mathbb{Z}$-Hodge structure ($\mathbb{Z}$MHS) is a triple
$(V,W_{\bullet},F^{\bullet})$ consisting of a (torsion-free) finitely
generated $\mathbb{Z}$-module $V$, a finite ascending filtration $W_{\bullet}$
of $V_{\mathbb{Q}}:=V\otimes_{\mathbb{Z}}\mathbb{Q}$ (called the _weight
filtration_) and a finite decreasing filtration $F^{\bullet}$ of
$V_{\mathbb{C}}$ (called the _Hodge filtration_) such that for each
$n\in\mathbb{Z}$,

$(\operatorname{Gr}_{n}^{W}V_{\mathbb{Q}},\operatorname{Gr}_{n}^{W}F^{\bullet})$

is a pure $\mathbb{Q}$-Hodge structure of weight $n$. A _graded polarization_
of a mixed $\mathbb{Z}$-Hodge structure is the datum of a polarization on the
pure $\mathbb{Q}$-Hodge structure
$\operatorname{Gr}^{W}V=\bigoplus_{n}\operatorname{Gr}^{W}_{n}V$.

The following is [76, 1.4, 1.5] and explains under which conditions a
homomorphism $h:\mathbb{S}_{\mathbb{C}}\to\operatorname{GL}(V_{\mathbb{C}})$
gives a MHS:

######  Lemma 4.1.

Let $V$ be a finite dimensional $\mathbb{Q}$-vector space. A morphism
$h:\mathbb{S}_{\mathbb{C}}\to\operatorname{GL}(V_{\mathbb{C}})$ defines a
$\mathbb{Q}$MHS on $V$ if and only if there exists a connected
$\mathbb{Q}$-algebraic subgroup $\mathbf{G}\subset\operatorname{GL}(V)$ such
that $h$ factors through $\mathbf{G}_{\mathbb{C}}$ and satisfies the following
conditions (where we write $W_{-1}$ for the unipotent radical of
$\mathbf{G}$):

  * •

The composition morphism
$\mathbb{S}_{\mathbb{C}}\to\mathbf{G}_{\mathbb{C}}\to(\mathbf{G}/W_{-1})_{\mathbb{C}}$
is defined over the reals;

  * •

Precomposing $\mathbb{S}\to(\mathbf{G}/W_{-1})_{\mathbb{R}}$ with the weight
homomorphism $\mathbb{G}_{m,\mathbb{R}}\to\mathbb{S}$ we obtain a cocharacter
of the center of $(\mathbf{G}/W_{-1})_{\mathbb{R}}$ defined over $\mathbb{Q}$;

  * •

The weight filtration on $\operatorname{Lie}\mathbf{G}$ defined by
$\operatorname{Ad}\circ h$ satisfies
$W_{0}(\operatorname{Lie}\mathbf{G})=\operatorname{Lie}\mathbf{G}$ and
$W_{-1}(\operatorname{Lie}\mathbf{G})=\operatorname{Lie}W_{-1}$.

######  Definition 4.2.

A _mixed Hodge datum_ is a pair $(\mathbf{G},D)$ where $\mathbf{G}$ is a
connected linear algebraic $\mathbb{Q}$-group (with unipotent radical
$\mathbf{W}_{-1}$) and $D$ is a
$\mathbf{G}(\mathbb{R})\mathbf{W}_{-1}(\mathbb{C})$-conjugacy class of
homomorphisms $\mathbb{S}\to\mathbf{G}(\mathbb{C})$, satisfying the conditions
of 4.1.

An important admissibility condition for variations of mixed Hodge structure
was introduced by Kashiwara in [55] (see also, for example, [48, Def. 3.3]).
From now on we denote by $\mathbb{Z}$VMHS the category of graded-polarizable
and admissible VMHS over $S$. When $\mathbb{Z}$ is replaced by $\mathbb{Q}$,
this is a Tannakian category, as discussed for example in [57, §2.4], and
references therein. (The fact that the category of admissible $\mathbb{Q}$VMHS
is abelian due to Kashiwara, see [55, §4.5 and Prop. 5.2.6] for the relevant
statements.)

######  Definition 4.3.

For every $s\in S$, the _Mumford-Tate_ group $\mathbf{G}_{s}(\mathbb{V})$ of
the Hodge structure $\mathbb{V}_{s}$ is the Tannakian group of
$\langle\mathbb{V}_{s}\rangle^{\otimes}$ of $\mathbb{Q}$MHS (here the exact
faithful $\mathbb{Q}$-linear tensor functor
$\operatorname{For}:\mathbb{Q}MHS\to\operatorname{Vec}_{\mathbb{Q}}$ is simply
the forgetful functor).

It is a connected $\mathbb{Q}$-algebraic group, and

  * •

reductive if $\mathbb{V}_{s}$ (or $\mathbb{V}$) is pure; in general

  * •

it is an extension of $\mathbf{G}_{s}(\textrm{Gr}^{W}V)$ by a unipotent group
(here $W$ denotes the weight filtration on $\mathbb{V}$).

We note that $\textrm{Gr}^{W}$ can be viewed as an exact functor [34, Them.
2.3.5(iv)] from $\mathbb{Q}$VMHS to $\mathbb{Q}$VHS. We have also a notion of
_Mumford–Tate group_ of the $\mathbb{Q}$VMHS $\mathbb{V}$, simply as the
Tannakian group of $\mathbb{V}$. Here we pick the fibre functor associated to
some base point $s\in S$, that is the functor

$\omega_{s}:\mathbb{Q}\operatorname{VMHS}\to\mathbb{Q}\operatorname{MHS}\to\operatorname{Vec}_{\mathbb{Q}},\hskip
20.00003pt\mathbb{V}\mapsto\mathbb{V}_{s}\mapsto\operatorname{For}(\mathbb{V}_{s}).$

We denote the Mumford–Tate group by $\mathbf{MT}(\mathbb{V},\omega_{s})$.
Thanks to [36, II, Thm. 3.2], the resulting group, by the general theory, will
not depend on such choice of a base point $s$ (that is there’s a _unique
natural_ isomorphism
$\mathbf{MT}(\mathbb{V},\omega_{s})\cong\mathbf{MT}(\mathbb{V},\omega_{s^{\prime}})$).
See also [1] for an equivalent point of view. By the general Tannakian
formalism we have a canonical inclusion
$\mathbf{MT}(\mathbb{V}_{s})\subset\mathbf{MT}(\mathbb{V},\omega_{s})$, at
every $s\in S$.

We define111It is easy to see that our definition agrees with the others in
the literature, even if it doesn’t often appear in this shape. Indeed usually
the fibre functor $\omega_{s}$ is implicit and one just writes
$\mathbf{MT}(\mathbb{V}_{s})\varsubsetneq\mathbf{MT}(\mathbb{V})$, or often
the inclusion is understood up to _parallel transport_. the following subset
of $S^{\operatorname{an}}=S(\mathbb{C})$.

$\operatorname{HL}(S,\mathbb{V}^{\otimes}):=\\{s\in
S^{\operatorname{an}}:\mathbf{MT}(\mathbb{V}_{s})\varsubsetneq\mathbf{MT}(\mathbb{V},\omega_{s})\\}.$
(4.1)

It is naturally a countable union of complex subspaces, and in fact Cattani,
Deligne and Kaplan [26, Thm. 1.1] proved the following in the pure case, and
Patrick, Pearlstein, and Schnell in the mixed case [23]. We also refer to [9,
Thm. 1.6] (see also [10]) for an alternative proof, using o-minimality, and
also to [7], for the mixed case.

######  Theorem 4.4.

Let $S$ be a smooth connected complex quasi-projective algebraic variety and
$\mathbb{V}$ be a $\mathbb{Z}$VMHS over $S$. Then
$\operatorname{HL}(S,\mathbb{V}^{\otimes})$ is a countable union of closed
irreducible algebraic subvarieties of $S$: the strict maximal special
subvarieties of $S$ for $\mathbb{V}$.

In general, the notion of _special subvariety_ is defined as follows:

######  Definition 4.5.

Let $(S,\mathbb{V})$ be a $\mathbb{Z}$VMHS. A _special_ subvariety $Y\subset
S$ is a closed irreducible complex algebraic subvariety which is maximal for
the Mumford-Tate group $\mathbf{G}_{Y}$ associated to
$(Y,{\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{Y}})$.

