1 Introduction and motivation
A cornerstone in the representation theory of lattices in Lie groups is the Margulis superrigidity theorem [Reference Margulis31, p. 2]. Indeed, the implications of superrigidity are profound, including arithmeticity of irreducible lattices in all non-compact semisimple Lie groups except
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
and
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
, that is, 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 [Reference Corlette14] and Gromov and Schoen [Reference Gromov and Schoen22]. While the harmonic maps method was extended to give a new proof in higher rank [Reference Mok, Siu and Yeung32], it is open (and quite relevant in the 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
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
and
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
that are not superrigid; see [Reference Johnson and Millson23, §5] for real hyperbolic examples and [Reference Mostow34, §22] or [Reference Klingler27, §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 [Reference Fisher20, Question 3.15].
Question 1.1. (D. Fisher)
Let
$\Gamma \subset G$
be a lattice where
$G= \operatorname {\mathrm {\operatorname {SO}}}(1,n)$
or
$\operatorname {\mathrm {\operatorname {SU}}}(1,n)$
. What conditions on a representation
$\rho : \Gamma \to \operatorname {\mathrm {GL}}_m(k)$
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 [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2, Reference Baldi and Ullmo8] to explore geometric conditions on a representation that allow one to use tools from dynamics and/or Hodge theory to prove new rigidity phenomena. The conditions we provide, which address Question 1.1 through the behavior of totally geodesic submanifolds under maps related to the representation, have also been considered in studying the infinite volume setting; see for example [Reference Kim and Oh25, Reference Kim and Oh26]. 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 and Maubon [Reference Koziarz and Maubon29] and many others);
-
• maximal for their limit set (Besson, Courtois and Gallot [Reference Besson, Courtois and Gallot10] and Shalom [Reference Shalom39]).
Here, we study a class of representations generalizing those shown to be superrigid in [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2], 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.
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
$\operatorname {\mathrm {\operatorname {PO}}}(q)$
. 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
$\operatorname {\mathrm {\operatorname {PU}}}(h)$
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. The associated locally symmetric spaces have an abundance of totally geodesic submanifolds, which are the main object of study in this paper.
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 [Reference Siu41, Problem (b), pp. 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
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
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 [Reference Cao and Mok13], which answered Question 1.2 in the affirmative for
$f : \Gamma \backslash \mathbb {B}^n \to \Lambda \backslash \mathbb {B}^m$
when
$m < 2 n$
. See [Reference Bader, Fisher, Miller and Stover2, Theorem 1.7] and the surrounding references for more on Siu’s analogous submersion 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 such that
$f(S_{\Gamma _1})$
does not lie in any smaller totally geodesic subvariety of
$S_{\Gamma _2}$
. Suppose that
$S_{\Gamma _1}$
contains a proper totally geodesic subvariety of positive dimension. Then, the following are equivalent.
-
(1) The map f is covering.
-
(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 positive-dimensional hypothesis on the subvarieties of
$S_{\Gamma _1}$
is to rule out the trivial counterexample of points. We offer two proofs of Theorem 1.3, one using homogeneous dynamics which gives a slightly weaker statement in §3, and the second using Hodge theory and functional transcendence in §4, as an application of Theorem 1.18 below.
For arithmetic lattices of simplest kind, Theorem 1.3 reduces Question 1.2 to checking a single dimension.
Corollary 1.4. If Question 1.2 has a positive answer when
$S_{\Gamma _1}$
is a two-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
$n \ge 2$
.
Remark 1.5. By [Reference Cao and Mok13], 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, Theorem 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 Theorem 1.17. 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 necessarily an isomorphism. Here, ‘generic’ means that they are not all contained in a finite union of maximal totally geodesic subvarieties of S.
From a dynamical point of view, the dimension at least two hypothesis in the previous results 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.
Theorem 1.6. Let
$S=\Gamma _n \backslash \mathbb {B}^n$
be a ball quotient with an immersive holomorphic map
$$ \begin{align*} f: S \longrightarrow \Gamma _m \backslash \mathbb{B}^m \end{align*} $$
such that
$f(S)$
does not lie in any smaller totally geodesic subvariety of
$\Gamma _m \backslash \mathbb {B}^m$
. Let
$\{C_\ell \}$
be a generic sequence of distinct closed geodesics in S in the sense that they are not all contained in a finite union of maximal totally geodesic subvarieties of 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 Corollary 4.4 for the proof. Note that Theorem 1.6 applies for non-arithmetic
$\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 this section with a final question that would generalize Theorem 1.6.
Question 1.7. Let
$\Gamma $
be any lattice in
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
,
$n>1$
, 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?
1.2 Rich representations and the main theorem
Let
$\Gamma \subset G$
be a torsion-free arithmetic lattice, where G is
$\operatorname {\mathrm {\operatorname {SU}}}(1,n)$
or
$\operatorname {\mathrm {\operatorname {SO}}}(1,n)$
, 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 case. Since 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 submanifolds of real dimension at least
$2$
. This implication 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 [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2, Reference Baldi and Ullmo8]. 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 distinct 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 those that ‘preserve’ a significant amount of the rich structure of
$\Gamma $
.
Definition 1.8. Let k be a local field and
${\mathbf H}$
be a semisimple k-algebraic group without compact factors. Let
$\rho : \Gamma \to {\mathbf H}(k)$
be a representation with unbounded, Zariski dense image. We say that
$\rho $
is geodesically rich (just rich when it is clear from context) if the following are satisfied:
-
•
$\Gamma $
is geodesically rich; and -
• there are infinitely many i so that
$\dim {\mathbf H}_i < \dim {\mathbf H}$
, where
${\mathbf H}_i$
is defined by (1.1)for each i, where closure is with respect to the Zariski topology.
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 1.9. Let
$\Gamma \subset G=\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
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 1.10. Do rich representations extend? Equivalently, are there non-trivial rich representations (that is, other than the case where
${\mathbf H}(k) = G$
and
$\rho $
is given by conjugation)?
Before stating one of our primary results toward a positive answer, we recall a definition that appears in [Reference Bader and Furman3] that will be used to give a generalization/abstract version of the method of [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2].
