†† _2020 Mathematics Subject Classification_. 14G35, 22E40, 03C64, 14P10,
32H02, 32M15.†† _Key words and phrases_. Complex hyperbolic lattices,
Superrigidity, Hodge theory and Mumford–Tate domains, Functional
transcendence, Unlikely intersections.

# Rich representations and superrigidity

Gregorio Baldi, Nicholas Miller, Matthew Stover, and Emmanuel Ullmo

###### Abstract.

We investigate and compare applications of the Zilber–Pink conjecture and
dynamical methods to rigidity problems for arithmetic real and complex
hyperbolic lattices. Along the way we obtain new general results about
reconstructing a variation of Hodge structures from its typical Hodge locus
that may be of independent interest. Applications to Siu’s immersion problem
are also discussed.

###### Contents

  1. 1 Introduction and Motivation
    1. 1.1 Siu’s immersion problem
    2. 1.2 Rich representations and the main theorem
    3. 1.3 Holomorphic and VHS results
    4. 1.4 Closed geodesics
    5. 1.5 Outline of paper
  2. 2 Dynamics background and compatibility
  3. 3 Dynamical proof for a variant of 1.3
  4. 4 The proof of 1.3
  5. 5 Hodge theoretic preliminaries and a quick introduction to Zilber–Pink
    1. 5.1 Hodge theory and period domains
    2. 5.2 Typical and atypical intersections
    3. 5.3 Some results on the distribution of the Hodge locus
  6. 6 General Hodge theoretic statement: from typical to atypical intersections
    1. 6.1 Isogenies between VHS
    2. 6.2 The main statement
  7. 7 Proofs of 1.11 and 1.14

##  1\. Introduction and Motivation

A cornerstone in the representation theory of lattices in Lie groups is
Margulis’s superrigidity theorem [28, p. 2]. Indeed, the implications of
superrigidity are profound, including arithmeticity of irreducible lattices in
all noncompact semisimple Lie groups except
$\operatorname{\operatorname{PO}}(1,n)$ and
$\operatorname{\operatorname{PU}}(1,n)$, i.e., for all but real and complex
hyperbolic lattices. Specifically, Margulis proved superrigidity of
irreducible lattices in semisimple Lie groups of real rank at least two. Using
the theory of harmonic maps, this was later extended to the rank one groups
$\mathrm{Sp}(1,n)$ and $\mathrm{F}_{4}^{(-20)}$ by Corlette [13] and
Gromov–Schoen [21]. While the harmonic maps method was extended to give a new
proof in higher rank [29], it is open (and quite relevant in context of the
blend of techniques used in this paper) whether the dynamical methods of
Margulis can give an independent proof in the rank one cases.

To be precise, let $\Gamma$ be a lattice in a Lie group $G$. A representation
$\rho$ from $\Gamma$ to a topological group $H$ is said to _extend_ if there
is a continuous homomorphism $\widetilde{\rho}:G\to H$ so that $\rho$ is the
restriction of $\widetilde{\rho}$ to $\Gamma$ under its lattice embedding in
$G$. Then $\Gamma$ is _superrigid_ if every unbounded, Zariski dense
representation of $\Gamma$ to a connected, adjoint simple algebraic group $H$
over a local field $k$ of characteristic zero extends. It is known that there
are lattices in $\operatorname{\operatorname{PO}}(1,n)$ and
$\operatorname{\operatorname{PU}}(1,n)$ that are not superrigid; [22, Sec. 5]
for real hyperbolic examples and [31, Sec. 22] or [24, Sec. 1] for some
results in the complex hyperbolic setting. The study of representations of
real and complex hyperbolic lattices is very difficult and rich in open
questions, and the extent to which superrigidity fails has yet to be made
precise. For example, one motivation for this paper is the first part of a
question asked by David Fisher [19, Qtn. 3.15]:

######  Question 1.1 (D. Fisher).

Let $\Gamma\subset G$ be a lattice where
$G=\operatorname{\operatorname{SO}}(1,n)$ or
$\operatorname{\operatorname{SU}}(1,n)$. What conditions on a representation
$\rho:\Gamma\to\operatorname{GL}_{m}(k)$ imply that $\rho$ extends or almost
extends? What conditions on $\Gamma$ imply that $\Gamma$ is arithmetic?

In particular, this paper builds on the techniques from [1, 2, 8] to explore
_geometric_ conditions on a representation that allow one to use tools from
dynamics and/or Hodge theory to prove new rigidity phenomena. There is some
history of success along these lines. For just one family of examples, some
classes of representations studied in the literature have _maximal_ geometric
behavior, where two distinct instances are:

  * •

Maximal for the _Toledo invariant_ (Koziarz–Maubon [26] and many others)

  * •

Maximal for their _limit set_ (Besson, Courtois and Gallot [9] and Shalom
[36])

Here we study a class of representations generalizing those shown to be
superrigid in [1, 2], namely those that are _geodesically rich_.

Before delving into technical definitions and stating our main results, we
discuss a primary application of our methods that can be stated much more
simply.

###  1.1. Siu’s immersion problem

An original motivation for studying the more general rigidity questions
considered in this paper is Siu’s _immersion problem_ [38, Prob. (b), p. 182].
Let $\mathbb{B}^{n}$ denote the unit ball in $\mathbb{C}^{n}$ with its Bergman
metric. If $\Gamma$ is a lattice in the holomorphic isometry group
$\operatorname{\operatorname{PU}}(1,n)$ of $\mathbb{B}^{n}$, then $S_{\Gamma}$
will denote the _ball quotient_ $\Gamma\backslash\mathbb{B}^{n}$.

######  Question 1.2 (Siu, 1985).

Is it true that every holomorphic embedding between compact ball quotients of
dimension at least $2$ must have totally geodesic image?

To our knowledge the only previous result is by Cao and Mok [12], which
answered 1.2 in the affirmative for
$f:\Gamma\backslash\mathbb{B}^{n}\to\Lambda\backslash\mathbb{B}^{m}$ when
$m<2n$. See [2, Thm. 1.7] and the surrounding references for more on Siu’s
analogous _submersive problem_. The following is our main contribution toward
a positive answer to Siu’s immersion problem.

######  Theorem 1.3.

Let $f:S_{\Gamma_{1}}\to S_{\Gamma_{2}}$ be an immersive holomorphic map
between arithmetic ball quotients of unequal dimension. Suppose that
$S_{\Gamma_{1}}$ contains a proper totally geodesic subvariety of positive
dimension. Then the following are equivalent:

  1. (1)

The map $f$ is a totally geodesic immersion.

  2. (2)

There is a generic sequence of positive-dimensional proper totally geodesic
subvarieties $Y_{\ell}\subset S_{\Gamma_{1}}$ such that each $f(Y_{\ell})$
lies is a proper totally geodesic subvariety $W_{\ell}\subset S_{\Gamma_{2}}$.

The unequal dimension hypothesis is simply to rule out coverings. The
positive-dimensional hypothesis on the subvarieties of $S_{\Gamma_{1}}$ is to
rule out the trivial counterexample of points, but note that $W_{\ell}$ could
be a point. We offer two proofs of 1.3, one using homogeneous dynamics which
gives a slightly weaker statement in Section 3, and the second using Hodge
theory and functional transcendence in Section 4, as an application of 1.15
below.

The arithmetic ball quotients with totally geodesic subvarieties of every
possible codimension are those of _simplest type_ , which are constructed
using hermitian forms over number fields. In this case, 1.3 reduces 1.2 to
checking a single dimension.

######  Corollary 1.4.

If 1.2 has a positive answer when $S_{\Gamma_{1}}$ is a $2$-dimensional
arithmetic ball quotient of simplest type, then it has a positive answer for
$S_{\Gamma_{1}}$ an arithmetic ball quotient of simplest type of any dimension
$n\geq 2$.

######  Remark 1.5.

By [12], the first open case is of a holomorphic embedding
$f:S_{\Gamma_{1}}\to S_{\Gamma_{2}}$ with $\dim S_{\Gamma_{1}}=2,\dim
S_{\Gamma_{2}}=4$. Here 1.3 implies that it suffices to prove that there is an
infinite collection of distinct totally geodesic curves $C_{i}\subset
S_{\Gamma_{1}}$ such that each $f(C_{i})$ lies in some strict totally geodesic
subvariety $Y_{i}\subset S_{\Gamma_{2}}$.

There are various similar generalizations that follow from 1.14. For example,
a period map $\psi:S_{\Gamma_{1}}\to\Gamma\backslash D$ such that the
$f^{-1}(Y_{i})$ are totally geodesic subvarieties of $S_{\Gamma_{1}}$ for a
generic sequence of Mumford–Tate subdomains $Y_{i}\subset\Gamma\backslash D$
is an isomorphism.

###  1.2. Rich representations and the main theorem

Let $\Gamma\subset G$ be a torsion-free arithmetic lattice, where $G$ is
$\operatorname{\operatorname{SU}}(1,n)$ or
$\operatorname{\operatorname{SO}}(1,n)$, and let $X$ denote the symmetric
space associated with $G$. Then $S_{\Gamma}$ will denote the manifold
$\Gamma\backslash X$, which is a smooth manifold that is also a smooth variety
in the complex hyperbolic case111Since both contexts are relevant to this
paper, we use the term submanifold when considering $S_{\Gamma}$ as a
differentiable manifold and subvariety when $S_{\Gamma}$ is complex hyperbolic
and the submanifold is also a subvariety for the natural algebraic structure..
We assume that $S_{\Gamma}$ contains one, and therefore infinitely many,
proper totally geodesic submanifolds222This claim follows easily from the fact
that the associated algebraic group commensurates the lattice. In the other
direction, if $S_{\Gamma}$ contains infinitely many maximal totally geodesic
submanifolds of dimension at least $2$, then $\Gamma$ is arithmetic; see [1,
2, 8]. of real dimension at least $2$ (see Section 1.4 for the case of closed
geodesics). When this is the case, we say that $S_{\Gamma}$ (or $\Gamma$) is
_(geodesically) rich_. Let $\\{S_{\Gamma_{i}}\\}_{i\in\mathbb{N}}$ be a
sequence of (immersed) maximal totally geodesic submanifolds, where maximal
means that the only totally geodesic submanifold strictly containing
$S_{\Gamma_{i}}$ is $S_{\Gamma}$ itself.

The group $\Gamma_{i}$ is the stabilizer in $\Gamma$ of a totally geodesic
subspace $Y_{i}$ of $X$, and the restriction of $\Gamma_{i}$ to $Y_{i}$
induces a surjective homomorphism from $\Gamma_{i}$ onto the fundamental group
of $S_{\Gamma_{i}}$. The kernel of this homomorphism encodes the action of
$\Gamma_{i}$ on the normal bundle to $Y_{i}$ in $X$. The main purpose of this
paper is to introduce the notion of (geodesically) rich representations, which
are morally the ones that “preserve” a significant amount of the rich
structure of $\Gamma$:

######  Definition 1.6.

Let $k$ be a local field and ${\mathbf{H}}$ be a semisimple $k$-algebraic
group without compact factors. Suppose that $\Gamma$ is geodesically rich and
that $\rho:\Gamma\to{\mathbf{H}}(k)$ is a representation with unbounded,
Zariski dense image. Define ${\mathbf{H}}_{i}$ by

(1.2.1)
$\mathbf{H}_{i}\coloneqq\overline{\rho(\Gamma_{i})}\subset{\mathbf{H}}$

for each $i$, where closure is with respect to the Zariski topology. Then
$\rho$ is _(geodesically) rich_ if there are infinitely many $i$ so that
$\dim{\mathbf{H}}_{i}<\dim{\mathbf{H}}$.