For us, from now on, all special subvarieties, so in particular all components
of the Hodge locus, will be equipped with their _reduced_ algebraic structure.

###  4.3 Period maps and atypical intersections

From now on we look only at admissible $\mathbb{Z}$VMHS. Then the data of such
a pair $(S,\mathbb{V})$ is the same datum as a _(mixed) period map_. Let
$(\mathbf{G},D)$ be a (connected, mixed) Hodge datum as in [57, Def. 3.3]
associated to $(S,\mathbb{V})$, and $\Gamma\backslash D$ the associated mixed
Mumford–Tate domain (called also Hodge variety by some authors). (Here and in
the sequel, we always assume that $\Gamma$ is a torsion free finite index
subgroup of $\mathbf{G}(\mathbb{Z})$). We will be mainly concerned with period
maps:

$\Psi:S^{\operatorname{an}}\to\Gamma\backslash D,$

that is locally liftable, holomorphic and horizontal maps (see [57, §3.5]).

######  Definition 4.6.

  1. 1.

A subvariety $Z$ of $S$ is said of positive period dimension for $\mathbb{V}$
if $\Psi(Z^{\operatorname{an}})$ has positive dimension.

  2. 2.

The _Hodge locus of positive period dimension_
$\operatorname{HL}(S,\mathbb{V}^{\otimes})_{\operatorname{pos}}$ is the union
of the special subvarieties of $S$ for $\mathbb{V}$ of positive period
dimension.

Here we follow the atypicality notion first introduced in [15], rather than
the one originally proposed by Klingler in [57].

######  Definition 4.7.

A special subvariety $Y\subset S$, with Hodge datum $(\mathbf{G}_{Y},D_{Y})$
(associated to $\mathbb{V}_{|Y^{\operatorname{sm}}}$) is _atypical_ if

$\operatorname{codim}_{\Gamma\backslash
D}\Psi(Y^{\operatorname{an}})<\operatorname{codim}_{\Gamma\backslash
D}\Psi(S^{\operatorname{an}})+\operatorname{codim}_{\Gamma\backslash
D}\Gamma_{Y}\backslash D_{Y},$ (4.2)

or equivalently,

$\dim\Psi(S^{\operatorname{an}})-\dim\Psi(Y^{\operatorname{an}})<\dim D-\dim
D_{Y}.$

Otherwise, it said to be _typical_. Here $\Gamma_{Y}$ simply denotes
$\Gamma\cap\mathbf{G}_{Y}(\mathbb{Q})$. In general, the induced map
$\Gamma_{Y}\backslash D_{Y}\to\Gamma\backslash D$ is known to be quasi-finite,
so the formal “codimension” makes sense.

######  Definition 4.8.

The _atypical Hodge locus_
$\operatorname{HL}(S,\mathbb{V}^{\otimes})_{\operatorname{atyp}}\subset\operatorname{HL}(S,\mathbb{V}^{\otimes})$
(resp. the _typical Hodge locus_
$\operatorname{HL}(S,\mathbb{V}^{\otimes})_{\operatorname{typ}}\subset\operatorname{HL}(S,\mathbb{V}^{\otimes})$)
is the union of the atypical (resp. strict typical) special subvarieties of
$S$ for $\mathbb{V}$.

In this paper we will be mainly concerned with the case of atypical special
subvarieties. In particular we want to investigate the following:

######  Conjecture 4.9 (Zilber–Pink conjecture for $\mathbb{Z}$VMHS).

$\operatorname{HL}(S,\mathbb{V}^{\otimes})_{\operatorname{atyp}}$ is
algebraic, i.e. there are only finitely many maximal (reduced) atypical
special subvarieties.

###  4.4 Weakly Special subvarieties in Hodge Theory

Essentially from 4.4, we observe that there are countably many special
subvarieties for $(S,\mathbb{V})$. It is also easy to observe that an
intersection of special subvarieties is a union of special subvarieties. In
this section we compare such notions and proprieties with the more general
concept of _weakly special subvarieties_ introduced in 2.3. These arguments
are essentially easy generalizations of the ones appearing in [15, 60] to the
mixed case and most of them appear in [48].

Let $\mathbb{V}$ be a $\mathbb{Z}$VMHS on $S$. For any closed point $s\in S$,
let $\mathbf{G}_{S,s}$ be the Mumford-Tate group of $(S,\mathbb{V})$ at
$\omega_{s}$, and $\mathbf{H}_{S,s}$ the algebraic monodromy at $s$ in the
sense of 2.2 applied to the $\mathbb{Q}$-local system
$\mathbb{V}_{\mathbb{Q}}$. We will need the following [1, Thm. 1] (see also
[34]):

######  Theorem 4.10 (André-Deligne Monodromy Theorem).

The monodromy group $\mathbf{H}_{S,s}$ is a normal subgroup of the derived
subgroup of $\mathbf{G}_{S,s}$.

We notice here that, in the general mixed case, the derived subgroup of
$\mathbf{G}_{S,s}$ is not a semisimple group.

######  Definition 4.11.

A subset $D^{\prime}$ of the classifying space $D$ is called _a weak Mumford-
Tate subdomain_ if there exists an element $t\in D^{\prime}$ and a normal
subgroup $\mathbf{N}$ of the derived group of $\mathbf{MT}(t)$ such that
$D^{\prime}=\mathbf{N}(\mathbb{R})^{+}\mathbf{N}^{u}(\mathbb{C})\cdot t$,
where $\mathbf{N}^{u}$ denotes the unipotent radical of $\mathbf{N}$.

A _weakly special subvariety_ of the Hodge variety $\Gamma\backslash D$ is an
irreducible component of the image in $\Gamma\backslash D$ of a weak Mumford-
Tate subdomain. The datum $((\mathbf{H},D_{H}),t)$ is called a _weak Hodge
subdatum_ of $(\mathbf{G},D)$. Given $(S,\mathbb{V})$ with associated period
map $\Psi:S\to\Gamma\backslash D$, every weakly special subvariety (in the
sense 2.3) of $(S,\mathbb{V})$ can be described as an irreducible component of
the preimage along $\Psi$ of a weakly special submanifold of the Hodge variety
$\Gamma\backslash D$; see [12, Cor. 2.9]. With this new description of weakly
special subvarieties in terms of the period domain $\Gamma\backslash D$, we
obtain the following:

######  Lemma 4.12.

A special subvariety $Y$ for $(S,\mathbb{V})$ is weakly special. The property
of being weakly special is closed under taking intersections and passing to
irreducible components.

The closure of special and weakly special subvarieties under intersection
gives natural notions of special and weakly special closures: the special
(resp. weakly special) closure of a variety $Y\subset S$ is the intersection
of all special (resp. weakly special) subvarieties in which it is contained.

To conclude we record the following fact about the monodromy of
$\mathbb{Z}$VMHS (actually the same holds true even for $\mathbb{C}$VMHS, see
[35, Prop. 1.13] and [87, Thm. 1.1] for related discussions and details). It
is essentially a corollary of 4.10 and [28, Lem. 3.3], which asserts that if
$G$ is a linear algebraic group whose quotient by its unipotent radical
$G^{u}$ is semisimple, then $G$ is sparse (recall 3.1).

######  Lemma 4.13.

The algebraic monodromy of a $\mathbb{Z}$VMHS is sparse.

###  4.5 Parameter spaces of mixed Hodge structures and weak atypicality

For our exposition we essentially follow [8, §3.5, and §6].

Let $V$ be a finite dimensional real vector space equipped with an increasing
filtration $W_{\bullet}$ and a collection of non-degenerate bilinear forms
$q_{k}:\operatorname{Gr}^{W}_{k}\otimes\operatorname{Gr}^{W}_{k}\to\mathbb{R}$
that are $(-1)^{k}$-symmetric. Fix a partition of the dimension of $V$ into
non-negative integers $h^{p,q}$ satisfying Hodge symmetry. For any integer $k$
we denote by $\widecheck{{\Omega}}_{k}$ the complex projective algebraic
variety parametrizing the $(q_{k})_{\mathbb{C}}$-isotropic filtrations
$\\{F^{\prime p}_{k}\\}$ on $\operatorname{Gr}^{W}_{k}V_{\mathbb{C}}$ with
$\dim F^{\prime p}_{k}=\sum_{r\geq p}h^{r,k-r}$. The complex group
$\textrm{Aut}(q_{k})$ acts transitively by algebraic automorphisms on
$\widecheck{{\Omega}}_{k}$.