Definition 1.11. 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
$U \le \operatorname {\mathrm {Ker}}(\tau )$
.
Remark 1.12. If k is non-Archimedean, then any pair
$(k,{\mathbf H})$
is strongly compatible with G.
Our main theorem establishes Conjecture 1.10 in two cases.
Theorem 1.13. Let
$\Gamma $
be an arithmetic lattice in G which is either
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
or
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
,
${\mathbf H}$
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)
${\mathbf H}$
is strongly compatible (in the sense of Definition 1.11) and k-simple; or -
(2)
$\Gamma $
is complex hyperbolic,
$\rho $
is complex geodesically rich, and
$\rho $
is cohomologically rigid (that is,
$H^1(\Gamma , \operatorname {Ad}_{{\mathbf H}} \circ \rho )=0$
),
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 part (1), which follows from the methods developed in [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2], is in §2. The proof of part (2), contained in Theorem 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 [Reference Baldi, Klingler and Ullmo7]. Regarding part (2), it might be possible to study some non-rigid representations via the so-called factorization theorem, see for example [Reference Simpson40, Theorem 10] and various more recent generalizations. The fact that
$(k, {\mathbf H})$
is always compatible when k is non-Archimedean has the following fairly direct corollary.
Corollary 1.14. Suppose that
$\Gamma \subset \operatorname {\mathrm {\operatorname {PU}}}(1,n)$
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.
Corollary 1.14 should be viewed as a complement to the fundamental recent work of Esnault and Groechenig [Reference Esnault and Groechenig18] that proves integrality of certain cohomologically rigid local systems. Note that Corollary 1.14 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 [Reference Toledo43], and hence not all representations of complex hyperbolic lattices are definable over a number field. As noticed in [Reference Bader, Fisher, Miller and Stover2], the version of Corollary 1.14 that implicitly follows from the results there makes the main theorem of that paper independent of [Reference Esnault and Groechenig18].
Remark 1.15. In contrast with Theorem 1.6, we now give a simple cautionary example regarding generalizing richness to closed geodesics. Let G be
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
for
$n \ge 3$
or
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
for
$n \ge 2$
, 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 [Reference Prasad and Rapinchuk37], and 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 $
, and hence this provides a fairly rich collection of one-dimensional maximal totally geodesic subspaces.
Representations as in Remark 1.15 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, and hence the representation could be considered rich in analogy with the higher-dimensional setting. While one might reasonably interpret this as saying the one-dimensional case is hopeless, Theorem 1.6 indicates that stronger geometric hypotheses can still lead to a rigidity theorem.
We now comment on how richness may differentiate between real and complex hyperbolic lattices. For example, we do not know the answer to the following.
Question 1.16. Is every faithful linear representation of an arithmetic lattice in
$\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
of simplest kind rich?
Bending deformations of lattices in
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
[Reference Johnson and Millson23, §5] easily imply that Question 1.16 is false for real hyperbolic lattices of simplest kind. However, Goldman–Millson rigidity [Reference Goldman and Millson21] implies that there are no analogous deformations in the complex hyperbolic setting. It seems that examples giving a negative answer to Question 1.16 would come from representations of a new kind. However, a positive answer would (for example) prove superrigidity of unbounded, Zariski dense, and faithful representations to connected, adjoint targets.
Since Theorem 1.13 proves that all rich representations to connected, adjoint k-simple groups over non-Archimedean 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 variation of Hodge structure results
To enter in the world of variations of Hodge structures (VHSs), we focus on complex hyperbolic lattices
$\Gamma \subset G=\operatorname {\mathrm {\operatorname {PU}}}(1,n)$
with
$n>1$
. See §5 for the various notions from Hodge theory used in this subsection. In this setting, we can restate Definition 1.9 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
$$ \begin{align} f: S_\Gamma \longrightarrow \Gamma_H\backslash X_H. \end{align} $$
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 equation (1.2) 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 1.17. 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 Theorem 1.17 builds on the main results of [Reference Baldi, Klingler and Ullmo7], namely the geometric part of the Zilber–Pink conjecture. In particular, the proof crucially uses the following result proved in §6, which may be of independent interest.
Theorem 1.18. 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 non-trivial. If
$$ \begin{align*} \mathrm{HL}(S,\mathbb{V}_1^\otimes)_{\mathrm{pos}, \mathrm{typ}}=\mathrm{HL}(S,\mathbb{V}_2^\otimes)_{\mathrm{pos}, \mathrm{typ}}\neq \emptyset, \end{align*} $$
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
$\mathrm {HL}(S,\mathbb {V}^\otimes )_{\mathrm {pos}, \mathrm {typ}}$
for the so-called typical Hodge locus of positive period dimension, as introduced in [Reference Baldi, Klingler and Ullmo7]. See Definition 6.1 for the precise definition of an isogeny in this context.
1.4 Outline of the paper
In §2, we prove the statement about compatibility. The rest of the paper is devoted to the complex hyperbolic case. In §3, we give the argument based on various forms of Ratner’s equidistribution theorem. In §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, §5 is devoted to all the required preliminaries from Hodge theory and the Zilber–Pink paradigm. In §6, we prove the most general statement on isogenies of VHSs, namely Theorem 1.18. Finally, we prove the main results about the extension of rich representations underlying a VHS in Theorem 7.
2 Dynamics background and compatibility
This section contains some background on the dynamical side, along with the proof of the first part of Theorem 1.13. Let G be either
$\operatorname {\mathrm {\operatorname {PO}}}(1,n)$
or
$\operatorname {\mathrm {\operatorname {PU}}}(1,m)$
. 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 [Reference Bader, Fisher, Miller and Stover1, Reference Bader, Fisher, Miller and Stover2].
Fixing a maximal compact subgroup K in G, the associated locally symmetric space is
$X = G/K$
. Fix a representative
$Y \subset X$
for each type (that is, G-orbit) of totally geodesic subspace of X with dimension at least two and call such a representative a standard subspace. 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
$g Y$
of the standard subspace Y has the property that
$g L g^{-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
$$ \begin{align*} \Gamma \backslash \Gamma g L \hookrightarrow \Gamma \backslash G \end{align*} $$
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 an 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
$T_i=g_iL_ig_i^{-1}$
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 $
limit of (the
$g_i$
translates of) the measures
$\mu _{\Gamma _i \backslash T_i}$
. All limits of measures in this paper will be in the weak-
$\ast $
sense.