Special consideration will be given in this paper to the case where
$S_{\Gamma}$ and the subspaces $S_{\Gamma_{i}}$ are all complex hyperbolic.

######  Definition 1.7.

Let $\Gamma\subset G=\operatorname{\operatorname{PU}}(1,n)$ be a complex
hyperbolic lattice and $\rho:\Gamma\to{\mathbf{H}}(k)$ be a rich
representation. Then $\rho$ is _complex (geodesically) rich_ if infinitely
many of the $S_{\Gamma_{i}}$ for which $\dim{\mathbf{H}}_{i}<\dim{\mathbf{H}}$
are complex hyperbolic.

Broadly, this paper is concerned with identifying situations where the
following question has a positive answer.

######  Question 1.8.

Do rich representations extend? Equivalently, are there any nontrivial rich
representations?

Before stating one of our primary results toward a positive answer, we recall
a definition that appears in [3] that will be used to give a
generalization/abstract version of the method of [1, 2].

######  Definition 1.9.

Fix a parabolic subgroup $P\subset G$ with unipotent radical $U$. A pair
$(k,{\mathbf{H}})$ as above is _strongly compatible_ with $G$ if for any
subgroup $\\{1\\}\neq{\mathbf{H}}^{\prime}\subset{\mathbf{H}}$ and every
continuous group homomorphism $\tau:P\to
N_{\mathbf{H}}({\mathbf{H}}^{\prime})/{\mathbf{H}}^{\prime}(k)$ we have that
$U\leq\operatorname{Ker}(\tau)$.

######  Remark 1.10.

If $k$ is nonarchimedean, then any pair $(k,{\mathbf{H}})$ is strongly
compatible with $G$.

Our main theorem establishes 1.8 in two cases:

######  Theorem 1.11.

Let $\Gamma$ be an arithmetic lattice in $G$ which is either
$\operatorname{\operatorname{PO}}(1,n)$ or
$\operatorname{\operatorname{PU}}(1,n)$, ${\mathbf{H}}$ be a connected,
adjoint algebraic group over a local field $k$ of characteristic zero, and
$\rho:\Gamma\to{\mathbf{H}}(k)$ be a rich representation. If either

  1. (1)

${\mathbf{H}}$ is strongly compatible (in the sense of 1.9) and $k$-simple, or

  2. (2)

$\Gamma$ is complex hyperbolic, $\rho$ is complex (geodesically) rich, and
$\rho$ is cohomologically rigid (i.e.,
$H^{1}(\Gamma,\operatorname{Ad}_{{\mathbf{H}}}\circ\rho)=0$),

then $\rho$ extends.

Notice that the former condition is just on the pair $(k,{\mathbf{H}})$,
whereas the latter depends only on $\rho$. The proof of (1), which follows
from the methods developed in [1, 2], is in Section 2. The proof of (2),
contained in Section 7, is related to the study of variations of Hodge
structures and in fact it will be related to the so-called _Zilber–Pink
conjecture_ [7]. Regarding (2), it might be possible to study some nonrigid
representations via the so called _factorization theorem_ , see for example
[37, Thm. 10] and various more recent generalizations. The fact that
$(k,{\mathbf{H}})$ is always compatible when $k$ is nonarchimedean has the
following fairly direct corollary.

######  Corollary 1.12.

Suppose that $\Gamma\subset\operatorname{\operatorname{PU}}(1,n)$ is an
arithmetic lattice and that $\rho:\Gamma\to{\mathbf{H}}(k)$ is a rich
representation to a $k$-simple algebraic group ${\mathbf{H}}$ defined over a
number field $k$. Then $\rho$ is integral.

1.12 should be viewed as a complement to the fundamental recent work of
Esnault and Groechenig [17] that proves integrality of certain cohomologically
rigid local systems. Note that 1.12 does not assume cohomological rigidity,
only that the representation is defined over a number field. Moreover, note
that there are homomorphisms from complex hyperbolic lattices onto surface
groups [40], hence not all representations of complex hyperbolic lattices are
definable over a number field. As noticed in [2], the version of 1.12 that
implicitly follows from the results there makes the main theorem of that paper
independent of [17].

We close this subsection with a comment on how richness may differentiate
between real and complex hyperbolic lattices. For example, we do not know the
answer to the following:

######  Question 1.13.

Is every faithful linear representation of an arithmetic lattice in
$\operatorname{\operatorname{PU}}(1,n)$ of simplest kind rich?

Bending deformations of lattices in $\operatorname{\operatorname{PO}}(1,n)$
[22, Sec. 5] easily imply that 1.13 is false for real hyperbolic lattices of
simplest kind. However, Goldman–Millson rigidity [20] implies that there are
no analogous deformations in the complex hyperbolic setting. It seems that
examples giving a negative answer to 1.13 would come from representations of a
new kind. On the other hand, a positive answer would (for example) prove
superrigidity of unbounded, Zariski dense, and faithful representations to
connected, adjoint targets.

Since 1.11 proves that all rich representations to connected, adjoint
$k$-simple groups over nonarchimedean fields of characteristic zero are
bounded, all other results in this paper consider the cases where $k$ is
$\mathbb{R}$ or $\mathbb{C}$. For the remainder of this introduction we
consider rich representations of complex hyperbolic lattices with coefficients
in $\mathbb{R}$.

###  1.3. Holomorphic and VHS results

To enter in the world of variations of Hodge structures, we focus on _complex
hyperbolic lattices_ $\Gamma\subset G=\operatorname{\operatorname{PU}}(1,n)$
with $n>1$. See Section 5 for the various notions from Hodge theory used in
this subsection. In this setting we can restate 1.7 in a more geometric way.
Suppose for example that $H$ is the automorphism group of a Hermitian
symmetric domain $X_{H}$, $\rho:\Gamma\to H$ takes values in a lattice
$\Gamma_{H}\subset H$, and is induced by taking the induced map on fundamental
groups for a holomorphic map

(1.3.1) $f:S_{\Gamma}\longrightarrow\Gamma_{H}\backslash X_{H}.$

Then the complex richness of $\rho$ translates into the following:
$f(S_{\Gamma_{i}})$ lies in a strict complex totally geodesic submanifold
$Y_{i}\subset\Gamma_{H}\backslash X_{H}$ for infinitely many $i$.

We remark here that the setting described in (1.3.1) is a special case of the
variation of Hodge structure (VHS, from now on) formalism. The next theorem
shows that rich representations underlying an integral pure polarized VHS
extend.

######  Theorem 1.14.

If $\rho$ is a complex rich representation underlying a direct
$\mathbb{C}$-factor of an irreducible $\mathbb{Q}$VHS, then $\rho$ extends.

The proof of 1.14 builds on the main results of [7], namely the _geometric
part of the Zilber–Pink conjecture_. In particular, the proof crucially uses
the following, which we prove in Section 6, and may be of independent
interest.

######  Theorem 1.15.

Let $S$ be a smooth quasi-projective complex variety supporting
$\mathbb{Z}$VHSs $\mathbb{V}_{1},\mathbb{V}_{2}$ whose algebraic monodromy
groups are $\mathbb{Q}$-simple and nontrivial. If

$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S,\mathbb{V}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset,$

then $\mathbb{V}_{1},\mathbb{V}_{2}$ are isogenous.

Given a smooth quasi-projective complex variety $S$ and ${\mathbb{V}}\to S$ a
variation of Hodge structures, we write
$\textnormal{HL}(S,\mathbb{V}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
for the so called _typical Hodge locus of positive period dimension_ , as
introduced in [7]. See 6.1 for the precise definition of an isogeny in this
context.

###  1.4. Closed geodesics

To conclude the introduction, we study the one-dimensional case avoided in our
previous results. From a dynamical point of view, the dimension at least two
hypothesis is natural and critical in that it allows one to access tools from
unipotent dynamics. From a Hodge theoretic point of view, one typically works
with a complex subvariety. However, there is still something one can say for
closed geodesics.

We begin with a simple cautionary example regarding generalizing richness to
closed geodesics. Let $G$ be $\operatorname{\operatorname{PO}}(1,n)$ for
$n\geq 3$ or $\operatorname{\operatorname{PU}}(1,n)$ for $n\geq 2$, and
suppose that $\Gamma_{1},\Gamma_{2}$ are arithmetic lattices in $G$ of
simplest type for which there is a surjection $\rho:\Gamma_{1}\to\Gamma_{2}$
with infinite kernel; such representations are known to exist in both
settings. Choose a maximal subgroup $\Delta\subset\Gamma_{2}$ associated with
a proper totally geodesic submanifold of $S_{\Gamma_{2}}$. Then
$\rho^{-1}(\Delta)$ contains the kernel of $\rho$, and therefore is Zariski
dense in $G$. Consequently, $\rho^{-1}(\Delta)$ contains a profinitely open
collection of regular semisimple elements [34], hence they preserve no totally
geodesic subspace of $X$ except their axis of translation, and thus the
associated closed geodesics on $S_{\Gamma}$ are maximal. In the complex
hyperbolic setting the associated closed geodesics are _generic_ in the sense
that their union is Zariski dense in $S_{\Gamma}$, hence this provides a
fairly rich collection of $1$-dimensional maximal totally geodesic subspaces.

Representations as in the previous paragraph are unbounded and Zariski dense
with connected, adjoint target, but they cannot extend since they have
infinite kernel. They have a well-distributed sequence of maximal closed
geodesics for which the associated elements in $\rho(\Gamma_{1})$ all preserve
a totally geodesic subspace of the target, hence the representation could be
considered rich in analogy with the higher-dimensional setting. While one
might reasonably interpret this as saying the $1$-dimensional case is
hopeless, our final result indicates that stronger geometric hypotheses can
still lead to a rigidity theorem.

######  Theorem 1.16.

Let $S=\Gamma_{n}\backslash\mathbb{B}^{n}$ be a ball quotient with an
immersive holomorphic map

$f:S\longrightarrow\Gamma_{m}\backslash\mathbb{B}^{m}.$

Let $\\{C_{\ell}\\}$ be a generic sequence of closed geodesics in $S$. Assume
that for each $\ell$ there exits a strict complex totally geodesic subvariety
$Y_{\ell}\subset\Gamma_{m}\backslash\mathbb{B}^{m}$ such that
$f(C_{\ell})\subset Y_{\ell}$. Then $f$ is a totally geodesic immersion.