Let $(V,W_{\bullet},F^{\bullet},q)$ be a mixed $\mathbb{Z}$-Hodge structure as
above, polarized by $q$. We denote by $D$ the corresponding _mixed period
domain_ : the set of decreasing filtrations $F^{\prime\bullet}$ of
$V_{\mathbb{C}}$ such that $(V_{\mathbb{R}},W_{\bullet},F^{\prime\bullet},q)$
is a real mixed Hodge structure graded-polarized by $q$ and such that

$\dim_{\mathbb{C}}(F^{\prime
p}\operatorname{Gr}_{p+q}^{W}\cap\overline{F^{\prime}}^{q}\operatorname{Gr}_{p+q}^{W})=\dim
F^{p}\operatorname{Gr}_{p+q}^{W}\cap\overline{F}^{q}\operatorname{Gr}_{p+q}^{W}=h^{p,q}.$

By definition, $D$ is a semi-algebraic open subset of the smooth projective
complex variety $\widecheck{{D}}$ that parametrizes the decreasing filtrations
of $V_{\mathbb{C}}$ by complex vector subspaces such that the filtration
induced on the graded pieces $\operatorname{Gr}^{W}_{k}V_{\mathbb{C}}$ is
inside $\widecheck{{\Omega}}_{k}$ for each $k$. In particular
$\widecheck{{D}}$ maps to the product of $\widecheck{{\Omega}}_{k}$.

Given a $\mathbb{Z}$VMHS $\mathbb{V}$ on $S$, we have associated a mixed Hodge
datum $(\mathbf{G}_{S},D_{S})$ as in 4.2 (we always assume that
$\mathbf{G}_{S}$ is the generic Mumford-Tate group of $\mathbb{V}$). In this
case, $\mathbf{G}_{S}$ comes with a faithful linear representation
$\rho:\mathbf{G}_{S}\to GL(V)$, and thanks to this representation, we obtain a
morphism from $D_{S}$ to the appropriate classifying space $\widecheck{{D}}$
introduced above.

##### Notation

We denote by $\widecheck{{D}}_{S}$ the Zariski closure of $D_{S}$ in
$\widecheck{{D}}$ (it comes with a transitive action of
$\mathbf{G}(\mathbb{C})$ and by
$\widecheck{{D}}^{0}_{S}\subset\widecheck{{D}}_{S}$ the monodromy orbit of
some base point $h_{0}\in D_{S}$, and write
$D^{0}_{S}=\widecheck{{D}}^{0}_{S}\cap D_{S}$. In particular to a
$\mathbb{Z}$VMHS $\mathbb{V}$ on $S$ we obtain the complete _Hodge-monodromy
datum_ :

$(\mathbf{H}_{S}\subset\mathbf{G}_{S},D^{0}_{S}\subset D_{S})$

######  Remark 4.14.

We note that $D_{S}$ (resp. $D_{S}^{\circ}$) is open in $\widecheck{{D}}_{S}$
(resp. $\widecheck{{D}}^{0}_{S}$), see [28, Thm. A.1].

In §7 we prove the so called _geometric part of Zilber-Pink_. For geometric
Zilber-Pink we will work with a finer notion of atypicality, that we now
recall. This corresponds to what, in [15, Def. 5.3] is called _monodromically
atypical_. See also Lem. 5.6, Ex 5.7, and Rmk. 5.7 in _op. cit._ for a more
detailed discussion, in the pure case, of such a definition and comparisons
with other variants. (The link between the two notions ultimately comes from
4.10.)

######  Definition 4.15.

A weakly special subvariety $Y$ for $(S,\mathbb{V})$ is called _monodromically
atypical_ if

$\dim\Psi(S^{\operatorname{an}})-\dim\Psi(Y^{\operatorname{an}})<\dim\widecheck{{D}}^{0}_{S}-\dim\widecheck{{D}}^{0}_{Y}.$
(4.3)

######  Remark 4.16.

Suppose $\Psi$ is quasi-finite, so every point is a weakly special subvariety.
Then unless $\Psi$ is dominant, each point is in fact a monodromically
atypical weakly special subvariety.

Geometric Zilber-Pink says, roughly, that there are finitely many _families_
of maximal monodromically atypical subvarieties. (The more precise statement
gives additional information about the Hodge-theoretic data defining those
families.) In the case where the ambient monodromy group of $S$ is
$\mathbb{Q}$-simple, this can be used to recover special cases of the true
Zilber-Pink conjecture for positive-dimensional atypical intersections, which
is why one often prefers the statement appearing in 1.3.

One subtlety about the generalization of the argument from [15] and other
prior work in the pure setting is that semisimplicity of monodromy is often
used (usually to say that a group, up to conjugation, has only finitely many
subgroups). This is not true in the general mixed case, so our extension will
require more care to ensure that atypical mixed intersections can still be
appropriately “over-parameterized” in line with the strategy discussed in
§1.2.

We also recall the Ax-Schanuel Theorem for period maps. It is implied by the
more general theorem 3.5. The original proof follows a series of papers, see
[11, 27, 48, 12].

######  Theorem 4.17 (Ax-Schanuel in the period domain).

Let $W\subset S\times\widecheck{{D}}^{0}$ be an algebraic subvariety. Let $U$
be an irreducible complex analytic component of $W\cap
S\times_{\Gamma\backslash D^{0}}D^{0}$ such that

$\operatorname{codim}_{S\times\widecheck{{D}}^{0}}U<\operatorname{codim}_{S\times\widecheck{{D}}^{0}}W+\operatorname{codim}_{S\times\widecheck{{D}}^{0}}(S\times_{\Gamma\backslash
D^{0}}D^{0})\;\;.$

Then the projection of $U$ to $S$ is contained in a strict weakly special
subvariety of $(S,\mathbb{V})$.

Notice that $S\times_{\Gamma\backslash D^{0}}D^{0}$ is simply the image of the
graph of $\tilde{\Phi}:\widetilde{S}\to D^{0}$ under $\widetilde{S}\times
D^{0}\to S\times D^{0}$. Then the intersection $W\cap
S\times_{\Gamma\backslash D^{0}}D^{0}$ can be identified with the intersection
in $S\times D^{0}$ between $W$ and the image of $\widetilde{S}$ in $S\times
D^{0}$ along the map $(\pi,\tilde{\Phi})$.

###  4.6 Relation with Period Torsors Associated to $\mathbb{V}$

The relationship between 4.17 and 3.7 is ultimately explained by the torsor
$P$ in 2.6 associated to the underlying local system of $\mathbb{V}$. Let
$(\mathbf{H}_{S},D^{0}_{S})$ be the weak Hodge datum associated to
$\mathbb{V}$. We may fix a basepoint $s_{0}\in S$, and regard this data as
being associated to the fibre $\mathbb{V}_{s_{0}}$. Then in particular,
$\widecheck{{D}}^{0}=\widecheck{{D}}^{0}_{S}$ is a variety parameterizing
mixed Hodge data at $s_{0}$. Recall that $P$ was constructed as subbundle of
the total space of the vector bundle
$\operatorname{\mathcal{H\kern-1.0pto\kern-2.0ptm}}(\mathcal{H},\mathcal{O}_{S}\otimes_{\mathbb{C}}\mathcal{H}_{s_{0}})$,
where $\mathcal{H}$ is the algebraic bundle underlying
$\mathbb{V}\otimes_{\mathbb{Z}}\mathcal{O}_{S^{\textrm{an}}}$. Then we obtain
a natural algebraic map $\nu:P\to\widecheck{{D}}^{0}$. Concretely, this map
acts as

$[\eta\in\textrm{Hom}(\mathcal{H}_{s},\mathcal{H}_{s_{0}})]\mapsto\eta(W_{\bullet,s},F^{\bullet}_{s}),$

where we use that both the Hodge flag and the filtration by weight are given
by algebraic subbundles of $\mathcal{H}$. This is a combination of the fact
that the weight filtration on $\mathbb{V}$ is given by local subsystems to
which we can apply the Riemann-Hilbert functor, together with the
corresponding result for the Hodge filtration on the pure graded quotients,
which is a consequence of work of Schmid [78]. To check that the map is well-
defined (the image lands inside $\widecheck{{D}}^{0}$ rather than some larger
flag variety), it suffices by $\mathbf{H}_{S}(\mathbb{C})$-invariance to check
this for a single leaf, where it follows from the fact that the associated
period map lands inside $\widecheck{{D}}^{0}$.