Theorem 2.1. Suppose that
$\Gamma \subset G$
is a geodesically rich lattice. 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 Definition 1.8,
$\rho $
is a homomorphism with unbounded, Zariski dense image. Following [Reference Bader, Fisher, Miller and Stover1, Theorem 1.6 and Remark 1.7], to prove that there exists a continuous extension
${\widetilde {\rho } : G\to {\mathbf H}(k)}$
of
$\rho $
, it suffices to find:
-
(1) a non-compact almost-simple, proper subgroup
$W \subset G$
; -
(2) a k-rational faithful, irreducible representation
${\mathbf H} \to \mathbf {SL} (V)$
on a finite-dimensional k-vector space V; and -
(3) a W-invariant measure
$\nu $
on
$\Gamma \backslash (G \times \mathbb {P}(V))$
that projects to the Haar measure
$\mu _{\Gamma \backslash G}$
on
$\Gamma \backslash G$
.
We now explain how richness implies the existence of items (1)–(3), which follows the argument given in [Reference Bader, Fisher, Miller and Stover1, §3] in a similar context.
Recall that by hypothesis,
$S_{\Gamma }$
contains infinitely many distinct 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
$W\le L_i\le N_G(W)$
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 Definition 1.8 that
${\mathbf H}_i=\overline {\rho (\Gamma _i)}$
is a k-defined non-compact 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 dth exterior power of the adjoint representation
$$ \begin{align*} \wedge^d(\operatorname{Ad}):{\mathbf H}(k)\longrightarrow\mathbf{GL}\bigg(\bigwedge\nolimits^d\mathfrak{h}\bigg), \end{align*} $$
where
$\mathfrak {h}$
is the Lie algebra of
${\mathbf H}(k)$
. The Lie algebra
$\frak {h}_i=\operatorname {\mathrm {Lie}}({\mathbf H}_i(k))$
determines a line
${\ell _i \subset \bigwedge \nolimits ^d\frak {h}}$
. The representation
$\wedge ^d(\operatorname {Ad})$
need not be irreducible; however, there exists a non-trivial summand
$\theta :{\mathbf H} \to \mathbf {GL}(V)$
, where V is a k-defined vector space, for which infinitely many
$\ell _i$
project non-trivially 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 non-trivially 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
$$ \begin{align*} (g,[v])\sim (\gamma g, \overline{\theta}(\gamma)[v]), \end{align*} $$
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:
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 item (3) since projection and weak-
$\ast $
limit commute. This completes the proof.
3 Dynamical proof for a variant of Theorem 1.3
The purpose of this section is to give a dynamical proof of the following variant of Theorem 1.3. We continue with the notation established in §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
to another arithmetic ball quotient of simplest kind. Suppose that
$Y_\ell $
is a sequence of distinct 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
$G=\operatorname {\mathrm {\operatorname {SU}}}(1,n)$
. 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 = \unicode{x3bb} \alpha $
for some constant
$\unicode{x3bb} $
, and 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 [Reference Koziarz and Maubon30, Corollary 1.5] (also see the related paper of Möller and Toledo [Reference Möller and Toledo33] and the more general work of Tayou and Tholozan [Reference Tayou and Tholozan42]) to both
$\beta $
and
$\omega ^{n-1}$
, we deduce that
$$ \begin{align} \frac{\int_{f(S_n)\cap Y_\ell} \beta}{\int_{f(S_n)\cap Y_\ell}\omega^{n-1}} \longrightarrow \frac{\int_{f(S_n)}\beta \wedge \omega}{\int_{f(S_n)}\omega^n} \end{align} $$
as
$\ell $
goes to infinity. By definition of the pullback, the left-hand side of equation (3.1) is also equal to
$$ \begin{align*} \frac{\int_{f^{-1}(Y_\ell)} f^*\beta}{\int_{f^{-1}(Y_\ell)}f^*\omega^{n-1}}=\frac{\int_{f^{-1}(Y_\ell)} \alpha^{n-1}}{\int_{f^{-1}(Y_\ell)}f^*\omega^{n-1}} \frac{\int_{f^{-1}(Y_\ell)}f^*\beta}{\int_{f^{-1}(Y_\ell)}\alpha^{n-1}}. \end{align*} $$
We want to compute the limits of
$({\int _{f^{-1}(Y_\ell )}f^*\beta }/{\int _{f^{-1}(Y_\ell )}\alpha ^{n-1}})$
and
$({\int _{f^{-1}(Y_\ell )}f^*\omega ^{n-1}}/ {\int _{f^{-1}(Y_\ell )}\alpha ^{n-1}})$
.
Since there are possibly several irreducible components, we write
$$ \begin{align*} f^{-1}(Y_\ell)=Z_{\ell,1}\cup \cdots \cup Z_{\ell,n_\ell}. \end{align*} $$
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 sequence
of probability measures on
$\Gamma _n\backslash G$
. 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 [Reference Bader, Fisher, Miller and Stover2, Lemma 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
$K=\mathrm {S}({\operatorname {U}}(1)\times {\operatorname {U}}(n))$
is a maximal compact subgroup of G, and a fixed embedding 
. Let
$\pi : \Gamma _n\backslash G\longrightarrow S_n$
be the natural projection associated with the right quotient by K. Then, any totally geodesic divisor Z on
$S_n$
is of the form
$$ \begin{align*} Z= \pi( \Gamma_n\backslash \Gamma_n g W) \end{align*} $$
for some
$g\in \operatorname {\mathrm {\operatorname {SU}}}(1,n)$
such that
$gWg^{-1}\cap \Gamma _n$
is a lattice in
$gWg^{-1}$
(see §2 and Remark 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
$g_{\ell , i}\in \operatorname {\mathrm {\operatorname {SU}}}(1,n)$
, where
$1\le i\le 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
Note that
$\pi _*\mu _\ell =\mu _{f^{-1}(Y_\ell )}$
by [Reference Bader, Fisher, Miller and Stover1, Lemma 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 [Reference Bader, Fisher, Miller and Stover1, Lemma 8.3] using quantitative non-divergence shows that there is no escape of mass and thus
$\mu $
is also a probability measure.
