Special subvarieties of non-arithmetic ball quotients and Hodge theory 9781400882595, 9783642514456, 9781400881918

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Annals of Mathematics 197 (2023), 159–220 https://doi.org/10.4007/annals.2023.197.1.3

Special subvarieties of non-arithmetic ball quotients and Hodge theory By Gregorio Baldi and Emmanuel Ullmo

Abstract Let Γ ⊂ PU(1, n) be a lattice and SΓ be the associated ball quotient. We prove that, if SΓ contains infinitely many maximal complex totally geodesic subvarieties, then Γ is arithmetic. We also prove an Ax–Schanuel Conjecture for SΓ , similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise SΓ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.

Contents 1. Introduction 2. Notation, terminology and recollections 3. Real Shimura data and general properties 4. Z-Hodge theory on non-arithmetic quotients and rigidity 5. Comparison between the Γ-special and Z-special structures 6. Finiteness of special subvarieties 7. Special points and their Zariski closure References

159 166 170 179 187 199 208 214

1. Introduction 1.1. Motivation. The study of lattices of semisimple Lie groups G is a field rich in open questions and conjectures. Complex hyperbolic lattices and their finite dimensional representations are certainly far from being understood. A lattice Γ ⊂ G is archimedean superrigid if for any simple noncompact Lie group G0 with trivial centre, every homomorphism Γ → G0 with Zariski dense image extends to a homomorphism G → G0 . Thanks to the work of Margulis [51], Corlette [16] and Gromov–Schoen [40], all lattices in simple Lie groups are Keywords: non-arithmetic lattices, rigidity, Hodge theory and Mumford–Tate domains, functional transcendence, unlikely intersections AMS Classification: Primary: 14G35, 22E40, 03C64, 14P10, 32H02. © 2023 Department of Mathematics, Princeton University. 159

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archimedean superrigid unless G is isogenous to either SO(1, n) or PU(1, n) for some n ≥ 1. Real hyperbolic lattices are known to be softer and more flexible than their complex counterpart, and non-arithmetic lattices in SO(1, n) can be constructed for every n [39]. Non-arithmetic complex hyperbolic lattices have been found only in PU(1, n), for n = 1, 2, 3 [60], [22]. In both cases one can consider the quotient by Γ of the symmetric space X associated to G. In the complex hyperbolic case we obtain a ball quotient SΓ = Γ\X that has a natural structure of a quasi-projective variety, as proven by Baily–Borel [5] in the arithmetic case, and by Mok [22] in general. By the commensurability criterion for arithmeticity of Margulis [51] we can decide whether Γ is arithmetic or not by looking at modular correspondences in the product SΓ ×SΓ . That is, Γ is arithmetic if and only if SΓ admits infinitely many totally geodesic correspondences. In this paper, by totally geodesic subvarieties we always mean complex subvarieties of SΓ of dimension > 0, whose smooth locus is totally geodesic with respect to the canonical Kähler metric. Even if Γ is arithmetic, SΓ may have no strict totally geodesic subvarieties. However, the existence of countably many Hecke correspondences implies that, if SΓ contains one totally geodesic subvariety, then it contains countably many of such. The aim of this paper is to study totally geodesic subvarieties of SΓ from multiple point of views, ultimately explaining how often and why they appear. 1.2. Main results. Let G be PU(1, n) for some n > 1, and let Γ ⊂ G be a lattice. Let X be the Hermitian symmetric space associated to G and SΓ be the quotient of X by Γ. We first look at maximal totally geodesic subvarieties of SΓ , that is, totally geodesic subvarieties that are not strictly contained in any totally geodesic subvariety different from SΓ . Our first main result is the following. Theorem 1.2.1. If Γ ⊂ G is non-arithmetic, then SΓ contains only finitely many maximal totally geodesic subvarieties. The problem was originally proposed informally by Reid and, independently, by McMullen for real hyperbolic lattices [23, Question 7.6], [52, Question 8.2]. For SO(1, n), this was recently proven by Bader, Fisher, Miller and Stover [4, Th. 1.1] and, for closed hyperbolic 3-manifolds, by Margulis and Mohammadi [54, Th. 1.1]. Around the same time Bader, Fisher, Miller and Stover announced also a proof of the finiteness of the maximal totally geodesic (both real and complex) subvarieties of a ball quotient. The proof then appeared in [3]. Such approaches build on superrigidity theorems and use results on equidistribution from homogeneous dynamics. As we explain in Section 1.3, we are able to interpret the problem as a phenomenon of unlikely intersections inside a period domain for polarised Z-variations of Hodge structure (VHS from now on) and deduce the finiteness from a strong Ax–Schanuel theorem established by Bakker