To recover 4.17 from 3.7 one pulls back the intersections appearing in the
statement of 4.17 along the map $\pi\times\nu:P\to
S\times\widecheck{{D}}^{0}$. We note that the image of a leaf
$\mathcal{L}\subset P$ which arises by parallel translating the identity
$\textrm{id}\in\textrm{Hom}(\mathbb{V}_{s_{0}},\mathbb{V}_{s_{0}})$ using the
integral structure of $\mathbb{V}$ has as image in
$S\times\widecheck{{D}}^{0}$ exactly the fibre product
$S\times_{\Gamma\backslash D^{0}}D^{0}$ appearing in 4.17. Since the map
$\pi\times\nu$ is surjective, the atypicality condition is preserved. See
indeed the discussion in [12].

###  4.7 Families of Weakly Specials in the Mixed Setting

In our applications of the results of §3.3, we will need the following fact.

######  Proposition 4.18.

Let $(S,\mathbb{V})$ be a $\mathbb{Z}$VMHS. Then there exists a countable
collection $\\{h_{i}:C_{i}\to B_{i}\\}_{i=1}^{\infty}$ of families of
irreducible algebraic varieties of $S$, all of whose fibres are weakly
special, and such that every weakly special subvarieties arises as a fibre in
some such family. Moreover, the maps $C_{i}\to S$ may be assumed quasi-finite.

###### Proof.

Let $Y\subset S$ be a weakly special subvariety with associated
monodromy/Hodge datum $(\mathbf{H}_{Y}\subset\mathbf{G}_{Y},D_{Y})$. Let
$Y^{\textrm{sp}}$ be the special closure of $Y$, or alternatively, the special
subvariety containing $Y$ defined by $D_{Y}$. Then we can construct a finite
cover $e:\widetilde{Y}^{\textrm{sp}}\to Y^{\textrm{sp}}$ and a period map
$\Xi:\widetilde{Y}^{\textrm{sp}}\to\Gamma_{\widetilde{Y}^{\textrm{sp}}/Y}\backslash
D_{\widetilde{Y}^{\textrm{sp}}/Y}$ associated to the quotient datum
$(\mathbf{G}_{\widetilde{Y}^{\textrm{sp}}/Y},D_{\widetilde{Y}^{\textrm{sp}}/Y})=(\mathbf{G}_{Y},D_{Y})/\mathbf{H}_{Y}$
(in the sense of [48, §5]). From [48, Prop. 5.1 (ii)], the irreducible
components $\widetilde{Z}$ of the fibres of $\Xi$ are weakly special
subvarieties of $\widetilde{Y}^{\textrm{sp}}$ for
${\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{\widetilde{Y}^{\textrm{sp}}}}$,
and their images $Z$ in $Y^{\textrm{sp}}$ are weakly special subvarieties of
$Y^{\textrm{sp}}$ for
${\left.\kern-1.2pt\mathbb{V}\vphantom{\big{|}}\right|_{Y^{\textrm{sp}}}}$
(use that the algebraic monodromy group is unchanged by passing to a finite
covering). Moreover, applying 4.12 to $Y^{\textrm{sp}}$ and the weakly special
closure of some such $Z$ one sees that these varieties are all weakly special
for $S$, and $Y$ is among their fibres.

Although a priori depending on $Y$, the construction above in fact depends
only on the choice of $Y^{\textrm{sp}}$ and a $\mathbb{Q}$-subgroup of
$\mathbf{G}_{Y}$; there are countably many such choices. From either the main
result of [8], or by first taking a proper compactification of $\Xi$ as in [8,
Prop. 2.4] and applying the Stein factorization, it follows that the fibres of
$\Xi$ belong to a common algebraic family $g:\widetilde{Y}^{\textrm{sp}}\to
T$. We conclude thanks to 4.19 below, applied to $(g,e)$. ∎

######  Lemma 4.19.

Let $e:N\to M$ be a quasi-finite map, and $g:Q\to T$ a family of subvarieties
of $N$. Then there exists a family $h:C\to B$, possibly over a disconnected
base, of subvarieties of $M$ whose fibres are exactly the closures of images
under $e$ of components of fibres of $g$.

###### Proof.

We may assume that $e$ is dominant, and that $N$, $M$ and $T$ are irreducible.
By stratifying $T$ with respect to the dimension of its fibres, we may assume
that all fibres have the same dimension. By removing a closed locus from $Q$,
one can further assume that the fibres of $g$ are equidimensional.

We observe that there exists some finite-type family $h:C\to B$ of
subvarieties of $M$, all of whose fibres are irreducible, and such that all
components of varieties of the form $\overline{e(g^{-1}(t))}^{\textrm{Zar}}$
occur as fibres. This can be deduced by observing that, with respect to a
fixed line bundle $\mathcal{L}$ on some projective compactification
$M\subset\overline{M}$, the varieties $\overline{e(g^{-1}(t))}^{\textrm{Zar}}$
all have uniformly bounded degree, so the locus of interest can be cut out of
a Hilbert scheme associated to $\overline{M}$. We then argue that the locus
$L$ of $b\in B$ such that $h^{-1}(b)$ agrees with a component of
$\overline{e(g^{-1}(t))}^{\textrm{Zar}}$ for some $t$ is constructible. It
suffices to consider the induced family $h^{\prime}:C^{\prime}\to B$ whose
fibre above $b$ is $e^{-1}(h^{-1}(b))$, and construct the locus in $T\times B$
where some component of $g^{-1}(t)$ contains some component of
$h^{\prime-1}(b)$. Since both families have equidimensional fibres, this can
be achieved by imposing a dimension condition on the intersection and so
constructibility of this locus follows from [80, Lemma 05F9]. Having done
this, we can partition $L$ by locally closed subvarieties of $B$, and replace
$B$ with the disjoint union of the strata in this partition. Afterwards, the
fibres of $h$ are exactly all those varieties of the form
$\overline{e(g^{-1}(t))}^{\textrm{Zar}}$ for some $t\in T$. ∎

##  5 Recap on orbit closures

In this section we survey some known results on the orbit closures, expanding
the discussion from the beginning of Section 1.1, and recall how they relate
to the map

$\Omega\mathcal{M}_{g}(\kappa)\to\Omega\mathcal{A}_{g,n}.$

We mainly follow [89, 44], but see also [91] for more background about the
ergodic viewpoint.

In the sequel, $\mathcal{M}_{g}$ denotes the moduli space of genus $g$ compact
Riemann surfaces and $\Omega\mathcal{M}_{g}\to\mathcal{M}_{g}$ the bundle
whose fibre over $[X]\in\mathcal{M}_{g}$ is the space of all (non-zero)
holomorphic 1-forms on $X$ (in particular it has dimension $4g-3$). It is a
rank $g$ (algebraic) vector bundle on $\mathcal{M}_{g}$ (with the zero section
removed), naturally equipped with a stratification by smooth (not necessarly
irreducible) algebraic subvarieties $\Omega\mathcal{M}_{g}(\kappa)$, where
$\kappa=(\kappa_{0},\dots,\kappa_{n})$ satisfies $\sum_{i}\kappa_{i}=2g-2$ and
denotes the multiplicities of zeros of $\omega$. Notice that
$\Omega\mathcal{M}_{g}(\kappa)$ has dimension $2g+n$, where
$n=\textrm{length}(\kappa)-1$. For example the only open stratum is
$\Omega\mathcal{M}_{g}(1,...,1)$ (where the 1 is repeated $2g-2$ times).

Below we use some rudiments of variational (mixed) Hodge theory. We have
already discussed such theory in greater generality in §4, but here we only
deal with variation of Hodge structures (VMHS) whose weight filtration has two
steps. In fact in the orbit closure setting all of our period spaces will
belong to the theory of mixed Shimura varieties of _Kuga type_ ; see [47] for
an introduction to this notion.

######  Remark 5.1.

In what follows, we often secretly view $\Omega\mathcal{M}_{g}$ and its strata
as orbifolds or Deligne-Mumford stacks. To avoid this, one can replace
$\Omega\mathcal{M}_{g}$ by a fixed covering over which the pairs $(X,\omega)$
glue together into a universal algebraic family and always work just with
algebraic varieties and variations of Hodge structure on this covering. This
will be harmless for our considerations (the finiteness/abundance of orbit
closures in a stratum can equivalently be detected on a finite covering).