See 4.4 for the proof. Note that 1.16 applies for nonarithmetic $\Gamma$,
since all finite volume ball quotients contain infinitely many free homotopy
classes of closed geodesics. That $f$ must take certain closed geodesics into
a proper totally geodesic subspace is a very strong hypothesis, but it is not
entirely unprecedented. Indeed one can show that some of the maps associated
with surjections between complex hyperbolic lattices take a certain
(necessarily finite) collection of totally geodesic subspaces of
$S_{\Gamma_{n}}$ to totally geodesic subspaces of $S_{\Gamma_{m}}$. We close
with a final question that would generalize 1.16.

######  Question 1.17.

Let $\Gamma$ be any lattice in $\operatorname{\operatorname{PU}}(1,n)$, $n>1$,
and $f:S_{\Gamma}\to\Gamma_{D}\backslash D$ be any period map (e.g.,
$\Gamma_{D}\backslash D$ is a hermitian symmetric space and $f$ is
holomorphic). If $f$ sends a generic sequence of closed geodesics
$\\{C_{\ell}\\}$ of $S_{\Gamma}$ to closed geodesics of $\Gamma_{D}\backslash
D$, is then $f(S_{\Gamma})$ a totally geodesic subvariety?

###  1.5. Outline of paper

In Section 2 we prove the statement about compatibility. The rest of the paper
is devoted to the complex hyperbolic case. In Section 3 we give the argument
based on various forms of Ratner’s equidistribution theorem. In Section 4 we
give a simple self-contained proof of the results about Siu’s immersion
problem. The rest of the paper uses more ideas and techniques from Hodge
theory. In particular Section 5 is devoted to all the needed preliminaries
from Hodge theory and the Zilber–Pink paradigm. In Section 6 we prove the most
general statement on isogenies of VHSs, namely 1.15. Finally we prove the main
results about the extension of rich representations underlying a VHS, see
Section 7.

### Acknowledgements

Baldi and Ullmo were partially supported by the NSF grant DMS-1928930, while
they were in residence at the MSRI in Berkeley, during the Spring 2023
semester. Miller was partially supported by Grant Number DMS-2005438/2300370
and Stover was partially supported by Grant Number DMS-2203555 from the
National Science Foundation. G.B. also thanks the IHES for the excellent
working conditions.

##  2\. Dynamics background and compatibility

This section contains some background on the dynamical side, along with the
proof of the first part of 1.11. Let $G$ be either
$\operatorname{\operatorname{PO}}(1,n)$ or
$\operatorname{\operatorname{PU}}(1,m)$. We first state some basic facts and
establish some notation that will be used throughout this paper. What is
described immediately below recaps facts that are given a detailed treatment
in [1, 2].

Fixing a maximal compact subgroup $K$ in $G$, the associated locally symmetric
space is $X=G/K$. Fix a _standard_ representative $Y\subset X$ for each type
(i.e., $G$-orbit) of totally geodesic subspace of $X$ with dimension at least
two. If $W$ is the subgroup of the stabilizer of $Y$ in $G$ generated by
unipotent elements, then the full stabilizer of $Y$ is the normalizer
$N_{G}(W)$, which is a compact extension of $W$.

The Haar measure on $G$ will be denoted $\mu_{G}$, and (after appropriate
scaling) given a lattice $\Gamma$ in $G$ this induces a probability measure
$\mu_{\Gamma\backslash G}$ on the quotient $\Gamma\backslash G$. Suppose that
the translate $gY$ of the standard subspace $Y$ has the property that
$gLg^{-1}\cap\Gamma$ defines a lattice $\Delta$ in $T=gLg^{-1}$ for some
$W\subseteq L\subseteq N_{G}(W)$. One obtains an immersion
$S_{\Delta}\looparrowright S_{\Gamma}$ induced by the embedding

$\Gamma\backslash\Gamma gL\hookrightarrow\Gamma\backslash G$

and an associated push-forward of (the $g$-translate of)
$\mu_{\Delta\backslash T}$ to a probability measure on $\Gamma\backslash G$.
Thus one obtains a homogeneous $W$-invariant measure on $\Gamma\backslash G$
with proper support. Upon taking an ergodic component of this measure, one can
moreover choose $L$ satisfying the above properties for which the
corresponding measure is $W$-ergodic. If $\\{S_{\Gamma_{i}}\\}$ is an infinite
sequence of distinct maximal totally geodesic subspaces of $S_{\Gamma}$, then
one can extract a subsequence so that they are all associated with
$G$-translates of the same $Y$, so $\Gamma_{i}$ is a lattice in
$T_{i}=g_{i}L_{i}g_{i}^{-1}$ with $W\subseteq L_{i}\subseteq N_{G}(W)$ for all
$i$. The maximality assumption ensures that $\mu_{\Gamma\backslash G}$ is a
weak-$\ast$ limit333All limits of measures in this paper will be in the
weak-$\ast$ sense. of (the $g_{i}$ translates of) the measures
$\mu_{\Gamma_{i}\backslash T_{i}}$.

Since it arises several times, we briefly recall what it means for
$\Gamma\subset G$ to be an arithmetic lattice of the simplest kind. These are
well-studied in the literature, so we are terse. In the real hyperbolic case,
this means that there is a totally real number field $F$ with integer ring
$\mathcal{O}_{F}$ and a quadratic form $q$ on $F^{n+1}$ so that $\Gamma$ is
commensurable with $\mathcal{G}(\mathcal{O}_{F})$, where $\mathcal{G}$ is the
$F$-algebraic group $\operatorname{\operatorname{PO}}(q)$. In the complex
hyperbolic case, there is a totally imaginary extension $E$ of the totally
real field $F$ so that $\Gamma$ is commensurable with
$\mathcal{G}(\mathcal{O}_{F})$, where $\mathcal{G}$ is now
$\operatorname{\operatorname{PU}}(h)$ for a hermitian form $h$ on $E^{n+1}$
(which is still an $F$-algebraic group). In each case, it is necessarily the
case that the form has signature $(1,n)$ at one place of $F$ and is definite
at all other places.

######  Theorem 2.1.

Suppose that $\Gamma\subset G$ is an arithmetic lattice so that the locally
symmetric space $S_{\Gamma}$ contains a proper immersed totally geodesic
subspace of dimension at least two. Let $k$ be a local field of characteristic
zero and ${\mathbf{H}}$ be a connected, adjoint $k$-algebraic group so that
the pair $(k,{\mathbf{H}})$ is compatible with $G$. If
$\rho:\Gamma\to{\mathbf{H}}(k)$ is a rich representation, then $\rho$ extends.

###### Proof.

By 1.6, $\rho$ is a homomorphism with unbounded, Zariski dense image.
Following Theorem 1.6 and Remark 1.7 of [1], to prove that there exists a
continuous extension ${\widetilde{\rho}:G\to{\mathbf{H}}(k)}$ of $\rho$, it
suffices to find

  1. (1)

a noncompact almost-simple, proper subgroup $W\subset G$,

  2. (2)

a $k$-rational faithful, irreducible representation
${\mathbf{H}}(k)\to\mathbf{SL}(V)$ on a finite dimensional $k$-vector space
$V$, and

  3. (3)

a $W$-invariant measure $\nu$ on $\Gamma\backslash(G\times\mathbb{P}(V))$ that
projects to Haar measure $\mu_{\Gamma\backslash G}$ on $\Gamma\backslash G$.

We now explain how richness implies the existence of (1)-(3), which follows
the argument given in [1, §3] in a similar context.

Recall that by hypothesis, $S_{\Gamma}$ contains infinitely many maximal
totally geodesic subspaces $S_{\Gamma_{i}}$. Passing to a subsequence, we
assume they are of the same type (in the sense described in this section).
This defines a sequence of homogeneous, $W$-ergodic measures $\\{\mu_{i}\\}$
with $\mathrm{supp}(\mu_{i})\neq\Gamma\backslash G$ for which
$\mu_{i}\to\mu_{\Gamma\backslash G}$, where $W\subset G$ is a fixed proper
subgroup generated by unipotent elements. This provides the $W$ in item (1)
above.

The condition that $\mu_{i}$ is $W$-ergodic and homogeneous is precisely the
condition that there exists $L_{i}\subset G$ with $W\leq L_{i}\leq N_{G}(W)$
and $g_{i}\in G$ for which $\Gamma\backslash\Gamma
g_{i}L_{i}\subset\Gamma\backslash G$ is closed, and the measure obtained by
push-forward is ergodic for the $W$-action. If $T_{i}=g_{i}L_{i}g_{i}^{-1}$,
then $\Gamma\backslash\Gamma g_{i}L_{i}$ being closed in $\Gamma\backslash G$
is equivalent to $\Gamma_{i}=T_{i}\cap\Gamma$ being a lattice in $T_{i}$. The
condition that $\mu_{i}$ is $W$-ergodic is equivalent to the condition that
$T_{i}$ is the identity component of the Zariski closure of
$g_{i}N_{G}(W)g_{i}^{-1}\cap\Gamma$.

Returning to the proof at hand, recall from 1.6 that
${\mathbf{H}}_{i}=\overline{\rho(\Gamma_{i})}$ is a $k$-defined noncompact
proper subgroup of ${\mathbf{H}}$ with
$\dim({\mathbf{H}}_{i})<\dim({\mathbf{H}})$ for infinitely many $i$. Passing
to a subsequence, we can assume that $\dim({\mathbf{H}}_{i})$ is constant, and
we denote this constant by $d$. Consider the $d^{th}$ exterior power of the
adjoint representation

$\wedge^{d}(\operatorname{Ad}):{\mathbf{H}}(k)\longrightarrow\mathbf{GL}\left(\bigwedge\nolimits^{d}\mathfrak{h}\right),$

where $\mathfrak{h}$ is the Lie algebra of ${\mathbf{H}}(k)$. The Lie algebra
$\mathfrak{h}_{i}=\operatorname{Lie}({\mathbf{H}}_{i}(k))$ determines a line
$\ell_{i}\subset\bigwedge\nolimits^{d}\mathfrak{h}$. The representation
$\wedge^{d}(\operatorname{Ad})$ need not be irreducible, however there exists
a nontrivial summand $\theta:{\mathbf{H}}(k)\to\mathbf{GL}(V)$, where $V$ is a
$k$-defined vector space, for which infinitely many $\ell_{i}$ project
nontrivially to $V$. This representation satisfies item (2) above.

Finally for item (3), we again pass to a subsequence to assume that all
$\ell_{i}$ project nontrivially to $V$ and therefore
$\theta(\rho(\Gamma_{i}))$ is contained in the stabilizer of $\ell_{i}$ for
infinitely many $i$. Let $x_{i}$ be the point corresponding to the image of
$\ell_{i}$ in the projectivization $\mathbb{P}(V)$ and we denote the
composition of $\theta\circ\rho$ with the projectivization by
$\overline{\theta}$.