Let
$\nu $
be an ergodic component of
$\mu $
. By Ratner’s theorem [Reference Ratner38],
$\nu $
is an homogeneous, W-invariant, W-ergodic measure and therefore is either a constant multiple of the Haar measure on
$\Gamma _n\backslash G$
or a constant multiple of the 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\le i\le 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 [Reference Koziarz and Maubon30, Corollary 1.5], but this time in
$S_n$
rather than in
$\mathbb {B}^m$
, we get
$$ \begin{align*} \frac{\int_{f^{-1}(Y_\ell)}f^*\beta}{\int_{f^{-1}(Y_\ell)}\alpha^{n-1}} \longrightarrow \frac{\int_{S_n}f^*\beta \wedge \alpha}{\int_{S_n} \alpha^n}. \end{align*} $$
When we apply this result to the case
$\beta =\omega ^{n-1}$
, we find that
$$ \begin{align*} \frac{\int_{f^{-1}(Y_\ell)}f^*\omega^{n-1}}{\int_{f^{-1}(Y_\ell)}\alpha^{n-1}} \longrightarrow \frac{\int_{S_n}f^*\omega^{n-1} \wedge \alpha}{\int_{S_n} \alpha^n}. \end{align*} $$
Putting everything together, we obtain
$$ \begin{align*} \frac{\int_{S_n} f^* \beta \wedge \alpha}{\int_{S_n} f^*\omega^{n-1} \wedge \alpha} = \frac{\int_{S_n} f^* \beta \wedge f^*\omega}{\int_{S_n} f^* \omega^n}. \end{align*} $$
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 = \unicode{x3bb} \alpha $
for
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 [Reference Baldi5, Reference Buium and Poonen12], which was again motivated by the Zilber–Pink philosophy.
Remark 3.3. Technically speaking, using [Reference Bader, Fisher, Miller and Stover2, Lemma 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
$g\in \operatorname {\mathrm {\operatorname {SU}}}(1,n)$
, 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 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 Lemma 3.2.
4 The proof of Theorem 1.3
We now prove Theorem 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
is an immersive holomorphic map to another arithmetic ball quotient. From now on, we always assume that
$f(S_n)$
is monodromy generic, that is, is not contained in any strict complex totally geodesic subvariety
$S^\prime $
of
$S_m$
. Let
$\Gamma _f$
be the graph of f in
$S_n \times S_m$
. We now prove the following equivalent formulation of Theorem 1.3.
Theorem 4.1. Let
$\{Z_\ell \}$
(respectively
$\{Y_\ell \}$
) be an infinite sequence of distinct complex totally geodesic subvarieties of
$S_n$
(respectively
$S_m$
). 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 [Reference Daw and Ren15] (see also the more general Theorem 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 [Reference Baldi, Klingler and Ullmo7, Reference Daw and Ren15])
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 Theorem 4.1
Define 
. 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,
$$ \begin{align} \operatorname{codim}_{S_n\times S_m}(U_\ell) < \operatorname{codim}_{S_n\times S_m}(\Gamma_f) + \operatorname{codim}_{S_n\times S_m}(Z_\ell \times Y_\ell). \end{align} $$
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
$$ \begin{align*} \operatorname{codim}_{S_n\times S_m}(U_\ell)&=n+m-d, \\ \operatorname{codim}_{S_n\times S_m}(\Gamma_f) + \operatorname{codim}_{S_n\times S_m}(Z_\ell \times Y_\ell) &= m+(n+m-d-m+e) \\ &= n+m-d+e. \end{align*} $$
Since
$e>0$
, equation (4.1) holds.
Since we assumed that
$f(S_n)$
is monodromy generic, we must prove that
$f(S_n)=S_m$
. Applying Theorem 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 monodromy generic, the second projection of
$S_0$
on
$S_m$
is surjective. Therefore,
$S_0$
is a strict totally geodesic subvariety of
$S_n\times S_m$
dominating both factors. This is possible only when
$S_n$
is commensurable with
$S_m$
and
$S_0$
is a modular correspondence. Since
$\Gamma _f \subset S_0$
have the same dimension and are irreducible, it follows that
$\Gamma _f=S_0$
, that is, f is a totally geodesic embedding.
Using a similar argument, we can also prove the following, where a sequence of geodesics is generic when they are not contained in a finite union of maximal totally geodesic subvarieties.
Corollary 4.4. Let 
be a finite-volume ball quotient with an immersive holomorphic map
to another finite-volume ball quotient. Let
$\{C_\ell \}$
be a generic sequence of distinct closed geodesics in
$S_n$
for which there exists 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 Corollary 4.4 uses the so-called Baby Ax–Lindemann theorem [Reference Ullmo and Yafaev44, Proposition 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
$\Gamma \subset \operatorname {\mathrm {\operatorname {PU}}}(1,n)$
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 Corollary 4.4
Note that we may reduce to the case that
$S_m$
is arithmetic. Indeed, if
$S_m$
were non-arithmetic, 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 (see [Reference Bader, Fisher, Miller and Stover2] or [Reference Baldi and Ullmo8]), 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 [Reference Borel11], the right-hand side is Zariski closed and
Thanks to Proposition 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 Theorem 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, our results on (closed) geodesics are inspired by the analogy between closed geodesics and CM points and the following. A point of a Shimura variety is called CM if its associated Mumford–Tate group is a torus.
Theorem 5.1. (Special case of André–Oort in
$S_1\times S_2$
, [Reference Pila, Shankar, Tsimerman, Esnault and Groechenig36])
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 [Reference Baldi, Klingler and Ullmo7, Reference Klingler, Ullmo and Yafaev28].