###  5.1 Recollection and first properties

We first recall some basic facts about _relative cohomology_. Let $(X,Z)$ be a
pair of a Riemann surface (i.e. an irreducible smooth projective algebraic
curve over $\mathbb{C}$) and a finite set of points
$Z=\\{z_{1},\dots,z_{n}\\}\subset X$ (that will usually be given by the zeros
of some (non-zero) $1$-form $\omega$ on $X$). We have a short exact sequence:

$0\to\tilde{H}^{0}(Z;\mathbb{Z})\to H^{1}(X,Z;\mathbb{Z})\to
H^{1}(X;\mathbb{Z})\to 0.$ (5.1)

Above, $\tilde{H}^{0}(Z;\mathbb{Z})$ is the _reduced cohomology_ : it denotes
the group of all formal linear combinations of points in $Z$, modulo the
element which takes each point in $Z$ with coefficient $1$ (it is the dual of
the reduced homology group). We will always refer to the middle term as
_relative cohomology_ and the the last term as _absolute cohomology_. Note
that the form $\omega$ naturally induces a class inside both
$H^{1}(X,Z(\omega);\mathbb{Z})$ and $H^{1}(X;\mathbb{Z})$ by integrating over
homological cycles and using homology-cohomology duality.

We have a version of (5.1) for flat bundles above
$\Omega\mathcal{M}_{g}(\kappa)$ (see also [44, 3.1.13]):

$0\to W_{0}\to H^{1}_{rel}\to H^{1}\to 0.$ (5.2)

The notation is justified by the fact that $W_{0}$ denotes the weight-zero
local subsystem for the mixed VHS structure on $H^{1}_{rel}$.

The $\operatorname{SL}_{2}(\mathbb{R})$-action on
$H^{1}(X,Z(\omega);\mathbb{C})$ is given by identifying $\mathbb{C}$ with
$\mathbb{R}^{2}$ and considering

$H^{1}(X,Z(\omega);\mathbb{C})\cong
H^{1}(X,Z(\omega);\mathbb{R})\otimes_{\mathbb{R}}\mathbb{R}^{2}.$

The action is now the natural $\operatorname{SL}_{2}(\mathbb{R})$ on the
$\mathbb{R}^{2}$ at the very end of the equation (it is less straightforward
to describe in a more global way the $\operatorname{SL}_{2}(\mathbb{R})$
action on the moduli space of translation surfaces).

We now recall the concept of developing map and monodromy. To do so, let
$\operatorname{Mod}(X,Z)$ be the mapping class group of diffeomorphisms of $X$
preserving the subset $Z$. We fix a base point
$s:=(X,\omega)\in\Omega\mathcal{M}_{g}(\kappa)$, and look at the developing
map/period coordinates

$\operatorname{Dev}:\widetilde{\Omega\mathcal{M}_{g}(\kappa)}\to
H^{1}(X,Z(\omega);\mathbb{C}).$ (5.3)

Concretely, the map sends $(X^{\prime},\omega^{\prime})$ to
$\omega^{\prime}\in H^{1}(X^{\prime},Z(\omega^{\prime});\mathbb{C})$ and then
applies the flat transport to obtain an element of
$H^{1}(X,Z(\omega);\mathbb{C})$. By a theorem of Veech [84, Thm. 7.5], locally
$\operatorname{Dev}$ is biholomorphic. Moreover $\operatorname{Dev}$ is
equivariant for a representation of

$\pi_{1}(\Omega\mathcal{M}_{g}(\kappa))\to\operatorname{Mod}(X,Z),$

where the mapping class groups acts on cohomology by

$L:\operatorname{Mod}(X,Z)\to\operatorname{Sp}(H^{1}(X,Z))\cong\operatorname{Sp}(H^{1}(X))\ltimes\operatorname{Hom}(H^{1}(X),\tilde{H}^{0}(Z;\mathbb{C})).$
(5.4)

All groups presented so far are naturally subgroups of
$\operatorname{Aut}(H^{1}_{rel,s})$: this is the group of automorphisms of
$H^{1}_{rel,s}$ that act as the identity on $W_{0,s}$ and act symplectically
on $H^{1}_{s}$. If we pick a splitting $H^{1}_{s}\to H^{1}_{rel,s}$, we obtain
a semidirect product decomposition
$\operatorname{Sp}(H^{1}_{s})\ltimes\textrm{Hom}(H^{1}_{s},W_{0,s})$. Indeed
we have

$0\to\textrm{Hom}(H^{1}_{s},W_{0,s})\to\operatorname{Aut}(H^{1}_{rel,s})\to\operatorname{Sp}(H^{1}_{s})\to
0.$

Recall that an element $\xi\in\textrm{Hom}(H^{1}_{s},W_{0,s})$ acts as a
unipotent automorphism of $H^{1}_{rel,s}$ by $v\mapsto v+\xi(p(v))$ (where
$p:H^{1}_{rel}\to H^{1}$ is the projection to absolute cohomology). All
automorphisms of $H^{1}_{rel,s}$ that act as the identity on $W_{0,s}$ and
symplectically on $H^{1}_{s}$ are in fact of this form.

The bundle $H^{1}_{rel}$ carries naturally the structure of an admissible,
graded-polarized, variation of $\mathbb{Z}$-mixed Hodge structures on (each
irreducible component of) $\Omega\mathcal{M}_{g}(\kappa)$. The weight
filtration on $\mathbb{V}$ has two steps (for facts and definitions about
mixed VHS, we refer to Section 4):

$0\to
W_{0}(\mathbb{V})\to\mathbb{V}\to\operatorname{Gr}_{1}^{W}(\mathbb{V})\cong
p^{*}R^{1}f_{*}\mathbb{Z}\to 0.$ (5.5)

Here $p:\Omega\mathcal{M}_{g}(\kappa)\to\mathcal{M}_{g}$ is just the
projection, and $f:\mathcal{C}_{g}\to\mathcal{M}_{g}$ is the universal family
of genus $g$ curves. The local system $W_{0}(\mathbb{V})$ becomes isomorphic
to $\mathbb{Z}(0)^{n}$ after passing to a finite covering. See also [59, §3.3
and §3.4] for a related discussion.

Finally, we refer to [61] (see also [44, 3.1.16]) for a description of the
connected components of $\Omega\mathcal{M}_{g}(\kappa)$, that we will
implicitly use when speaking of atypical orbit closures.

###  5.2 Structure of orbit closures

We recall a special case of a theorem of Eskin, Mirzakhani, and Mohammadi [41,
Thm. 2.1] see also [44, Thm. 1]:

######  Theorem 5.2 (Linearity/topological rigidity).

Each orbit closure $\mathcal{N}\subset\Omega\mathcal{M}_{g}(\kappa)$ is
locally in period coordinates a _linear manifold_ , i.e. any sufficiently
small open $U\subset\mathcal{N}$ is such that $\operatorname{Dev}(U)$ is an
open set inside a linear subspace (see (5.3) for the definition of the
developing map).

Let $\mathcal{N}\subset\Omega\mathcal{M}_{g}(\kappa)$ be an orbit closure, and
denote by $T\mathcal{N}$ its tangent bundle. The linearity of orbit closures,
as recalled in 5.2, allows one to realize $T\mathcal{N}$ as a local subsystem
of $H^{1}_{rel}$. As a consequence we have another short exact sequence of
real bundles above $\mathcal{N}$:

$0\to W_{0}(T\mathcal{N})\to T\mathcal{N}\to H^{1}(T\mathcal{N})\to 0.$ (5.6)

To explain the notation: $H^{1}(T\mathcal{N})$ is just the image of
$T\mathcal{N}\subset H^{1}_{rel}$ in $H^{1}$ (the absolute cohomology), and
$W_{0}(T\mathcal{N})=T\mathcal{N}\cap W_{0}$.

Let $T\subset\operatorname{SL}_{2}(\mathbb{R})$ denote the upper triangular
group.