The _suspension space_ , $\Gamma\backslash(G\times\mathbb{P}(V))$ is formed
via the identification

$(g,[v])\sim(\gamma g,\overline{\theta}(\gamma)[v]),$

which is naturally a fiber bundle over $\Gamma\backslash G$. Since the right
$G$-action on the first factor of $G\times\mathbb{P}(V)$ commutes with this
$\Gamma$-action, it descends to a right $G$-action on the suspension space.
Recalling the notation from the beginning of the proof, the fact that
$\Gamma\backslash\Gamma T_{i}$ is closed in $\Gamma\backslash G$ is equivalent
to the natural map $\Gamma_{i}\backslash T_{i}\to\Gamma\backslash\Gamma T_{i}$
being a homeomorphism. As $\overline{\theta}(\Gamma_{i})$ fixes $\\{x_{i}\\}$,
it is straightforward that $\Gamma_{i}\backslash T_{i}$ is isomorphic to
$\Gamma_{i}\backslash(T_{i}\times\\{x_{i}\\})$ under the same action as above.
Recalling that $g_{i}L_{i}=T_{i}g_{i}$, commutativity with the right
$G$-action yields the following commutative diagram:

$\textstyle{\Gamma_{i}\backslash\left(L_{i}\times\\{x_{i}\\}\right)g_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{\Gamma_{i}\backslash\left(T_{i}\times\\{x_{i}\\}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\Gamma\backslash\left(G\times\mathbb{P}(V)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma\backslash\Gamma
g_{i}L_{i}=\Gamma\backslash\Gamma
T_{i}g_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\Gamma\backslash\Gamma
T_{i}\cong\Gamma_{i}\backslash
T_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma\backslash
G}$

If $\sigma_{i}$ denotes the induced map $\sigma_{i}:\Gamma\backslash\Gamma
g_{i}L_{i}\hookrightarrow\Gamma\backslash(G\times\mathbb{P}(V))$, then
$\nu_{i}=(\sigma_{i})_{*}\mu_{i}$ provides a $W$-invariant, ergodic measure on
$\Gamma\backslash(G\times\mathbb{P}(V))$. Again passing to a subsequence, let
$\nu_{\infty}$ denote any weak-$\ast$ limit of the $\nu_{i}$. Then any ergodic
component of $\nu_{\infty}$ projects to $\mu_{\Gamma\backslash G}$ and hence
satisfies (3) since projection and weak-$\ast$ limit commute. This completes
the proof. ∎

##  3\. Dynamical proof for a variant of 1.3

The purpose of this section is to give a dynamical proof of the following
variant of 1.3. We continue with the notation established in Section 2.

######  Theorem 3.1.

Let $S_{n}=\Gamma_{n}\backslash\mathbb{B}^{n}$ be a smooth complex ball
quotient that is arithmetic of simplest kind, and suppose that there is
immersive holomorphic map

$f:S_{n}\longrightarrow S_{m}\coloneqq\Gamma_{m}\backslash\mathbb{B}^{m}$

to another arithmetic ball quotient of simplest kind. Suppose that $Y_{\ell}$
is a sequence of irreducible totally geodesic divisors on $S_{m}$ for which
the top dimensional irreducible components of $f^{-1}(Y_{\ell})$ are totally
geodesic subvarieties of $S_{n}$. Then $f$ is a totally geodesic immersion.

###### Proof.

Throughout this proof, let $G=\operatorname{\operatorname{SU}}(1,n)$. Let
$\omega$ be the Kähler form on $S_{m}$ descending from the $G$-invariant form
on $\mathbb{B}^{m}$ and $\alpha$ be the analogous form on $S_{n}$. We will
prove that $f^{*}\omega=\lambda\alpha$ for some constant $\lambda$, hence the
induced metric on $S_{n}$ is a multiple of the locally symmetric metric. This
implies that $f$ must be a totally geodesic immersion and thus, under the
standard normalizations, $f^{*}\omega=\alpha$.

A simple transversality argument implies that $f^{-1}(Y_{\ell})$ has top-
dimensional irreducible components with codimension one for all but finitely
many $\ell$. Let $\beta$ be a positive $(n-1,n-1)$ form on $S_{m}$ with
compact support. The $Y_{\ell}$ equidistribute in $S_{m}$, so by applying [27,
Cor. 1.5]444Also see the related paper of Möller–Toledo’s [30] and the more
general work of Tayou–Tholozan [39]. to both $\beta$ and $\omega^{n-1}$ we
deduce that

(3.0.1) $\frac{\int_{f(S_{n})\cap Y_{\ell}}\beta}{\int_{f(S_{n})\cap
Y_{\ell}}\omega^{n-1}}\longrightarrow\frac{\int_{f(S_{n})}\beta\wedge\omega}{\int_{f(S_{n})}\omega^{n}},$

as $\ell$ goes to infinity. By definition of the pullback, the left hand side
of Equation (3.0.1) is also equal to

$\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\beta}{\int_{f^{-1}(Y_{\ell})}f^{*}\omega^{n-1}}=\frac{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}{\int_{f^{-1}(Y_{\ell})}f^{*}\omega^{n-1}}\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\beta}{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}.$

We want to compute the limit of
$\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\beta}{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}$
and of
$\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\omega^{n-1}}{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}$.

Since there are possibly several irreducible components, we write

$f^{-1}(Y_{\ell})=Z_{\ell,1}\cup....\cup Z_{\ell,n_{\ell}}.$

Now we use Ratner’s theorem in $S_{n}$, using the fact that $f^{-1}(Y_{\ell})$
consists of totally geodesic divisors. Until this point we only used that we
had some typical intersections, now we redo the same computations using what
we know about $S_{n}$ and $f^{-1}(Y_{\ell})$.

######  Lemma 3.2.

Let $\\{\mu_{f^{-1}(Y_{\ell})}\\}$ be the following sequence of probability
measures on $\Gamma_{n}\backslash G$

$\mu_{f^{-1}(Y_{\ell})}\coloneqq\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}}\mu_{Z_{\ell,i}}.$

Then the $\mu_{f^{-1}(Y_{\ell})}$ weak-$\ast$ converge to $\mu_{S_{n}}$.

###### Proof.

We need to recall the description of the totally geodesic divisors on
$S_{n}=\Gamma_{n}\backslash\mathbb{B}^{n}$. We refer to [2, Lem. 8.2] for a
more detailed description and for a proof of the following facts. We have an
isomorphism $\mathbb{B}^{n}=G/K$, where
$K=\mathrm{S}(\operatorname{\operatorname{U}}(1)\times\operatorname{\operatorname{U}}(n))$
is a maximal compact subgroup of $G$, and a fixed embedding
$W\coloneqq\operatorname{\operatorname{SU}}(1,n-1)\subset G$. Let
$\pi:\Gamma_{n}\backslash G\longrightarrow S_{n}$ be the natural projection
associated to the right quotient by $K$. Then any totally geodesic divisor $Z$
on $S_{n}$ is of the form

$Z=\pi(\Gamma_{n}\backslash\Gamma_{n}gW)$

for some $g\in\operatorname{\operatorname{SU}}(1,n)$ such that
$gWg^{-1}\cap\Gamma_{n}$ is a lattice in $gWg^{-1}$ (see Section 2 and 3.3 at
the end of the proof).

Using this description we can write
$Z_{\ell,i}=\pi(\Gamma_{n}\backslash\Gamma_{n}g_{\ell,i}W)$ for some
$g_{\ell,i}\in\operatorname{\operatorname{SU}}(1,n)$, where $1\leq i\leq
n_{\ell}$ for each $\ell$. Let $\mu_{\ell,i}$ be the $W$-invariant probability
measure on $\Gamma_{n}\backslash G$ with support
$\Gamma_{n}\backslash\Gamma_{n}g_{\ell,i}W$ and define

$\mu_{\ell}\coloneqq\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}}\mu_{\ell,i}.$

Note that $\pi_{*}\mu_{\ell}=\mu_{f^{-1}(Y_{\ell})}$ by [1, Lem. 8.2(1)]. As
each $\mu_{\ell,i}$ is $W$-invariant so is $\mu_{\ell}$ and consequently any
weak-$\ast$ limit $\mu$ of a subsequence of the $\mu_{\ell}$ is a
$W$-invariant measure. Moreover, the similar argument to [1, Lem. 8.3] using
quantitative non-divergence shows that there is no escape of mass and $\mu$ is
also a probability measure.

Let $\nu$ be an ergodic component of $\mu$. By Ratner’s theorem [35], $\nu$ is
a homogeneous, $W$-invariant, $W$-ergodic measure and therefore is either a
constant multiple of Haar measure on $\Gamma_{n}\backslash G$ or a constant
multiple of Haar measure on a proper closed $W$-orbit
$V\subsetneq\Gamma_{n}\backslash G$. However, note that for any such closed
$W$-orbit, $\mu_{\ell,i}(V)=0$ for sufficiently large $\ell$ and all $1\leq
i\leq n_{\ell}$ and hence the latter cannot occur. Therefore only the former
possibility occurs and hence $\nu=\mu_{\Gamma_{n}\backslash G}$. Since
pushforward and weak-$\ast$ limits commute, we conclude that
$\mu_{f^{-1}(Y_{\ell})}\to\mu_{S_{n}}$ as required. ∎

As a consequence, again by upgrading Ratner to a statement about forms as in
[27, Cor. 1.5], but this time in $S_{n}$ rather than in $\mathbb{B}^{m}$, we
get

$\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\beta}{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}\longrightarrow\frac{\int_{S_{n}}f^{*}\beta\wedge\alpha}{\int_{S_{n}}\alpha^{n}}.$

When we apply this result to the case $\beta=\omega^{n-1}$, we find that

$\frac{\int_{f^{-1}(Y_{\ell})}f^{*}\omega^{n-1}}{\int_{f^{-1}(Y_{\ell})}\alpha^{n-1}}\longrightarrow\frac{\int_{S_{n}}f^{*}\omega^{n-1}\wedge\alpha}{\int_{S_{n}}\alpha^{n}}.$

Putting everything together, we obtain

$\frac{\int_{S_{n}}f^{*}\beta\wedge\alpha}{\int_{S_{n}}f^{*}\omega^{n-1}\wedge\alpha}=\frac{\int_{S_{n}}f^{*}\beta\wedge
f^{*}\omega}{\int_{S_{n}}f^{*}\omega^{n}}.$

Since the above is true for all positive $(n-1,n-1)$ forms $\beta$ with
compact support on $S_{m}$ and $f^{*}:\Omega^{p}(S_{m})\to\Omega^{p}(S_{n})$
is locally surjective because $f$ is an immersion, we deduce that
$f^{*}\omega=\lambda\alpha$ for

$\lambda\coloneqq\frac{\int_{S_{n}}f^{*}\omega^{n}}{\int_{S_{n}}f^{*}\omega^{n-1}\wedge\alpha}>0.$

As explained at the beginning of the proof, this implies that $f$ is indeed a
totally geodesic immersion, as required. ∎

The “double equidistribution computation” employed above is reminiscent of an
argument appearing in [11, 5], which was again motivated by the Zilber–Pink
philosophy.