Remark 5.2. It may 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 [Reference Mumford35, Corollary 1, p. 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 Theorem 5.1 and shows that a version of Siu’s immersion problem (Question 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 (respectively analytic) variety, by a subvariety
$Y \subset S$
, we always mean a closed algebraic (respectively 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
$$ \begin{align*} V_{\mathbb{C}}= \bigoplus_{p\in \mathbb{Z}} F^{p} \oplus \overline{F^{n-p}}. \end{align*} $$
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. When V is polarized, this is a reductive algebraic group.
In the two sections below, we recall the setting and the main results from [Reference Baldi, Klingler and Ullmo7]. We refer also to the companion [Reference Baldi, Klingler and Ullmo6] (especially §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
$$ \begin{align} \Phi: S^{\mathrm{an}} \longrightarrow \Gamma \backslash D, \end{align} $$
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
${\mathbf G}^{\mathrm {ad}}$
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
$\Gamma \subset {\mathbf G}^{\mathrm {ad}}(\mathbb {R})^+$
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
$$ \begin{align} \Phi: S^{\mathrm{an}} \longrightarrow \Gamma \backslash D \cong \Gamma_1 \backslash D_1 \times \cdots \times \Gamma_r \backslash D_r \times D_{\mathrm{triv}}, \end{align} $$
where the projection of
$\Phi (S^{\mathrm {an}})$
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) A subvariety Z of S is said to be of positive period dimension for
${\mathbb V}$
if
$\Phi (Z^{\mathrm {an}})$
has positive dimension, that is, if
$\dim _{\mathbb {C}}\Phi (Z^{\mathrm {an}})>0$
. -
(2) The Hodge locus of positive period dimension,
$\mathrm {HL}(S, {\mathbb V}^\otimes )_{\mathrm {pos}}$
, 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
$Z^{\mathrm {an}}$
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
$\Phi (S^{\mathrm {an}})$
and
$\Gamma ^\prime \backslash D^\prime $
do not intersect generically along
$\Phi (Z^{\mathrm {an}})$
. That is, Z is atypical when
$$ \begin{align} \operatorname{codim}_{\Gamma\backslash D} \Phi(Z^{\mathrm{an}}) < \operatorname{codim}_{\Gamma\backslash D} \Phi(S^{\mathrm{an}}) + \operatorname{codim}_{\Gamma\backslash D} \Gamma^\prime\backslash D^\prime. \end{align} $$
Otherwise, Z is said to be typical. The atypical Hodge locus
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {atyp}} \subset \mathrm {HL}(S, {\mathbb V}^\otimes )$
(respectively the typical Hodge locus
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {typ}} \subset \mathrm {HL}(S, {\mathbb V}^\otimes )$
) is the union of the atypical (respectively strict typical) special subvarieties of S for
${\mathbb V}$
.
Remark 5.5. Since it will be important in the following, 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
$$ \begin{align*} Z=\Phi^{-1}(\Gamma^\prime \backslash D^\prime)^0=\Phi^{-1}(\Gamma" \backslash D")^0 \end{align*} $$
for some bigger
$\Gamma " \backslash D"$
, 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 [Reference Baldi, Klingler and Ullmo7], we expect the following conjecture.
Conjecture 5.6. (Zilber–Pink conjecture for the atypical Hodge locus, strong version)
The atypical Hodge locus
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {atyp}}$
is a finite union of atypical special subvarieties of S for
${\mathbb V}$
.
Conjecture 5.7. (Density of the typical Hodge locus)
If
$\mathrm {HL}(S, {\mathbb V}^\otimes )_{\mathrm {typ}}$
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 [Reference Baldi, Klingler and Ullmo7, Theorems 6.1, 10.1, and Remark 10.2]. 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
${\mathbf G}_Z^{\mathrm {ad}}$
decomposes as a non-trivial product
${\mathbf H}^{\mathrm {ad}}_Z \times {\mathbf L}_Z$
, 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) and Z is Hodge generic in a special subvariety
$$ \begin{align*} \Phi_{|Z^{\mathrm{an}}}= (\Phi_{{\mathbf H}_{Z}}, \Phi_{{\mathbf L}_{Z}}): Z^{\mathrm{an}} \longrightarrow \Gamma_{{\mathbf G}_{Z}}\backslash D_{{\mathbf G}_{Z}}= \Gamma_{{\mathbf H}_{Z}}\backslash D_{{\mathbf H}_{Z}} \times \Gamma_{{\mathbf L}_{Z}}\backslash D_{{\mathbf L}_{Z}} \subset \Gamma \backslash D, \end{align*} $$
$\Phi ^{-1}(\Gamma _{{\mathbf G}_{Z}}\backslash D_{{\mathbf G}_{Z}})^0$
of S for
$\Phi $
which is monodromically typical and therefore typical.
We refer also to [Reference Baldi and Urbanik9, Theorem 7.1] for an effective proof of a more general statement (proved using differential geometry rather than o-minimality).
Theorem 5.9. If the typical Hodge locus
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {pos},\mathrm {typ}} $
is non-empty, then
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {pos},\mathrm {typ}}$
is analytically dense in S.
The above statement already hides an application of Theorem 5.8, as explained in [Reference Baldi, Klingler and Ullmo7, Remark 10.2]. Indeed, it is proved by first showing that
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {pos}}$
is dense, then invoking Theorem 5.8 to say that
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {pos},\mathrm {atyp}}$
is algebraic, and therefore is exactly
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {pos},\mathrm {typ}}$
, which is dense. In future applications, we will actually use this finer version. See also [Reference Khelifa and Urbanik24, Theorem 1.6 and Remark 1.7] for a related discussion.
Remark 5.10. Hodge theory actually gives a simple combinatorial criterion to decide whether
$\mathrm {HL}(S,{\mathbb V}^\otimes )_{\mathrm {typ}}$
is empty or not. Indeed, see the recent work [Reference Eterović and Scanlon19, Reference Khelifa and Urbanik24].