######  Theorem 5.3 (Isolation property, [41, Thm. 2.3], see also [44, Thm.
4.1.8.]).

For every sequence of orbit closures $(\mathcal{N}_{i})$ there are
$T$-invariant measures $\mu_{i}$, and after passing to a subsequence
$(\mathcal{N}_{i},\mu_{i})$ there is another linear immersed submanifold
$\mathcal{M}$, with finite $T$-invariant measure $\mu$ such that
$\mathcal{N}_{i}\subset\mathcal{M}$ and $\mu_{i}\to\mu$.

Note that the above result implies the same statement with $T$ replaced by
$\operatorname{SL}_{2}(\mathbb{R})$, which is how we will use it; see [44,
Rem. 4.1.9 (ii)].

We can now recall the following theorem of Filip [43] (which builds on the
earlier work [42], as well as earlier work of Wright [88] and Möller [67]),
see also [44, Thm. 2 and Thm. 4.4.2]. We denote by
$K=K_{\mathcal{N}}\subset\mathbb{R}$ the smallest field over which
$\mathcal{N}$ is $K$-linear in the sense of 5.2. We have:

######  Theorem 5.4 (S. Filip).

Let $\mathcal{N}\subset\Omega\mathcal{M}_{g}(\kappa)$ be an orbit closure.
There exists, up to isogeny, a factor $\mathcal{F}\subset\mathcal{J}$ of the
relative Jacobian over $\mathcal{N}$, and a subgroup $\mathcal{S}$ of the free
abelian group on the zeros of $1$-forms, such that:

  1. 1.

$\mathcal{F}$ admits real multiplication by $K$ (i.e.
$K=\operatorname{End}^{0}(\mathcal{F}_{\mathcal{N}})$), which is, in
particular, a totally real number field;

  2. 2.

The Abel-Jacobi map, possibly twisted by real multiplication,
$AJ:\mathcal{S}\to\mathcal{F}$ assumes torsion values;

  3. 3.

For each $(X,\omega)\in\mathcal{N}$, $\omega$ is an eigenform for the real
multiplication on the fibre of $\mathcal{F}$ above $X$ (see also [44,
§4.2.17]).

Furthermore, these conditions, together with a dimension bound (c.f. 5.8),
characterize $\mathcal{N}$.

(See [44, §4.5] for more details on the _twisted torsion_ condition.)

######  Remark 5.5.

Notice also that in [44, Thm. 2 and Thm. 4.4.2], the factor $\mathcal{F}$ is
implicitly understood as a $\mathbb{Q}$-_factor_ of the associated pure
$\mathbb{Z}$VHS. See also [67, Thm. 2.7], for a related statement highlighting
the possibility of the splitting up to isogeny.

######  Corollary 5.6.

Each orbit closure is algebraic, that is a Zariski closed subvariety
$\Omega\mathcal{M}_{g}$. In fact, orbit closures are defined over
$\overline{\mathbb{Q}}$, and their Galois conjugates are again orbit closures.

From 5.4, one may introduce _real multiplication, torsion and eigenform_
invariants (see Section 6.1, for a geometric description). In particular, we
write

  * •

$r=r_{\mathcal{N}}$, the _(cylinder) rank_ $\frac{1}{2}\dim
H^{1}(T\mathcal{N})$ (which is in fact an integer, since $T\mathcal{N}$ is
known to be symplectic thanks to [3]);

  * •

$d=d_{\mathcal{N}}$, to simply denote the degree of the number field defining
the linear equations of $\mathcal{N}$ (that is $[K:\mathbb{Q}]$);

  * •

And finally $t=t_{\mathcal{N}}$ the _torsion corank_ is $\dim
W_{0}(T\mathcal{N})$.

With these notations, the dimension bound appearing at the end of 5.4 is just

$\dim\mathcal{N}=2r_{\mathcal{N}}+t_{\mathcal{N}}.$

######  Remark 5.7.

Combing 5.6 and 5.3, we observe that the Zariski closure of a collection of
orbit closures is a finite union of orbit closures

######  Remark 5.8.

Let us comment on the “dimension bound” appearing in 5.4 (c.f. [43, Rem. 5.5]
and the discussion in the proof of [43, Cor. 5.7]). Any locus $L$ satisfying
the first three conditions in 5.4 will always have dimension _at most_ $2r+t$,
where $r$ and $t$ are integers naturally associated to the data. This occurs
for the following reasons.

  * (1)

The third condition in 5.4 implies that, over $L$, the form $\omega$ lies in
an eigensubspace of $H^{1}_{\textrm{abs}}$ which has dimension $2r$, and hence
imposes $2g-2r$ linear conditions.

  * (2)

For holomorphic forms over $L$ lying in such an eigensubspace, the second
condition in 5.4 implies linear conditions on the relative periods (c.f. [43,
Rem. 1.5]). As explained in [44, §4.5.4], the number of such conditions
imposed on periods of $\omega$ is $n-t$, where $n$ is the number of marked
points associated to the ambient stratum $\Omega\mathcal{M}_{g}(\kappa)$.

  * (3)

The conditions are independent, so since
$\textrm{rank}\,H^{1}_{\textrm{rel}}=2g+n$ one finds in flat coordinates over
$L$ that the form $\omega$ lies in a subspace of dimension $2r+t$.

  * (4)

The restriction of the developing map (5.3) to $L$ is locally injective, so
this implies $\dim L\leq 2r+t$.

###  5.3 Orbit closures and relative dynamical (a)typicality

In this subsection we first recall the definition of (a)typicality appearing
in 1.2. Actually, rather than giving the intrinsic definition as in [40, 44]
in terms of _algebraic hulls_ , we give one that captures the essential
property of algebraic hull computations which will be relevant to us.

######  Definition 5.9.

At each $p=(X,\omega)\in\mathcal{N}$, we have the _tautological 2-dimensional
subspace_
$T(p):=\operatorname{Span}(Re(\omega),Im(\omega))\subset(H^{1}_{rel})_{p}$
(and analogously for $H^{1}_{abs}$).

We remark here that the above construction gives a
$\mathbf{SL}_{2}(\mathbb{R})$-invariant subbundle, which is not flat. See also
[44, §5.3.7.].

######  Definition 5.10.

Given an orbit closure $\mathcal{N}$, the _algebraic hull_ for the
$\mathbf{GL}_{2}(\mathbb{R})$-action on $H^{1}_{rel,\mathcal{N}}$ is the
subgroup of the monodromy group (at a fixed point $p\in\mathcal{M}$)
$\mathbf{H}_{\mathcal{N},\mathbb{R}}$ stabilizing the tautological plane
$T=T(p)$, introduced in 5.9.

The following is implicit in [40, Thm. 1.3], and the (a)typical terminology
can be found, explicitly, in [44, §5.1.13] (see also §5.1.16 and Thm. 5.4.5 in
_op. cit._):

######  Definition 5.11 (Eskin-Filip-Wright).

Let $\mathcal{N}\subset\mathcal{M}$ be a suborbit closure. We say that
$\mathcal{N}$ is _(dynamically) typical relatively to $\mathcal{M}$_ if the
algebraic hull of $\mathcal{N}$ is equal, up to compact factors, to the
algebraic hull of $\mathcal{M}$. Otherwise, $\mathcal{N}$ is said
_(dynamically) atypical relatively to $\mathcal{M}$_.

We are finally ready to recall the theorem stated in the introduction (to ease
the notation, we omit the parenthetical “dynamically”): See 1.2 Our goal is to
introduce an a priori different notion of (a)typicality defined directly from
their Hodge-theoretic description, rather than a dynamical description. Even
this Hodge-theoretic viewpoint is strongly influenced by Filip’s results
described above, and it is essentially taken from [44]. As we will see below,
some extra care is needed when $\mathcal{M}$ is different from the ambient
stratum of $\Omega\mathcal{M}_{g}$. (In particular our proof will use the
results from Section 5.2 but not the ones on equidistribution of algebraic
hulls from [40].)

##  6 Finiteness of atypical orbit closures and density of the typical ones

In this section we study orbit closures via various _period maps_. Denote by
$\mathcal{A}_{g}=\operatorname{Sp}_{2g}(\mathbb{Z})\backslash\mathbb{H}_{g}$
the moduli space of principally polarized abelian varieties of dimension $g$,
and by $j$ the Torelli morphism $j:\mathcal{M}_{g}\to\mathcal{A}_{g},\
X\mapsto(J(X),\theta_{X})$ which associates to a genus $g$ curve $X$ its
Jacobian $J(X)$ (naturally equipped with its principal polarization
$\theta_{X}$). For every $n$, let $\mathcal{A}_{g,n}\to\mathcal{A}_{g}$ denote
the fibration whose fibre over a point $[A]\in\mathcal{A}_{g}$ is
$\operatorname{Sym}^{[n]}A$, the unordered $n$-tuples of points of $A$.