######  Remark 3.3.

Technically speaking, using [2, Lem. 8.2] one only obtains the fact that any
totally geodesic divisor $Z$ is of the form
$Z=\pi(\Gamma_{n}\backslash\Gamma_{n}gL)$ for some
$g\in\operatorname{\operatorname{SU}}(1,n)$, where $L$ is an intermediate
subgroup $W\subseteq L\subseteq N_{G}(W)$ and $gLg^{-1}\cap\Gamma_{n}$ is a
lattice in $gLg^{-1}$. Moreover, there are many instances where it is
necessary to take $L$ strictly larger than $W$ in order for
$gLg^{-1}\cap\Gamma_{n}$ to be a lattice in $gLg^{-1}$. However, in the case
where $\Gamma_{n}$ is a lattice of simplest kind, one can always take $L=W$
which is what we are implicitly using in 3.2.

##  4\. The proof of 1.3

We now prove 1.3 using some Hodge theoretic features behind the theory of
Shimura varieties. This can serve as a warm up case for later arguments that
use the full power of the theory of variations of Hodge structures, though it
only requires the language used in previous sections, allowing us to defer the
introduction of more technical definitions and results.

Suppose that $S_{n}=\Gamma_{n}\backslash\mathbb{B}^{n}$ is an arithmetic ball
quotient and that

$f:S_{n}\longrightarrow S_{m}\coloneqq\Gamma_{m}\backslash\mathbb{B}^{m}$

is an immersive holomorphic map to another arithmetic ball quotient. Let
$\Gamma_{f}$ be the graph of $f$ in $S_{n}\times S_{m}$. We now prove the
following equivalent formulation of 1.3.

######  Theorem 4.1.

Let $\\{Z_{\ell}\\}$ (resp. $\\{Y_{\ell}\\}$) be an infinite sequence of
complex totally geodesic subvarieties of $S_{n}$ (resp. $S_{m}$). If
$f(Z_{\ell})\subset Y_{\ell}$ for all $\ell$, then $f$ is a totally geodesic
immersion.

We first state the _geometric Zilber–Pink conjecture_ , which is a theorem in
our special case. In fact in the more general setting of arbitrary Shimura
varieties this is a theorem of Daw and Ren [14] (see also the more general
5.8, which will be used later). First, recall that an intersection of
subvarieties is _atypical_ if its dimension is smaller than the expected
dimension coming from a differential topology codimension count.

######  Theorem 4.2 (Special case of [14] and [7]).

Let $Y$ be a closed irreducible subvariety of $S_{n}\times S_{m}$. If $Y$ has
a Zariski dense set of atypical intersections with totally geodesic
subvarieties of positive dimension, then $Y$ is contained in a strict complex
totally geodesic subvariety $S_{0}\subset S_{n}\times S_{m}$.

###### Proof of 4.1.

We can and do assume that $f(S_{n})$ is _Hodge generic_ , i.e., is not
contained in any strict complex totally geodesic subvariety $S^{\prime}$ of
$S_{m}$, as $S^{\prime}$ would be a ball quotient of dimension $m^{\prime}<m$
and the result for $S_{m}$ is implied immediately if we prove it instead for
$S^{\prime}$.

Define $U_{\ell}\coloneqq\Gamma_{f}\cap(Z_{\ell}\times Y_{\ell})$. The key
calculation in the proof is the following.

######  Lemma 4.3.

The intersection $U_{\ell}$ between $\Gamma_{f}$ and $Z_{\ell}\times Y_{\ell}$
is atypical for infinitely many $\ell$, that is,

(4.0.1) $\operatorname{codim}_{S_{n}\times
S_{m}}(U_{\ell})<\operatorname{codim}_{S_{n}\times
S_{m}}(\Gamma_{f})+\operatorname{codim}_{S_{n}\times S_{m}}(Z_{\ell}\times
Y_{\ell}).$

###### Proof.

We can pass to a subsequence and assume that the dimensions of $Z_{\ell}$ and
$Y_{\ell}$ are independent of $\ell$. Set $d=\dim Z_{\ell}$ and $\dim
Y_{\ell}=m-e$. We then have

$\displaystyle\operatorname{codim}_{S_{n}\times S_{m}}(U_{\ell})$
$\displaystyle=n+m-d$ $\displaystyle\operatorname{codim}_{S_{n}\times
S_{m}}(\Gamma_{f})+\operatorname{codim}_{S_{n}\times S_{m}}(Z_{\ell}\times
Y_{\ell})$ $\displaystyle=m+(n+m-d-m+e)$ $\displaystyle=n+m-d+e$

Since $e>0$, Equation (4.0.1) holds. ∎

Applying 4.2 to $\Gamma_{f}\subset S_{n}\times S_{m}$, we conclude that
$\Gamma_{f}$ is contained in a strict totally geodesic subvariety $S_{0}$ of
$S_{n}\times S_{m}$. As $f(S_{n})$ is Hodge generic, the second projection of
$S_{0}$ on $S_{m}$ is surjective. As $\Gamma_{f}$ is isomorphic to $S_{n}$
under the first projection, we deduce that $S_{0}$ is also isomorphic to
$S_{n}$ by the first projection and that $f$ is a totally geodesic embedding.
∎

Using a similar argument we can also prove the following.

######  Corollary 4.4.

Let $S_{n}\coloneqq\Gamma_{n}\backslash\mathbb{B}^{n}$ be a finite-volume ball
quotient with an immersive holomorphic map

$f:S_{n}\longrightarrow S_{m}\coloneqq\Gamma_{m}\backslash\mathbb{B}^{m}$

to another finite-volume ball quotient. Let $\\{C_{\ell}\\}$ be a generic
sequence of closed geodesics in $S_{n}$ for which there exits a strict complex
totally geodesic subvariety $Y_{\ell}\subset S_{m}$ such that
$f(C_{\ell})\subset Y_{\ell}$ for all $\ell$. Then $f$ is a totally geodesic
immersion.

The proof of 4.4 uses the so called _Baby Ax–Lindemann theorem_ [41, Prop.
2.6], which we recall below in the special case we need.

######  Proposition 4.5.

Let $Z\subset\mathbb{B}^{n}$ be a real totally geodesic subvariety (of any
real dimension). Let $\Gamma\subset\operatorname{\operatorname{PU}}(1,n)$ be
an arithmetic lattice, and $\pi:\mathbb{B}^{n}\to
S_{\Gamma}=\Gamma\backslash\mathbb{B}^{n}$ be the associated locally Hermitian
space. Then the Zariski closure of $\pi(Z)$ is a complex totally geodesic
subvariety of $S_{\Gamma}$.

###### Proof of 4.4.

Note that $S_{m}$ must be arithmetic. Indeed, if $S_{m}$ were nonarithmetic
the fact that $\\{C_{\ell}\\}$ is generic implies that, after passing to
maximal proper totally geodesic subvarieties, $f(S_{n})$ is contained in a
finite union of totally geodesic subvarieties. Since the map is a smooth
immersion, the image is contained in a single such subvariety $S^{\prime}$.
Replacing $S_{m}$ with $S^{\prime}$, we see that $S^{\prime}$ must have
infinitely many distinct maximal totally geodesic subvarieties, and hence is
arithmetic.

By assumption, $C_{\ell}\subset f^{-1}(Y_{\ell})$. Since $f$ is in fact an
algebraic morphism by Borel’s extension theorem [10], the right-hand side is
Zariski closed and

$Z_{\ell}\coloneqq\overline{C_{\ell}}\subset f^{-1}(Y_{\ell})\subsetneq
S_{n}.$

Thanks to 4.5, $\overline{C_{\ell}}$ is a complex totally geodesic subvariety
of $S_{n}$, which we denote by $Y_{\ell}$. Since the sequence $\\{C_{\ell}\\}$
is generic, we can apply 4.1 to conclude the proof. ∎

##  5\. Hodge theoretic preliminaries and a quick introduction to Zilber–Pink

We begin this section by recalling the André–Oort and Zilber–Pink conjectures
and some formalism from Hodge theory. Loosely speaking, the results on
(closed) geodesics are inspired by the analogy between closed geodesics and CM
points555A point of a Shimura variety is called _CM_ if its associated
Mumford–Tate group is a torus. and the following.

######  Theorem 5.1 (Special case of André–Oort in $S_{1}\times S_{2}$, [33]).

A holomorphic map $f:S_{1}\to S_{2}$ between Shimura varieties sending a
Zariski dense set of CM points of $S_{1}$ to CM points of $S_{2}$, is a
morphism of Shimura varieties. In particular, it is totally geodesic.

For this we refer the reader to [25, 7].

######  Remark 5.2.

It might be interesting to compare this to what happens with abelian
varieties. If $A,B$ are complex abelian varieties, any morphism $f:A\to B$ has
totally geodesic image. This follows from a simple _Rigidity lemma_ , as in
[32, Cor 1, page 43]. If $f$ sends a torsion point of $A$ to a torsion point
of $B$, then the aforementioned corollary shows that $f$ is in fact a
homomorphism. This latter statement is the analogue of 5.1 and shows that a
version of Siu’s immersion problem, 1.2, holds for abelian varieties.

###  5.1. Hodge theory and period domains

We begin with some general notation and conventions. In what follows, an
algebraic variety $S$ is a reduced scheme of finite type over the field of
complex numbers, but may be reducible. If $S$ is an algebraic (resp. analytic)
variety, by a subvariety $Y\subset S$ we always mean a _closed_ algebraic
(resp. analytic) subvariety.

A $\mathbb{Q}$-Hodge structure of weight $n$ on a finite dimensional
$\mathbb{Q}$-vector space $V$ is a decreasing filtration $F^{\bullet}$ on the
complexification $V_{\mathbb{C}}$ such that

$V_{\mathbb{C}}=\bigoplus_{p\in\mathbb{Z}}F^{p}\oplus\overline{F^{n-p}}.$

The category of pure $\mathbb{Q}$-Hodge-structures is Tannakian and moreover
semisimple when considering polarizable Hodge structures, as we frequently
will. The Mumford–Tate group $\mathbf{MT}(V)\subset\mathbf{GL}(V)$ 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, $\mathbf{MT}(V)$ is the smallest
$\mathbb{Q}$-algebraic subgroup of $\mathbf{GL}(V)$ whose base-change to
$\mathbb{R}$ contains the image of
$h:\mathbb{S}\to\mathbf{GL}(V_{\mathbb{R}})$. It is also the stabilizer in
$\mathbf{GL}(V)$ of the Hodge tensors for $V$. As $V$ is polarized, this is a
reductive algebraic group.

In the two sections below we recall the setting and the main results from [7].
We refer also to the companion [6] (especially Section 2 in _op. cit._) for a
quick introduction to this circle of ideas.