6 General Hodge theoretic statement: from typical to atypical intersections
6.1 Isogenies between VHSs
For an introduction to Tannakian categories, we refer for example to the article of Deligne and Milne [Reference Deligne, Milne, Ogus and Shih17]. The category of
$\mathbb {Q}$
VHS on a smooth quasi-projective base S, which 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 equivalence of tensor categories
$\langle {\mathbb V}_{1,\mathbb {Q}} \rangle ^\otimes \cong \langle {\mathbb V}_{2,\mathbb {Q}} \rangle ^\otimes $
, where
$ \langle {\mathbb V}_{i,\mathbb {Q}} \rangle ^\otimes $
denotes the smallest Tannakian subcategory of
$\mathbb {Q}$
VHSs containing
${\mathbb V}_{i,\mathbb {Q}}$
.
Of course, the Tannakian categories
$\langle {\mathbb V}_{i,\mathbb {Q}} \rangle ^\otimes $
appearing above are equivalent (as tensor categories) to the category of finite-dimensional representations of their generic Mumford–Tate group. (The equivalence is realized by the functor of
$\otimes $
-automorphisms of the fiber functor). It can happen that two VHSs have isomorphic Mumford–Tate groups, but the isomorphism does not induce an equivalence of tenor categories, cf. the article of Deligne and Milne [Reference Deligne, Milne, Ogus and Shih17, Definition 1.10 and Theorem 2.11].
Remark 6.2. If two complex principally polarized abelian varieties
$A,B$
are isogenous in the usual sense (either via a polarized or an unpolarized isogeny), then the Hodge structures
$H^1(A,\mathbb {Z}), H^1(B,\mathbb {Z})$
are also isogenous in the sense of Definition 6.1 (here, we take as base S the spectrum of
$\mathbb {C}$
). 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
$$ \begin{align*} V\subset \Gamma_1\backslash D_1\times \Gamma_2\backslash D_2 \end{align*} $$
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 [Reference Baldi, Klingler and Ullmo7, Definition 3.23] for more details). In this situation, V is a Mumford–Tate domain, that is,
$V=\Gamma \backslash D$
, and we have that
$D\simeq D_1\simeq D_2$
and that
$\Gamma _1$
is commensurable with
$\Gamma _2$
. 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) there is an isogeny between
${\mathbb V}_1$
and
${\mathbb V}_2$
; -
(2) the Hodge data
$({\mathbf G}_1,D_1)$
,
$({\mathbf G}_2,D_2)$
are isomorphic. 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 and a period map
$$ \begin{align*} \Gamma\backslash D\subset \Gamma_1\backslash D \times \Gamma_2\backslash D \end{align*} $$
$\psi _0: S\to \Gamma \backslash D$
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 fiber
$W_{\mathbb {Q}}$
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,
$$ \begin{align*} \langle {\mathbb V}_{1,\mathbb{Q}} \rangle^\otimes\cong \langle {\mathbb V}_{2,\mathbb{Q}} \rangle^\otimes \cong \langle {\mathbb V}_{1,\mathbb{Q}} \oplus {\mathbb V}_{2,\mathbb{Q}} \rangle^\otimes \end{align*} $$
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
$$ \begin{align*} \psi_0: S\longrightarrow \Gamma\backslash D\subset \Gamma_1\backslash D_1 \times \Gamma_2\backslash D_2, \end{align*} $$
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
$$ \begin{align*} \mathrm{HL}(S,\mathbb{V}_1^\otimes)_{\mathrm{pos}, \mathrm{typ}}=\mathrm{HL}(S,\mathbb{V}_2^\otimes)_{\mathrm{pos}, \mathrm{typ}}\neq \emptyset, \end{align*} $$
then
$\mathbb {V}_1$
is isogenous to
$\mathbb {V}_2$
. As a consequence,
$\mathrm {HL}(S,\mathbb {V}_1^\otimes )=\mathrm {HL}(S,\mathbb {V}_2^\otimes )$
.
Some stronger variants with the weakly special Hodge locus replacing the Hodge locus can be given. Also, the same proof works if there exists a Zariski dense subset of components of
$\mathrm {HL}(S,\mathbb {V}_1^\otimes )_{\mathrm {pos}, \mathrm {typ}}$
that is contained in
$\mathrm {HL}(S,\mathbb {V}_2^\otimes )_{\mathrm {pos}, \mathrm {typ}}$
.
Remark 6.5. By Theorem 5.9, the condition
$ \mathrm {HL}(S,\mathbb {V}_i^\otimes )_{\mathrm {pos}, \mathrm {typ}}\neq \emptyset $
implies that
$\mathrm {HL}(S,\mathbb {V}_i^\otimes )_{\mathrm {pos}, \mathrm {typ}}$
is analytically dense in S. Conjecture 5.7 predicts that
$ \mathrm {HL}(S,\mathbb {V}_i^\otimes )_{\mathrm {typ}}$
should be analytically dense whenever it is non-empty. The condition
$$ \begin{align*} {\mathrm{HL}(S,\mathbb{V}_1^\otimes)_{\mathrm{pos}, \mathrm{typ}}=\mathrm{HL}(S,\mathbb{V}_2^\otimes)_{\mathrm{pos}, \mathrm{typ}}\neq \emptyset} \end{align*} $$
is therefore quite restrictive. Moreover, by [Reference Baldi, Klingler and Ullmo7, Theorem 1.5], the levels of the
$\mathbb {Z}$
VHS
${\mathbb V}_1$
and
${\mathbb V}_2$
must be
$1$
(the Shimura case) or
$2$
.
To better explain Theorem 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 
for the associated VHS. If
$$ \begin{align} \mathrm{HL}(S,{\mathbb V}_1^\otimes)_{\mathrm{pos}, \mathrm{typ}}=\mathrm{HL}(S,{\mathbb V}_2^\otimes)_{\mathrm{pos}, \mathrm{typ}}\neq \emptyset, \end{align} $$
then
$\mathcal {A}_1$
is isogenous to
$\mathcal {A}_2$
as principally polarized S-abelian schemes.