The varieties $\mathcal{A}_{g}$ and $\mathcal{A}_{g,n}$ can be interpreted as
mixed period spaces (or mixed Shimura varieties). For $\mathcal{A}_{g}$ this
is the standard Siegel space description, and a detailed construction of
$\mathcal{A}_{g,1}$ as a mixed Shimura variety appears in [47, §2]. In general
one has a natural bijection

$\mathcal{A}_{g,n}\simeq\\{(H,E):H\in\mathcal{A}_{g},\hskip
5.0ptE\in\textrm{Ext}^{1}_{\textrm{MHS}}(H,\mathbb{Z}^{n}))\\},$

where we view $\mathcal{A}_{g}$ as the moduli space for principally polarized
pure weight one $\mathbb{Z}$-Hodge structures (of type $(g,g)$), and we view
$\mathbb{Z}^{n}$ as a pure weight zero Hodge structure; see [44, 4.3.14].
Using (5.2) and applying this bijection, one obtains an algebraic mixed period
map $\varphi:\Omega\mathcal{M}_{g}(\kappa)\to\mathcal{A}_{g,n}$ which sends
$(X,\omega)$ to the isomorphism class of the extension described by (5.1). The
data of $\varphi$ is equivalent to the data of the mixed $\mathbb{Z}$VHS
$H^{1}_{\textrm{rel}}$ (at least when viewed as a map of orbifolds). Note that
$\varphi$ is quasi-finite, c.f. the discussion at the end of [59, §3.4]. We
will typically work with the factorization

$\Omega\mathcal{M}_{g}(\kappa)\xrightarrow{\Omega\varphi}\Omega\mathcal{A}_{g,n}\to\mathcal{A}_{g,n},$

of $\varphi$, where the first arrow is obtained by seeing the differential
$\omega$ on the Jacobian of $X$.

######  Remark 6.1.

With the language of (mixed) Shimura varieties one can equivalently describe
$\mathcal{A}_{g,n}$ as follows. The classifying space for the extensions of a
weight one $\mathbb{Z}$VHS of dimension 2g by $\mathbb{Z}(0)$ is the mixed
Shimura variety $\mathbb{A}_{g}$. It is the universal principally polarized
abelian variety of dimension $g$ over $\mathcal{A}_{g}$. Finally

$\mathcal{A}_{g,n}\cong\mathbb{A}_{g}\times_{\mathcal{A}_{g}}\mathbb{A}_{g}\times_{\mathcal{A}_{g}}\dots\times_{\mathcal{A}_{g}}\mathbb{A}_{g}$

(product of $n$ factors). In fact, $\mathcal{A}_{g,n}$ is a _mixed Shimura
variety of Kuga type_ , see [47, §2] for more details on this. This means
that, with the notation of §4, that $W_{-2}=0$ and $W_{-1}$ is the unipotent
radical of its generic Mumford–Tate group, which is simply given by
$\mathbb{G}_{a}^{2g\cdot n}$.

###  6.1 Orbit closures as (a)typical intersections, inspired by Filip

In this section we give a geometric reinterpretation of the conditions
appearing in 5.4. Here we sligthly depart from the exposition of [44] (which
was already implicit in [43]), and explain the difference in Section 6.1.1.

We will think of an orbit closure $\mathcal{N}$ (with associated invariants
$r,d,t$ as in Section 5.2) as the reduced subvariety underlying an irreducible
component of $(\Omega\varphi)^{-1}(E[\mathcal{N}])$, where
$E[\mathcal{N}]\subset\Omega\mathcal{A}_{g,n}$ is an algebraic subvariety
constructed using the conditions appearing in 5.4. In accordance with the
structure of 5.4, the variety $E[\mathcal{N}]$ is constructed in three stages,
depicted in the following diagram:

$\leavevmode\hbox to205.04pt{\vbox
to52.74pt{\pgfpicture\makeatletter\hbox{\hskip
102.52225pt\lower-26.41928pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-102.52225pt}{-26.31944pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathcal{N}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.05554pt\hfil&\hfil\hskip 38.81245pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.50694pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${E[\mathcal{N}]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
14.81248pt\hfil&\hfil\hskip 40.22913pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys<EMAIL_ADDRESS>(6.1)

We now give details and definitions regarding the objects appearing in the
above diagram.

  1. 1.

Let $S[\mathcal{N}]\subset\mathcal{A}_{g}$ be the smallest weakly special
subvariety of $\mathcal{A}_{g}$ containing the image of $\mathcal{N}$. It lies
in the locus $R[\mathcal{N}]$ of $g$-dimensional principally polarized abelian
varieties that have real multiplication _of the same type of $\mathcal{N}$_
(in the sense of 5.4). The latter is a special subvariety of $\mathcal{A}_{g}$
whose the generic Mumford-Tate group is isomorphic to
$\left(\operatorname{Res}_{K/\mathbb{Q}}{\operatorname{GSp}_{2r,K}}\right)\times\operatorname{GSp}_{2(g-dr)}$.

  2. 2.

Let $M[\mathcal{N}]\subset\mathcal{A}_{g,n}$ be the smallest weakly special
subvariety of $\mathcal{A}_{g,n}$ containing $\mathcal{N}$. It lies in
$T[\mathcal{N}]$, the bundle over $R[\mathcal{N}]$ consisting of $n$-tuples of
points satisfying the same torsion conditions as $\mathcal{N}$.

  3. 3.

Let $E[\mathcal{N}]$ be the the bundle of $1$-forms over $M[\mathcal{N}]$
inside in the $K$-eigenspace containing the restriction of the section
$\omega$ to $\mathcal{N}$.

The notation $S[\mathcal{N}]$ is chosen as we think of the variety
$S[\mathcal{N}]$ as giving the _Shimura condition_. Similarly,
$M[\mathcal{N}]$ gives the _mixed-Shimura condition_ , and $E[\mathcal{N}]$
includes also the eigenform condition in addition to the previous two.

The conditions $R[\mathcal{N}]$ and $T[\mathcal{N}]$ are simply geometric
reinterpretations222We stress here the fact that the first two conditions of
5.4 are written in terms of rational Hodge structures (as already noticed and
used by Filip). At first sight, the _twist by real multiplication_ appearing
in the condition on the Abel-Jacobi variety could look like a statement about
$K$-Hodge structures, but this becomes a $\mathbb{Q}$-condition after
including Galois conjugate conditions. of the first two items in 5.4, whereas
the conditions $S[\mathcal{N}]$ and $M[\mathcal{N}]$ may in principle be
stronger. The condition $E[\mathcal{N}]$ corresponds to the third condition in
5.4. One can think of $M[\mathcal{N}]$ as a variety that records all mixed
Hodge-theoretic conditions present on $\mathcal{N}$, and $E[\mathcal{N}]$ as a
variety that records, in addition to $M[\mathcal{N}]$, the third condition in
5.4.

We now give the _intersection theoretic_ (a)typicality condition that will be
suitable for our method. The reader interested only in the
$\mathcal{M}=\Omega\mathcal{M}_{g}(\kappa)$ case can skip the rest of this
section, and just consult equation (6.2) below.

######  Definition 6.2 (Relative (a)typicality — Intersection theoretic
version).

Let $\mathcal{M}$ be a orbit closure in some stratum
$\Omega\mathcal{M}_{g}(\kappa)$ (possibly equal to component of the latter).
Then an orbit closure $\mathcal{N}\subset\mathcal{M}$ is said _(intersection
theoretically) atypical_ (relative to $\mathcal{M}$) if

$\operatorname{codim}_{E[\mathcal{M}]}\mathcal{N}<\operatorname{codim}_{E[\mathcal{M}]}(E[\mathcal{N}])+\operatorname{codim}_{E[\mathcal{M}]}(\mathcal{M}),$

and _(intersection theoretically) typical_ (relative to $\mathcal{M}$)
otherwise.

The codimensions appearing in 6.2 are formal, but one can make sense of them
by taking the image of $\mathcal{N}$ in $\Omega\mathcal{A}_{g,n}$. For most of
this section, we will just work with the intersection-theoretic notion of
(a)typical.