###  5.2. Typical and atypical intersections

To understand a VHS ${\mathbb{V}}\to S$, we consider the associated
holomorphic period map

(5.2.1) $\Phi:S^{\textnormal{an}}\longrightarrow\Gamma\backslash D,$

which completely describes ${\mathbb{V}}$. We let $({\mathbf{G}},D)$ denote
the generic Hodge datum of ${\mathbb{V}}$ and $\Gamma\backslash D$ the
associated Hodge variety. The Mumford–Tate domain $D$ decomposes as a product
$D_{1}\times\cdots\times D_{k}$ according to the decomposition of the adjoint
group ${\mathbf{G}}^{\textnormal{ad}}$ into a product
${\mathbf{G}}_{1}\times\cdots\times{\mathbf{G}}_{k}$ of simple factors, where
some ${\mathbf{G}}_{i}$ may be $\mathbb{R}$-anisotropic. Replacing $S$ by a
finite étale cover and reordering the factors if necessary, the lattice
$\Gamma\subset{\mathbf{G}}^{\textnormal{ad}}(\mathbb{R})^{+}$ decomposes as a
direct product $\Gamma\cong\Gamma_{1}\times\cdots\times\Gamma_{r}$ with $r\leq
k$, where $\Gamma_{i}\subset{\mathbf{G}}_{i}(\mathbb{R})^{+}$ is an arithmetic
lattice for each $i$. Writing $D_{\mathrm{triv}}=D_{r+1}\times\cdots\times
D_{k}$ for the product of factors where the monodromy is trivial, which
includes all of the factors $D_{i}$ for which ${\mathbf{G}}_{i}$ is
$\mathbb{R}$-anisotropic, the period map can be written as

(5.2.2) $\Phi:S^{\textnormal{an}}\longrightarrow\Gamma\backslash
D\cong\Gamma_{1}\backslash D_{1}\times\cdots\times\Gamma_{r}\backslash
D_{r}\times D_{\mathrm{triv}},$

where the projection of $\Phi(S^{\textnormal{an}})$ to $D_{\mathrm{triv}}$ is
a point.

We distinguish between special subvarieties $Z$ of _zero period dimension_ ,
which are geometrically elusive, and those of _positive period dimension_.

######  Definition 5.3.

  1. (1)

A subvariety $Z$ of $S$ is said to be of positive period dimension for
${\mathbb{V}}$ if $\Phi(Z^{\textnormal{an}})$ has positive dimension, that is,
$\dim_{\mathbb{C}}\Phi(Z^{\textnormal{an}})>0$.

  2. (2)

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

Using period maps, special subvarieties can also be defined as intersection
loci. Indeed, a closed irreducible subvariety $Z\subset S$ is special for
${\mathbb{V}}$ precisely when $Z^{\textnormal{an}}$ coincides with an analytic
irreducible component $\Phi^{-1}(\Gamma^{\prime}\backslash D^{\prime})^{0}$ of
$\Phi^{-1}(\Gamma^{\prime}\backslash D^{\prime})$, for
$({\mathbf{G}}^{\prime},D^{\prime})\subset({\mathbf{G}},D)$ the generic Hodge
subdatum of $Z$ and $\Gamma^{\prime}\backslash
D^{\prime}\subset\Gamma\backslash D$ the associated Hodge subvariety. We will
equivalently say that $Z$ is special for $\Phi$.

######  Definition 5.4.

Let $Z=\Phi^{-1}(\Gamma^{\prime}\backslash D^{\prime})^{0}\subset S$ be a
special subvariety for ${\mathbb{V}}$ with generic Hodge datum
$({\mathbf{G}}^{\prime},D^{\prime})$. Then $Z$ is said to be _atypical_ if
$\Phi(S^{\textnormal{an}})$ and $\Gamma^{\prime}\backslash D^{\prime}$ do not
intersect generically along $\Phi(Z^{\textnormal{an}})$. That is, $Z$ is
atypical when

(5.2.3) $\operatorname{codim}_{\Gamma\backslash
D}\Phi(Z^{\textnormal{an}})<\operatorname{codim}_{\Gamma\backslash
D}\Phi(S^{\textnormal{an}})+\operatorname{codim}_{\Gamma\backslash
D}\Gamma^{\prime}\backslash D^{\prime}.$

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

######  Remark 5.5.

Since it will be important in the sequel, we remark here that the notion of
typicality and atypicality is always computed with respect to the smallest
special subvariety of $\Gamma\backslash D$, usually called the special
closure. It can, and does, happen that

$Z=\Phi^{-1}(\Gamma^{\prime}\backslash
D^{\prime})^{0}=\Phi^{-1}(\Gamma^{\prime\prime}\backslash
D^{\prime\prime})^{0}$

for some bigger $\Gamma^{\prime\prime}\backslash D^{\prime\prime}$, but
computing codimensions with the latter would give the wrong conclusions.

Let ${\mathbb{V}}$ be a polarizable $\mathbb{Z}$VHS on an irreducible smooth
quasi-projective variety $S$, from [7] we expect:

######  Conjecture 5.6 (Zilber–Pink conjecture for the atypical Hodge locus,
strong version).

The atypical Hodge locus
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{atyp}}$ is a finite
union of atypical special subvarieties of $S$ for ${\mathbb{V}}$.

######  Conjecture 5.7 (Density of the typical Hodge locus).

If $\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{typ}}$ is not empty
then it is analytically dense in $S$.

###  5.3. Some results on the distribution of the Hodge locus

We now recall some results from [7], namely Theorem 6.1, Theorem 10.1, and
Remark 10.2 therein. To simplify the statements the reader can first assume
that the generic Mumford–Tate group is simple, however later it will be used
in a product situation.

######  Theorem 5.8 (Geometric Zilber–Pink).

Let ${\mathbb{V}}$ be a polarizable $\mathbb{Z}$VHS on a smooth connected
complex quasi-projective variety $S$, with generic Hodge datum
$({\mathbf{G}},D)$ and $Z$ be an irreducible component of the Zariski closure
of the union of the atypical special subvarieties of positive period dimension
in $S$. Then either

  * (a)

$Z$ is a maximal atypical special subvariety, or

  * (b)

the adjoint Mumford–Tate group ${\mathbf{G}}_{Z}^{\textnormal{ad}}$ decomposes
as a nontrivial product
${\mathbf{H}}^{\textnormal{ad}}_{Z}\times{\mathbf{L}}_{Z}$, $Z$ contains a
Zariski-dense set of fibers of $\Phi_{{\mathbf{L}}_{Z}}$ which are atypical
weakly special subvarieties of $S$ for $\Phi$, where (possibly up to an étale
covering)

$\Phi_{|Z^{\textnormal{an}}}=(\Phi_{{\mathbf{H}}_{Z}},\Phi_{{\mathbf{L}}_{Z}}):Z^{\textnormal{an}}\longrightarrow\Gamma_{{\mathbf{G}}_{Z}}\backslash
D_{{\mathbf{G}}_{Z}}=\Gamma_{{\mathbf{H}}_{Z}}\backslash
D_{{\mathbf{H}}_{Z}}\times\Gamma_{{\mathbf{L}}_{Z}}\backslash
D_{{\mathbf{L}}_{Z}}\subset\Gamma\backslash D,$

and $Z$ is Hodge generic in a special subvariety
$\Phi^{-1}(\Gamma_{{\mathbf{G}}_{Z}}\backslash D_{{\mathbf{G}}_{Z}})^{0}$ of
$S$ for $\Phi$ which is monodromically typical and therefore typical.

######  Theorem 5.9.

If the typical Hodge locus
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
is nonempty then
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
is analytically dense in $S$.

The above statement already hides an application of 5.8, as explained in [7,
Rmk. 10.2]. Indeed it is proved by first showing that
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{pos}}$ in dense then
invoking 5.8 to say that
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{atyp}}$
is algebraic, and therefore is exactly
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$,
which is dense. In future applications we will actually use this finer
version. See also [23, Thm. 1.6 and Rmk. 1.7] for a related discussion.

######  Remark 5.10.

Hodge theory actually gives a simple combinatorial criterion to decide whether
$\textnormal{HL}(S,{\mathbb{V}}^{\otimes})_{\textnormal{typ}}$ is empty or
not. Indeed see the recent work [18, 23].

##  6\. General Hodge theoretic statement: from typical to atypical
intersections

###  6.1. Isogenies between VHS

For an introduction to Tannakian categories, we refer for example to the
article of Deligne and Milne [16]. The category of $\mathbb{Q}$VHS on a smooth
quasi-projective base $S$, that we denote by $\mathbb{Q}VHS/S$, is Tannakian
(we can fix a fiber functor $\omega_{s}$ corresponding to some base point
$s\in S$). Given a $\mathbb{Z}VHS$ ${\mathbb{V}}$, we denote the associated
$\mathbb{Q}VHS$ by ${\mathbb{V}}_{\mathbb{Q}}$.

######  Definition 6.1.

Let $\mathbb{V}_{1},{\mathbb{V}}_{2}\in\mathbb{Z}VHS/S$. We say that
${\mathbb{V}}_{1}$ and ${\mathbb{V}}_{2}$ are _isogenous_ if there is an
isomorphism
$\langle{\mathbb{V}}_{1,\mathbb{Q}}\rangle^{\otimes}\cong\langle{\mathbb{V}}_{2,\mathbb{Q}}\rangle^{\otimes}$
as Tannakian categories, where
$\langle{\mathbb{V}}_{i,\mathbb{Q}}\rangle^{\otimes}$ denotes the smallest
Tannakian subcategory of $\mathbb{Q}VHS$’s containing
${\mathbb{V}}_{i,\mathbb{Q}}$.

######  Remark 6.2.

Two complex principally polarized abelian varieties $A,B$ that are isogenous
in the usual sense (either via a polarized or an unpolarized isogeny), are
also isogenous in the sense of 6.1. However the converse is in general not
true, for example $H^{1}(A,\mathbb{Z})$ is isogenous to $H^{1}(A\times
A,\mathbb{Z})$. However, for two principally polarized $g$-dimensional abelian
varieties with Mumford–Tate group $\mathbf{GSp}_{2g}$ the two notions agree,
since this is the case when the Mumford–Tate group is as big as possible.

Note that isogeny is an equivalence relation which we denote by
${\mathbb{V}}_{1}\simeq{\mathbb{V}}_{2}$. Let $\Gamma_{1}\backslash D_{1}$ and
$\Gamma_{2}\backslash D_{2}$ be two Mumford–Tate Domains. A _modular
correspondence_ between $\Gamma_{1}\backslash D_{1}$ and $\Gamma_{2}\backslash
D_{2}$ is a subvariety

$V\subset\Gamma_{1}\backslash D_{1}\times\Gamma_{2}\backslash D_{2}$

such that the two projections $p_{1}:V\rightarrow\Gamma_{1}\backslash D_{1}$
and $p_{2}:V\rightarrow\Gamma_{2}\backslash D_{2}$ are finite étale surjective
maps (see also [7, Def. 3.23] for more details). In this situation $V$ is a
Mumford–Tate domain, that is $V=\Gamma\backslash D$, we have that $D\simeq
D_{1}\simeq D_{2}$ and that $\Gamma_{1}$ and $\Gamma_{2}$ are commensurable.
The following proposition translates the isogeny relation from a Tannakian
statement to a more useful one on period maps.