Proof. We will use Remark 6.2 and the following fact: given two principally polarized families of abelian varieties
$h_i:\mathcal {A}_i \to S$
, one has
$$ \begin{align*} \operatorname{\mathrm{Hom}}_S (\mathcal{A}_1, \mathcal{A}_2)= \operatorname{\mathrm{Hom}} (R_1{h_1,_*}\mathbb{Z},R_1{h_2,_*}\mathbb{Z}), \end{align*} $$
where the second
$\operatorname {\mathrm {Hom}}$
is in the category of
$\mathbb {Z}$
VHS; see [Reference Deligne16, Corollary 4.4.15]. Indeed, Theorem 6.4 implies that the two
$\mathbb {Z}$
VHSs
${\mathbb V}_i$
for
$i\in \{1,2\}$
are isogenous in the sense of Definition 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 [Reference Baker and DeMarco4] in complex dynamics, which shows 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 [Reference Yuan and Zhang45]. Both proofs mentioned above use equidistribution, but here we work with positive dimensional subvarieties. The statement without pos is deeper and would follow from Conjecture 5.6.
Proof of Theorem 6.4
The main step in the proof is the following lemma, where we do not need to assume that the generic Mumford–Tate group of
${\mathbb V}_i$
is
$\mathbb {Q}$
-simple.
Lemma 6.8. Assuming the hypothesis of Theorem 6.4, let
$f_i: S\to \Gamma _i \backslash D_i$
be the period map associated with
$\mathbb {V}_i$
,
$i=1,2$
, and let
$$ \begin{align*} f_1\times f_2: S \longrightarrow \Gamma_1 \backslash D_1 \times \Gamma_2 \backslash D_2 \end{align*} $$
be the period map associated with
${\mathbb V}_1\oplus {\mathbb V}_2$
. Then,
$\mathrm {HL}(S,({\mathbb V}_1\oplus {\mathbb V}_2)^{\otimes })_{\mathrm {pos}, \mathrm {atyp}}$
is analytically dense in S.
Proof. As
$\mathrm {HL}(S,{\mathbb V}_i^{\otimes })_{\mathrm {pos},\mathrm {typ}}\neq \emptyset $
, by Theorem 5.9, there exists sequences of special subvarieties
$\{Z_{i,\ell }\}_{\ell \in \mathbb {N}}$
in
$\Gamma _i\backslash D_i$
and a sequence
$\{W_{\ell }\}_{\ell \in \mathbb {N}}$
of components of both
$\mathrm {HL}(S,{\mathbb V}_1^{\otimes })_{\mathrm {pos},\mathrm {typ}}$
and
$\mathrm {HL}(S,{\mathbb V}_2^{\otimes })_{\mathrm {pos},\mathrm {typ}}$
, such that
$$ \begin{align*} W_\ell= f_1^{-1}(Z_{1,\ell})^0=f_2^{-1}(Z_{2,\ell})^0 \end{align*} $$
for some components
$f_i^{-1}(Z_{i,\ell })^0$
of
$f_i^{-1}(Z_{i,\ell })$
and for which
$\bigcup _\ell W_\ell $
is Zariski dense in S. The fact that the
$W_\ell $
are again typical intersections, guaranteed by Theorem 5.9, implies that
$W_\ell $
is Hodge generic for
${\mathbb V}_i$
in
$Z_{i,\ell }$
(see also the discussion in Remark 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
$$ \begin{align*} \operatorname{codim}_{\Gamma_i \backslash D_i} f_i(W_\ell) = \operatorname{codim}_{\Gamma_i \backslash D_i} Z_{i,\ell} + \operatorname{codim}_{\Gamma_i \backslash D_i}(f_i(S) ) \end{align*} $$
for each
$i\in \{1,2\}$
. That is, if 
, 
, and 
, then
$$ \begin{align} d_i - w_i=d_i -z_i+ d_i -s_i. \end{align} $$
Note that we do not assume that our period maps are immersive. Notice that
$$ \begin{align*} W_\ell \subset (f_1\times f_2) ^{-1} (Z_{1,\ell} \times Z_{2,\ell}) \varsubsetneq S. \end{align*} $$
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
$\mathrm {HL}(S,({\mathbb V}_1\oplus {\mathbb V}_2)^{\otimes })$
for all
$\ell $
. For this, we need to show that
$$ \begin{align*} \operatorname{codim}_{ \Gamma_1 \backslash D_1 \times \Gamma_2 \backslash D_2}((f_1 \times f_2)(W_\ell^\prime)) &< \operatorname{codim}_{ \Gamma_1 \backslash D_1 \times \Gamma_2 \backslash D_2}(Z_{1,\ell}\times Z_{2,\ell}) \\ &\quad + \operatorname{codim}_{ \Gamma_1 \backslash D_1 \times \Gamma_2 \backslash D_2}((f_1\times f_2)(S)). \end{align*} $$
Summing equation (6.2) for
$i=1,2$
, we have
$$ \begin{align} d_1+d_2 - w_1-w_2=d_1+d_2 -z_1-z_2+ d_1+d_2 -s_1 -s_2. \end{align} $$
This implies the requisite equation by noticing that
$$ \begin{align*} \max \{\dim (f_1 (Y)),\dim (f_2 (Y)) \} \leq \dim ((f_1 \times f_2) (Y)) \leq \dim (f_1 (Y))+\dim (f_2 (Y)) \end{align*} $$
for any Y in S.
Lemma 6.9. Each of the following hold.
-
(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) If the Mumford–Tate group
${\mathbf G}_i$
of
${\mathbb V}_i$
is
$\mathbb {Q}$
-simple, then
${\mathbf G}_1^{\mathrm {ad}}\cong {\mathbf G}_2^{\mathrm {ad}}$
,
$D_1\simeq D_2$
. Denoting the corresponding Hodge datum by
$({\mathbf G},D)$
, we have that
$(f_1\times f_2)(S) $
is Hodge generic in a modular correspondence
$$ \begin{align*} \Gamma\backslash D\subset \Gamma_1 \backslash D \times \Gamma_2 \backslash D. \end{align*} $$
Proof. By Lemma 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 Theorem 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
${\mathbf G}_1^{\mathrm {ad}}\cong {\mathbf G}_2^{\mathrm {ad}}$
. This finishes the proof of the second part of the lemma.
Theorem 6.4 is now a consequence of the characterization of isogenous
$\mathbb {Z}$
VHSs given in Proposition 6.3.