######  Proposition 6.3 (Filip + $\epsilon$).

Let $\mathcal{N}\subset\mathcal{M}=\Omega\mathcal{M}_{g}(\kappa)$ be a
suborbit closure. Then for all but finitely many exceptional orbit closures,
the following are equivalent:

  1. 1.

$\mathcal{N}$ is dynamically typical relatively to $\mathcal{M}$ (see 5.11);

  2. 2.

$\mathcal{N}$ is intersection theoretically typical relatively to
$\mathcal{M}$ (see 6.2);

  3. 3.

$\mathcal{N}$ belongs to the three cases of [44, Prop. 5.1.6]: torus covering,
Hilbert modular surface, Weierstrass curve.

###### Proof.

Filip [44, §5.1], by a direct computation, proved that 1 is equivalent to 3.
It’s easy to see that if $\mathcal{N}$ is intersection theoretically typical,
then it is dynamically typical. For the converse, we argue as follows. The
first step is to show that the monodromy datum on the stratum $\mathcal{M}$ is
as big as possible, as recalled in 6.5 below. The Hilbert modular surface
(resp. Weierstrass curve) cases happen only when $g=2$, and $t=n=1$ (resp.
$g=2$, and $t=n=1$) and on checks directly by computing their monodromy [44,
Thm. 5.3.15] that our definitions agree. To conclude, it is enough to check
that all but finitely many torus covers $\mathcal{N}$ (also known as _square-
tiled surfaces_) are typical in our definition. Outside of finitely many
exceptional $\mathcal{N}$, that the monodromy is maximal follows from the
arguments in [44, §5.4.6] (it’s stated only for absolute cohomology, but same
argument works for relative cohomology as well). ∎

######  Remark 6.4.

One famous example of torus covering with smaller monodromy group than
expected is the “Eierlegende Wollmilchsau” [52].

######  Lemma 6.5.

The monodromy group $\mathbf{H}_{\Omega\mathcal{M}_{g}(\kappa)}$ of the
$\mathbb{Z}$VMHS $\mathbb{V}$ on $\Omega\mathcal{M}_{g}(\kappa)$ is
$\operatorname{Sp}_{2g}\ltimes\mathbb{G}_{\operatorname{a}}^{2g\cdot n}$. (In
particular it is equal to the derived subgroup of the generic Mumford–Tate
group $\mathbf{G}$ of $\mathbb{V}$).

###### Proof.

It is well known that (5.4) is surjective. The result then follows from the
fact that the image of

$\pi_{1}(\Omega\mathcal{M}_{g}(\kappa))\to\operatorname{Mod}(X,Z)$

has finite index, as established in [24, Thm. A]. See also [44, §3.1.14 and
3.1.16], for related discussions on connected components of
$\Omega\mathcal{M}_{g}(\kappa)$ (here we just care about the monodromy, not
the image of the representations). ∎

####  6.1.1 Comparison with Filip’s notation

This section gives more details about the case
$\mathcal{M}=\Omega\mathcal{M}_{g}(\kappa)$. In Filip’s notation, the
following is the key diagram to see orbit closures as intersections.

${\mathcal{N}}$${\mathcal{E}\Omega\mathcal{A}_{r,d,t}}$${\mathcal{E}\mathcal{A}_{r,d,t}}$${\mathcal{E}\mathcal{A}_{r,d}}$${\Omega\mathcal{M}_{g}(\kappa)}$${\Omega\mathcal{A_{g,n}}}$${\mathcal{A}_{g,n}}$${\mathcal{A}_{g}}$

.

In the notation of the previous section,
$\mathcal{E}\mathcal{A}_{r,d}=R[\mathcal{N}]$,
$\mathcal{E}\mathcal{A}_{r,d,t}=M[\mathcal{N}]$, and
$\mathcal{E}\Omega\mathcal{A}_{r,d,t}$ is nothing more than the eigenform
bundle above $\mathcal{E}\mathcal{A}_{r,d,t}$ corresponding to the third
condition in 5.4. In the case where
$\mathcal{M}=\Omega\mathcal{M}_{g}(\kappa)$, an orbit closure
$\mathcal{N}\subset\mathcal{M}$ is (intersection theoretically) atypical iff

$\operatorname{codim}_{\Omega\mathcal{A}_{g,n}}\mathcal{N}<\operatorname{codim}_{\Omega\mathcal{A}_{g,n}}(\mathcal{E}\Omega\mathcal{A}_{r,d,t}=E[\mathcal{N}])+\operatorname{codim}_{\Omega\mathcal{A}_{g,n}}\Omega\mathcal{M}_{g}(\kappa).$
(6.2)

To get back the dimensions appearing in [44, 5.1.3] one just needs to add on
the RHS
$0=\dim\Omega\mathcal{M}_{g}(\kappa)-\dim\Omega\mathcal{M}_{g}(\kappa)$ (in
this way $\operatorname{codim}_{\Omega\mathcal{M}_{g}(\kappa)}\mathcal{N}$
will appear).

###  6.2 The Eskin-Filip-Wright finiteness from the intersection-theoretic
viewpoint

We are now ready to prove our main result. That is to give a new proof, in our
unifying setting, of the finiteness parts of 1.1 (and equivalently 1.2), as
described in Section 1.1. More precisely, with the vocabulary of 6.2:

######  Theorem 6.6.

Let $\mathcal{M}\subset\Omega\mathcal{M}_{g}$ be an orbit closure (possibly
equal to an irreducible component of some stratum
$\Omega\mathcal{M}_{g}(\kappa)$). Then $\mathcal{M}$ contains at most finitely
many (strict) orbit closures $\mathcal{N}$ that are:

  1. 1.

Maximal, i.e. the only suborbit closures of $\mathcal{M}$ containing
$\mathcal{N}$ are $\mathcal{N}$ and $\mathcal{M}$;

  2. 2.

Atypical (relative to $\mathcal{M}$) in the sense of 6.2.

In particular in each (connected component of each) stratum
$\Omega\mathcal{M}_{g}(\kappa)$, all but finitely many orbit closures have
rank 1 and degree at most 2.

The above gives the first part of 1.5 and the proof provided below is the
starting point for the _effective finiteness_ we will prove in the final
sections. After discussing some preliminaries, in §6.2.3 we give our proof.

For the rest of the section, let
$\mathcal{M}\subset\Omega\mathcal{M}_{g}(\kappa)$ be some fixed orbit closure.
We fix a base point $s_{0}:=(X,\omega)\in\mathcal{M}$. We write
$(\mathbf{H},D^{0}\subset\widecheck{{D}}^{0})=(\mathbf{H}_{\mathcal{M}},D^{0}_{\mathcal{M}}\subset\widecheck{{D}}^{0}_{\mathcal{M}})$
for the associated weak Shimura (or Hodge) datum associated to $\mathcal{M}$.
For example
$\mathbf{H}=\operatorname{Sp}_{2g}\ltimes\mathbb{G}_{\operatorname{a}}^{2g\cdot
n}$ if $\mathcal{M}=\Omega\mathcal{M}_{g}(\kappa)$, where $n=|\kappa|-1$.

####  6.2.1 Rephrasing the atypical condition in the period torsor

We now let $(\mathcal{H},\nabla)$ be the algebraic vector bundle with regular
singular connection associated to $\mathbb{V}=H^{1}_{\textrm{rel}}$, and
consider the bundle $P$ and map $\nu$ constructed in §4.6. The graded quotient
of $\mathcal{H}$ which corresponds to $H^{1}_{\textrm{abs}}$ we denote
$\mathcal{H}_{\textrm{abs}}$. Then $\omega$ is an algebraic section of
$\mathcal{H}_{\textrm{abs}}$, and we then obtain any algebraic map
$r:P\to\mathcal{H}_{abs,s_{0}}$ given by

$[\eta\in\textrm{Hom}(\mathcal{H}_{s},\mathcal{H}_{s_{0}})]\mapsto\eta(\omega_{s}).$

######  Lemma 6.7.

The map $r$ lands inside a unique eigenspace
$E_{\mathcal{M}}=E_{\mathcal{M},s_{0}}\subset\mathcal{H}_{\textrm{abs},s_{0}}$
for the field of real multiplication $K_{\mathcal{M}}$ associated to
$\mathcal{M}$. The complex algebraic variety