######  Proposition 6.3.

Let ${\mathbb{V}}_{1},{\mathbb{V}}_{2}$ be $\mathbb{Z}VHS$ on $S$, with
associated Hodge data $({\mathbf{G}}_{i},D_{i})$ and period map $\psi_{i}$.
The following are equivalent:

  1. (1)

There is an isogeny between ${\mathbb{V}}_{1}$ and ${\mathbb{V}}_{2}$;

  2. (2)

We have $({\mathbf{G}}_{1},D_{1})$, $({\mathbf{G}}_{2},D_{2})$ are isomorphic
as Hodge datum. Using $({\mathbf{G}},D)$ for the corresponding Hodge datum and
$\psi_{1}:S\to\Gamma_{1}\backslash D$, $\psi_{2}:S\to\Gamma_{2}\backslash D$
for the period maps associated with ${\mathbb{V}}_{1}$, ${\mathbb{V}}_{2}$
(respectively), then there exists a modular correspondence

$\Gamma\backslash D\subset\Gamma_{1}\backslash D\times\Gamma_{2}\backslash D$

and a period map $\psi_{0}:S\to\Gamma\backslash D$ such that if
$p_{i}:\Gamma\backslash D\to\Gamma_{i}\backslash D$ are the natural
projections, then $\psi_{i}=p_{i}\circ\psi_{0}$ for $i\in\\{1,2\\}$.

###### Proof.

Let ${\mathbb{V}}$ be a $\mathbb{Z}$VHS on $S$ with generic Hodge datum
$({\mathbf{G}},D)$. Then the Tannakian category generated by
${\mathbb{V}}_{\mathbb{Q}}$ is determined by $({\mathbf{G}},D)$ as any
${\mathbb{W}}$ in $\langle{\mathbb{V}}_{\mathbb{Q}}\rangle^{\otimes}$ is, by
definition of a Tannakian category, given by a representation of
${\mathbf{G}}$ on the fibre $W_{\mathbb{Q}}$ of ${\mathbb{W}}_{\mathbb{Q}}$ at
a point $s\in S$ and the Hodge structure on $W_{\mathbb{Q}}$ is determined by
some Hodge morphism $\alpha:\mathbb{S}\rightarrow{\mathbf{G}}$ belonging to
$D$. Therefore if $({\mathbf{G}}_{1},D_{1})\cong({\mathbf{G}}_{2},D_{2})$,
then ${\mathbb{V}}_{1},{\mathbb{V}}_{2}$ are isogenous.

If ${\mathbb{V}}_{1},{\mathbb{V}}_{2}$ are isogenous, they have isomorphic
generic Mumford–Tate groups, denoted by ${\mathbf{G}}$, under the
interpretation of the Mumford–Tate group in terms of Tannakian categories
recalled above. Thus

$\langle{\mathbb{V}}_{1,\mathbb{Q}}\rangle^{\otimes}\cong\langle{\mathbb{V}}_{2,\mathbb{Q}}\rangle^{\otimes}\cong\langle{\mathbb{V}}_{1,\mathbb{Q}}\oplus{\mathbb{V}}_{2,\mathbb{Q}}\rangle^{\otimes}$

and therefore the generic Mumford–Tate group of
${\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2}$ is also ${\mathbf{G}}$.

The $\mathbb{Z}$VHS ${\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2}$ corresponds to a
period map

$\psi_{0}:S\longrightarrow\Gamma\backslash D\subset\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2},$

where $({\mathbf{G}},D)$ is the generic Hodge subdatum associated with
${\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2}$. Moreover, if the natural projections
are $p_{i}:\Gamma\backslash D\to\Gamma_{i}\backslash D_{i}$, we have
$\psi_{i}=p_{i}\circ\psi_{0}$. Let $\alpha_{i}:\mathbb{S}\to{\mathbf{G}}$ be
Hodge morphisms such that $D_{i}={\mathbf{G}}(\mathbb{R})^{+}\alpha_{i}$. Then
there exists an embedding of ${\mathbf{G}}$ in
${\mathbf{G}}\times{\mathbf{G}}$ such that $D$ is the
${\mathbf{G}}(\mathbb{R})^{+}$-orbit of $\alpha=(\alpha_{1},\alpha_{2})$.
Therefore $D\simeq D_{1}\simeq D_{2}$. ∎

###  6.2. The main statement

######  Theorem 6.4.

Let $S$ be a smooth quasi-projective variety and
$\mathbb{V}_{1},\mathbb{V}_{2}$ two pure polarized $\mathbb{Z}$VHSs on $S$.
Assume that the generic Mumford–Tate groups of ${\mathbb{V}}_{1}$ and
${\mathbb{V}}_{2}$ are $\mathbb{Q}$-simple. If

$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S,\mathbb{V}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset,$

then $\mathbb{V}_{1}$ is isogenous to $\mathbb{V}_{2}$. As a consequence,
$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})=\textnormal{HL}(S,\mathbb{V}_{2}^{\otimes})$.

We remark here that, in fact, some stronger variants with the _weakly special
Hodge locus_ replacing the Hodge locus can be given.

######  Remark 6.5.

By 5.9, the condition
$\textnormal{HL}(S,\mathbb{V}_{i}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset$
implies that
$\textnormal{HL}(S,\mathbb{V}_{i}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
is analytically dense in $S$. 5.7 predicts that
$\textnormal{HL}(S,\mathbb{V}_{i}^{\otimes})_{\textnormal{typ}}$ should be
analytically dense whenever it is nonempty. The condition
$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S,\mathbb{V}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset$
is therefore quite restrictive. Moreover by [7, Thm. 1.5], the level of the
$\mathbb{Z}$VHS ${\mathbb{V}}_{1}$ and ${\mathbb{V}}_{2}$ must be $1$ (the
Shimura case) or $2$.

To better explain 6.4, we give a simple and more concrete corollary about two
principally polarized families of abelian varieties $h_{i}:\mathcal{A}_{i}\to
S$.

######  Corollary 6.6.

Fix $g\geq 2$. Let $S$ be a smooth quasi projective variety and
$h_{i}:\mathcal{A}_{i}\to S$ be two principally polarized $g$-dimensional
families of abelian varieties whose monodromy group is $\mathbf{Sp}_{2g}$. Set
${\mathbb{V}}_{i}\coloneqq{R^{1}h_{i,*}\mathbb{Z}}$ for the associated VHS. If

(6.2.1)
$\textnormal{HL}(S,{\mathbb{V}}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S,{\mathbb{V}}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset,$

then $\mathcal{A}_{1}$ is isogenous to $\mathcal{A}_{2}$ as principally
polarized $S$-abelian schemes.

###### Proof.

We will use 6.2 and the following fact: Given two principally polarized
families of abelian varieties $h_{i}:\mathcal{A}_{i}\to S$, one has
$\operatorname{Hom}_{S}(\mathcal{A}_{1},\mathcal{A}_{2})=\operatorname{Hom}(R_{1}{h_{1},_{*}}\mathbb{Z},R_{1}{h_{2},_{*}}\mathbb{Z})$;
see [15, Cor. 4.4.15]. Indeed 6.4 implies that the two $\mathbb{Z}$VHSs
${\mathbb{V}}_{i}$ for $i\in\\{1,2\\}$ are isogenous in the sense of 6.1, and
the above facts imply that this notion of isogeny translates precisely to the
traditional one. ∎

######  Remark 6.7.

The above statement bears some similarities with work of Baker and DeMarco [4]
in complex dynamics, who show that for any fixed $a,b\in\mathbb{C}$ and any
integer $d\geq 0$, the set of $c\in\mathbb{C}$ for which both $a$ and $b$ are
preperiodic for $z^{d}+c$ is infinite if and only if $a^{d}=b^{d}$. Also see
generalizations like in the work of Yuan and Zhang [42]. Both proofs mentioned
above use equidistribution but here we work with positive dimensional
subvarieties. The statement without _pos_ is deeper and would follow from 5.6.

###### Proof of 6.4.

The main step in the proof is the following lemma for which we do not need to
assume that the generic Mumford–Tate group of ${\mathbb{V}}_{i}$ is
$\mathbb{Q}$-simple.

######  Lemma 6.8.

Let $f_{i}:S\to\Gamma_{i}\backslash D_{i}$ be the period map associated with
$\mathbb{V}_{i}$, $i=1,2$, and let

$f_{1}\times f_{2}:S\longrightarrow\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}$

be the period map associated with ${\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2}$.
Then
$\textnormal{HL}(S,({\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2})^{\otimes})_{\textnormal{pos},\textnormal{atyp}}$
is analytically dense in $S$.

###### Proof.

As
$\textnormal{HL}(S,{\mathbb{V}}_{i}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset$,
by 5.9 there exists sequences of special subvarieties
$\\{Z_{i,\ell}\\}_{\ell\in\mathbb{N}}$ in $\Gamma_{i}\backslash D_{i}$ and a
Zariski dense sequence $\\{W_{\ell}\\}_{\ell\in\mathbb{N}}$ of components of
both
$\textnormal{HL}(S,{\mathbb{V}}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
and
$\textnormal{HL}(S,{\mathbb{V}}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
such that

$W_{\ell}=f_{1}^{-1}(Z_{1,\ell})^{0}=f_{2}^{-1}(Z_{2,\ell})^{0}$

for some components $f_{i}^{-1}(Z_{i,\ell})^{0}$ of $f_{i}^{-1}(Z_{i,\ell})$.
The fact that $W_{\ell}$ are again typical intersections, guaranteed by 5.9,
implies that $W_{\ell}$ is Hodge generic for ${\mathbb{V}}_{i}$ in
$Z_{i,\ell}$ (see also the discussion in 5.5). By passing to a subsequence, we
assume that $Z_{i,\ell}$ has fixed dimension $z_{i}$.

Since $W_{\ell}$ is realized in two ways as a typical intersection, we have

$\operatorname{codim}_{\Gamma_{i}\backslash
D_{i}}f_{i}(W_{\ell})=\operatorname{codim}_{\Gamma_{i}\backslash
D_{i}}Z_{i,\ell}+\operatorname{codim}_{\Gamma_{i}\backslash D_{i}}(f_{i}(S)).$

for each $i\in\\{1,2\\}$. That is, if $w_{i}\coloneqq\dim f_{i}(W)$,
$d_{i}\coloneqq\dim\Gamma_{i}\backslash D_{i}$, and $s_{i}\coloneqq\dim
f_{i}(S)$, then

(6.2.2) $d_{i}-w_{i}=d_{i}-z_{i}+d_{i}-s_{i}.$

Note that we do not assume that our period maps are immersive. Notice that

$W_{\ell}\subset(f_{1}\times f_{2})^{-1}(Z_{1,\ell}\times
Z_{2,\ell})\varsubsetneq S.$

Let $W_{\ell}^{\prime}$ be an irreducible component of $(f_{1}\times
f_{2})^{-1}(Z_{1,\ell}\times Z_{2,\ell})$ containing $W_{\ell}$. We claim that
$W_{\ell}^{\prime}$ is an atypical intersection of positive period dimension
in the Hodge locus
$\textnormal{HL}(S,({\mathbb{V}}_{1}\oplus{\mathbb{V}}_{2})^{\otimes})$ for
all $\ell$. For this, we need to show that

$\operatorname{codim}_{\Gamma_{1}\backslash D_{1}\times\Gamma_{2}\backslash
D_{2}}(f_{1}\times
f_{2}(W_{\ell}^{\prime}))<\operatorname{codim}_{\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}}(Z_{1,\ell}\times
Z_{2,\ell})+\operatorname{codim}_{\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}}(f_{1}\times f_{2}(S)).$

Summing Equation (6.2.2) for $i=1,2$ we have

(6.2.3)
$d_{1}+d_{2}-w_{1}-w_{2}=d_{1}+d_{2}-z_{1}-z_{2}+d_{1}+d_{2}-s_{1}-s_{2}.$

This implies the requisite equation by noticing that

$\max\\{\dim(f_{1}(Y)),\dim(f_{2}(Y))\\}\leq\dim(f_{1}\times
f_{2}(Y))\leq\dim(f_{1}(Y))+\dim(f_{2}(Y)).$

for any $Y$ in $S$. ∎

######  Lemma 6.9.

Each of the following hold:

  1. (1)

If the monodromy of ${\mathbb{V}}_{i}$ is equal to the derived subgroup of its
Mumford–Tate group ${\mathbf{G}}_{i}$, then $f_{1}\times f_{2}(S)$ is not
Hodge generic in $\Gamma_{1}\backslash D_{1}\times\Gamma_{2}\backslash D_{2}$.