Remark 6.10. Theorem 6.4 (or, more precisely, a slight generalization of it) implies Theorem 4.1 for 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 Definition 5.4, a sub-Shimura variety is always in the typical Hodge locus
$$ \begin{align*} \mathrm{HL}(S_n,\mathbb{V}_1^\otimes)_{\mathrm{pos}}=\mathrm{HL}(S_n,\mathbb{V}_1^\otimes)_{\mathrm{pos}, \mathrm{typ}} \end{align*} $$
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 on
$S_m$
. Once this translation from immersions to Hodge theory is made, the proof of Theorem 4.1 is just a special case of the one given in Theorem 6.4.
7 Proofs of Theorems 1.13 and 1.17
In this section, we prove the second part of Theorem 1.13 and then Theorem 1.17 following a similar argument. Let
$\Gamma \subset \operatorname {PU}(1,n)$
for
$n>1$
be a lattice of the simplest kind, which we can assume to be neat (that is, no element of
$\Gamma $
has a non-trivial root of unity as an eigenvalue under the adjoint representation). Consider the smooth quasi-projective variety
$S_\Gamma $
and let
$$ \begin{align*} \rho: \Gamma \longrightarrow {\mathbf H}(k) \end{align*} $$
be a cohomologically rigid representation, that is,
$H^1(\Gamma , \operatorname {Ad}_{\mathbf H} \rho )=0$
, where
$$ \begin{align*} \operatorname{Ad}_{\mathbf H}: {\mathbf H}(k) \longrightarrow \operatorname{Aut}(\mathfrak{h}) \end{align*} $$
is the adjoint representation over some local field k. Assuming that
$\rho $
is geodesically rich in the sense of Definition 1.8, we would like to show that
$\rho $
extends. The proof is a generalization of the arguments from [Reference Baldi and Ullmo8, §§4 and 5], improved by combining them with §6.
Proof of Theorem 1.13(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 [Reference Baldi and Ullmo8, §4]. Moreover,
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 [Reference Baldi and Ullmo8, §4], it is done by invoking the recent work of Esnault and Groechenig [Reference Esnault and Groechenig18]. However, here we can give a self-contained proof by applying Corollary 1.14.
From here forward, we assume that
$\rho (\Gamma )$
is Zariski dense in the k-group
${\mathbf H}$
and will then prove that
${\mathbf H}(K)\cong G$
. In particular, from the above, we obtain a natural K-form of
${\mathbf H}$
(that we denote by the same symbol
${\mathbf H}$
). 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
$\widehat {{\mathbf H}}\subseteq \operatorname {Res}_{K/\mathbb {Q}} \mathbf {H}$
and consider the
$\mathbb {Z}$
-local system
$$ \begin{align*} \hat{\rho}: \Gamma &\longrightarrow \widehat{\mathbf H} (\mathbb{R}) \\ \gamma &\longmapsto \hat{\rho} (\gamma)=(\sigma_i(\rho(\gamma))). \end{align*} $$
Again, by cohomological rigidity of
$\rho $
, we have that
$\hat {\rho }$
naturally underlies a
$\mathbb {Z}$
VHS
$\widehat {\mathbb {V}}$
(compare with [Reference Baldi and Ullmo8, Theorem 4.2.4]). As recalled in §5,
$\widehat {\mathbb {V}}$
corresponds to a period map
$$ \begin{align*} \psi : S_{\Gamma}^{\mathrm{an}}\longrightarrow \widehat{\mathbf H} (\mathbb{Z}) \backslash D. \end{align*} $$
Now that we have a
$\mathbb {Z}$
VHS, as in Remark 6.10, we use the results of §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 with some faithful linear rational representation of
$\mathbf {G}$
. By construction,
$\mathrm {HL}(S_\Gamma , \mathbb {V}_1^{\otimes })_{\mathrm {pos}}$
is the union of the complex totally geodesic subvarieties of
$S_{\Gamma }$
, and it is equal to
$\mathrm {HL}(S_\Gamma , \mathbb {V}_1^{\otimes })_{\mathrm {pos},\mathrm {typ}}$
; 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. Indeed, let
$S_{\Gamma _i}\subset S_\Gamma $
be a totally geodesic subvariety such that
$\mathbf {H}_i=\overline {\rho (\Gamma _i)}$
has dimension smaller than
$\dim \mathbf {H}$
. Then,
$\dim \widehat {\mathbf {H}}_i:=\overline {\hat {\rho }(\Gamma _i)} < \dim \widehat {\mathbf {H}}$
. Moreover,
$S_{\Gamma _i}$
lies in the
$ \mathrm {HL}(S_\Gamma , \mathbb {V}_1^{\otimes })_{\mathrm {pos},\mathrm {typ}}$
, as well as
$\mathrm {HL}(S_\Gamma , \widehat {\mathbb {V}}^{\otimes })_{\mathrm {pos},\mathrm {typ}}$
. Since this is true for a sequence of i, (the proof of) Theorem 6.4 implies that
$\mathbb {V}_1,\widehat {\mathbb {V}} $
are isogenous, which means that the period domain
$\widehat {\mathbf H} (\mathbb {Z}) \backslash D$
is isomorphic to
$S_\Gamma $
and that
$\psi $
is isogenous to the identity (as in Proposition 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 Theorem 1.17
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 Corollary 1.14,
${\mathbb V}$
must be a
$\mathbb {Z}$
VHS, and the richness assumption implies that there is a generic sequence of subvarieties
$S_{\Gamma _i}\in \mathrm {HL}(S_\Gamma , {\mathbb V}^{\otimes })_{\mathrm {pos},\mathrm {typ}}\cap \mathrm {HL}(S_\Gamma , {\mathbb V}_1^\otimes )_{\mathrm {pos},\mathrm {typ}}$
. Therefore, the results follow once again from Theorem 6.4.
Acknowledgements
G.B. and E.U. were partially supported by the NSF grant DMS-1928930, while they were in residence at the MSRI in Berkeley, during the Spring 2023 semester. N.M. was partially supported by Grant Number DMS-2005438/2300370 and M.S. was partially supported by Grant Number DMS-2203555 from the National Science Foundation. G.B. also thanks the IHES for the excellent working conditions.