  2. (2)

If the Mumford–Tate group ${\mathbf{G}}_{i}$ of ${\mathbb{V}}_{i}$ is
$\mathbb{Q}$-simple, then
${\mathbf{G}}_{1}^{\textnormal{ad}}\cong{\mathbf{G}}_{2}^{\textnormal{ad}}$,
$D_{1}\simeq D_{2}$. Denoting by $({\mathbf{G}},D)$ the corresponding Hodge
datum, we have that $f_{1}\times f_{2}(S)$ is Hodge generic in a modular
correspondence

$\Gamma\backslash D\subset\Gamma_{1}\backslash D\times\Gamma_{2}\backslash D.$

###### Proof.

By 6.8, $S$ has a Zariski dense set of positive dimensional atypical
components of the Hodge locus for the period map $f_{1}\times f_{2}$.
Therefore $S$ satisfies conclusion (a) or (b) of 5.8. If conclusion (a) holds,
then $f_{1}\times f_{2}(S)$ is not Hodge generic in $\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}$, as desired. If the monodromy of
${\mathbb{V}}_{i}$ is equal to the derived subgroup of its Mumford–Tate group
${\mathbf{G}}_{i}$, then the projection of $f_{i}(S)$ on every factor of
$\Gamma_{i}\backslash D_{i}$ is positive dimensional and Hodge generic. The
same is true for the period map $f_{1}\times f_{2}$. Therefore conclusion (b)
is not satisfied. This finishes the proof of the first part of the lemma.

If the Mumford–Tate group ${\mathbf{G}}_{i}$ of ${\mathbb{V}}_{i}$ is
$\mathbb{Q}$-simple, then the monodromy of ${\mathbb{V}}_{i}$ is equal to the
derived subgroup of its Mumford–Tate group ${\mathbf{G}}_{i}$ and by the first
part, $f_{1}\times f_{2}(S)$ is not Hodge generic in $\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}$. Moreover in this situation, the only
strict special subvarieties of $\Gamma_{1}\backslash
D_{1}\times\Gamma_{2}\backslash D_{2}$ whose projections on both factors are
surjective are the modular correspondences. This forces $D_{1}\simeq D_{2}$
and
${\mathbf{G}}_{1}^{\textnormal{ad}}\cong{\mathbf{G}}_{2}^{\textnormal{ad}}$.
This finishes the proof of the second part of the lemma. ∎

6.4 is now a consequence of the characterization of isogenous $\mathbb{Z}$VHSs
given in 6.3. ∎

######  Remark 6.10.

6.4 implies 4.1 on ball quotients, which we now explain. On the smaller ball
quotient $S_{n}$, which we assume to be arithmetic for simplicity, we have a
$\mathbb{Z}$VHS $\mathbb{V}_{1}$ induced by any faithful linear representation
of $\operatorname{GU}(1,n)$. The Hodge locus for such a VHS does not depend on
the linear representation by a theorem of Deligne and it is equal to the union
of all sub-Shimura varieties. With the terminology introduced in 5.4, a sub-
Shimura variety is always in the typical Hodge locus. In particular

$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos}}=\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$

and this locus is equal to the union of all totally geodesic subvarieties. Now
a map $f:S_{n}\to S_{m}$ gives another VHS $\mathbb{V}_{2}$ by pulling back
the natural one we have on $S_{m}$. The intersection theoretic assumptions
appearing in 4.1 force
$\textnormal{HL}(S,\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S,\mathbb{V}_{2}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$.
Therefore we conclude by applying 6.4.

##  7\. Proofs of 1.11 and 1.14

In this section we prove the second part of 1.11 and then 1.14 following a
similar argument. Let $\Gamma\subset\operatorname{PU}(1,n)$ for $n>1$ be a
lattice of the first kind, which we can assume to be neat. Consider the smooth
quasi-projective variety $S_{\Gamma}$ and let

$\rho:\Gamma\longrightarrow{\mathbf{H}}(k)$

be a cohomologically rigid representation, i.e.,
$H^{1}(\Gamma,\operatorname{Ad}_{\mathbf{H}}\rho)=0$, where

$\operatorname{Ad}_{\mathbf{H}}:{\mathbf{H}}(k)\longrightarrow\operatorname{Aut}(\mathfrak{h})$

is the adjoint representation over some local field $k$. Assuming that $\rho$
is geodesically rich in the sense of 1.6, we would like to show that $\rho$
extends. The proof is a generalization of the arguments from [8, Sec. 4 and
5], improved by combining them with Section 6.

###### Proof of 1.11 (2).

The proof proceeds in two steps. The first is to build a rich $\mathbb{Z}$VHS,
using the rigidity assumption, and the second is to use richness to finish the
proof.

From the vanishing $H^{1}(\Gamma,\operatorname{Ad}_{\mathbf{H}}\rho)=0$ we
have that $\rho$ underlies a $\mathbb{C}$VHS. This is essentially due to
Simpson and is explained in detail in [8, Sec. 4]. Moreover,

$\mathbb{Z}(\rho)\coloneqq\mathbb{Z}[\operatorname{tr}\operatorname{Ad}_{\mathbf{H}}(\rho(\gamma)):\gamma\in\Gamma]$

has a natural embedding in $\mathcal{O}_{K}$, the ring of integers of a
totally real number field $K$. There are two ways to show that
$\mathbb{Z}(\rho)$ lies in $\mathcal{O}_{K}$, as opposed to the ring of
$S$-integers for some finite set of finite places $S$ of $K$. In [8, Sec. 4]
it is done by invoking the recent work of Esnault and Groechenig [17].
However, here we can give a self-contained proof by applying 1.12.

From here forward we assume that $\rho(\Gamma)$ is Zariski dense in
${\mathbf{H}}(k)$ and will then prove that ${\mathbf{H}}(k)\cong G$. In
particular, from the above we obtain a natural $K$-form $\mathbf{H}\subset$ of
${\mathbf{H}}$. Let $\sigma_{i}:K\to\mathbb{R}$ be the places of $K$, ordered
in such a way that $\sigma_{1}$ is the identity for the given lattice
embedding. Let
$\widehat{{\mathbf{H}}}\subset=\operatorname{Res}_{K/\mathbb{Q}}\mathbf{H}\subset$
and consider the $\mathbb{Z}$-local system

$\displaystyle\hat{\rho}:\Gamma$
$\displaystyle\longrightarrow\widehat{\mathbf{H}}\subset(\mathbb{R})$
$\displaystyle\gamma$
$\displaystyle\longmapsto\hat{\rho}(\gamma)=(\sigma_{i}(\rho(\gamma))).$

Again by cohomological rigidity of $\rho$, we have that $\hat{\rho}$ naturally
underlies a $\mathbb{Z}$VHS $\widehat{\mathbb{V}}$ (compare with [8, Thm.
4.2.4]). As recalled in Section 5, $\widehat{\mathbb{V}}$ corresponds to a
period map

$\psi:S_{\Gamma}^{\textnormal{an}}\longrightarrow\widehat{\mathbf{H}}\subset(\mathbb{Z})\backslash
D.$

Now that we have a $\mathbb{Z}$VHS, as in 6.10 we use the results of Section 6
to prove the theorem. Let $S_{\Gamma}=\Gamma\backslash\mathbb{B}^{n}$ be the
associated ball quotient, and $\mathbb{V}_{1}$ the $\mathbb{Z}$VHS on
$S_{\Gamma}$ (associated to some faithful linear rational representation of
$\mathbf{G}$). By construction
$\textnormal{HL}(S_{\Gamma},\mathbb{V}_{1}^{\otimes})_{\textnormal{pos}}$ is
the union of the complex totally geodesic subvarieties of $S_{\Gamma}$, and it
is equal to
$\textnormal{HL}(S_{\Gamma},\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$
(in particular it is non-empty). It is straightforward to see that the fact
that $\rho$ is complex rich implies that $\hat{\rho}$ is complex rich as well.
In particular, it follows that

(7.0.1)
$\textnormal{HL}(S_{\Gamma},\mathbb{V}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S_{\Gamma},\widehat{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{typ}}$

6.4 implies that $\mathbb{V}_{1},\widehat{\mathbb{V}}$ are isogenous which
means that the period domain
$\widehat{\mathbf{H}}\subset(\mathbb{Z})\backslash D$ is isomorphic to
$S_{\Gamma}$, and that $\psi$ is isogenous to the identity (as in 6.3).
Therefore $\hat{\rho}$ extends, which is only possible if the initial
representation $\rho$ extends. That is, $\psi$ is an isomorphism. This
concludes the proof of the desired statement. ∎

###### Proof of 1.14.

The proof follows the same argument given above. The only difference is we are
given a $\mathbb{Q}$VHS ${\mathbb{V}}$ on the algebraic variety $S_{\Gamma}$.
By applying 1.12, ${\mathbb{V}}$ must be a $\mathbb{Z}$VHS, and the richness
assumption implies that

$\textnormal{HL}(S_{\Gamma},{\mathbb{V}}^{\otimes})_{\textnormal{pos},\textnormal{typ}}=\textnormal{HL}(S_{\Gamma},{\mathbb{V}}_{1}^{\otimes})_{\textnormal{pos},\textnormal{typ}}\neq\emptyset,$

as in (7.0.1) above. Therefore the results follows, once more, from 6.4. ∎

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CNRS, IMJ-PRG, Sorbonne Université, 4 place Jussieu, 75005 Paris, France

E-mail address, G. Baldi<EMAIL_ADDRESS><EMAIL_ADDRESS>

Department of Mathematics, University of Oklahoma, Norman, OK 73019

E-mail address, N. Miller<EMAIL_ADDRESS>

Department of Mathematics, Temple University, Philadelphia, PA 19122

E-mail address, M. Stover<EMAIL_ADDRESS>

I.H.E.S., Université Paris-Saclay, CNRS, Laboratoire Alexandre Grothendieck.
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