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and Tsimerman [7]. Indeed we will see that Theorem 1.2.1 is predicted by a conjecture of Klingler [45, Conj. 1.9], which was our main motivation for studying such finiteness statements. The main novelty is the use of Z-Hodge theory, rather than R-Hodge theory. As a by-product of our strategy, in Section 6.2, we give a new proof of the Margulis commensurability criterion for arithmeticity for complex hyperbolic lattices in PU(1, n), at least for n > 1 (without using any superrigidity techniques). Moreover our approach gives an explicit description of the totally geodesic subvarieties in terms of the Hodge locus of a Z-VHS giving, for example, Corollary 1.3.3 as a simple application. Our second main result looks at totally geodesic subvarieties from the functional transcendence point of view, in the sense of [48]. Whenever a complex algebraic variety S has a semi-algebraic universal cover π : X → S, one can formulate Ax–Schanuel and Zilber–Pink type conjectures. For more details, see [74, §2.2] and [48]. For example an abstract Ax–Lindemann–Weierstrass would assert the following. Let V be a semi-algebraic subvariety of X, and let S 0 be the Zariski closure of π(V ). Then S 0 is bi-algebraic; that is, the irreducible components of π −1 (S 0 ) are semi-algebraic in X. As in the case of Shimura varieties, we prove that totally geodesic subvarieties are the same as bi-algebraic subvarieties and have a natural group theoretical description in terms of real sub-Shimura datum; see Definition 3.1.6. We prove the non-arithmetic Ax–Schanuel conjecture, generalising the Ax–Schanuel conjecture for Shimura varieties [47], [57] to non-arithmetic ball quotients. Theorem 1.2.2. Let W ⊂ X × SΓ be an algebraic subvariety and Π ⊂ X × SΓ be the graph of π : X → SΓ . Let U be an irreducible component of W ∩ Π such that codim U < codim W + codim Π, the codimension being in X × SΓ or, equivalently, dim W < dim U + dim SΓ . If the projection of U to SΓ is not zero dimensional, then it is contained in a strict totally geodesic subvariety of SΓ . Mok [56] has developed a different perspective on functional transcendence problems for ball quotients, and using methods of several complex variables, algebraic geometry and Kähler geometry recently managed to prove a form of Ax–Lindemann–Weierstrass, which follows from the above theorem. See also the earlier work of Chan and Mok [12] regarding compact ball quotients. Finally notice that both statements depend only on the commensurability class of Γ. Therefore, we may and do replace Γ by a finite index subgroup to assume that all lattices we consider are torsion free.

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1.3. Strategy. For both Theorems 1.2.1 and 1.2.2, the starting point is to embed SΓ in a period domain for polarised integral VHS. Let n be an integer > 1, and let G = PU(1, n). We start with the key observation that lattices in G satisfy a form of infinitesimal rigidity. (This is known to fail when n = 1.) Namely, Garland and Raghunathan [34], building on the work initiated by Calabi–Vesentini and Weil [10], [78], [79], proved that the first Eilenberg–MacLane cohomology group of a lattice Γ in G with respect to the adjoint representation is trivial. Therefore, thanks to the work of Simpson [68], Γ determines a totally real number field K and a K-form of G, which we denote by G. Since X is a Hermitian symmetric space, SΓ supports a natural polarised variation of C-Hodge structure, which we denote by V. Using a recent work of Esnault and Groechenig [32], and Simpson’s theory [68], we prove. Theorem 1.3.1. For every γ ∈ Γ, the trace of Ad(γ) lies in the ring of integers of a totally real number field K . As a consequence, up to conjugation by G, b on SΓ . Γ lies in G(OK ). Moreover V induces a Z-variation of Hodge structure V b By the theory of period “ Let G/Q be the generic Mumford–Tate group of V. b induces a commutative diagram domains and period maps of Griffiths [37], V in the complex analytic category X

ψ˜

πZ

π

SΓ an

D = DG “

ψ

“ G(Z)\D,

b Corollary 5.2.2 “ where ψ : SΓ an → G(Z)\D is the period map associated to V. “ is the Weil restriction from K to Q of G. Such a map ψ shows that G generalises the theory of modular embeddings of triangle groups [82], [14] and Deligne–Mostow lattices [15]. From a functional transcendence point of view, there are now two bialgebraic structures on SΓ (in the sense of [45, Def. 7.2]): on the one hand, the one coming from the fact that its universal covering X is semi-algebraic and on the other, the one coming from the fact that SΓ supports a polarised integral VHS. Depending on the special structure we choose, we have two, a priori different, definitions of bi-algebraic subvarieties. For example, a totally geodesic subvariety is bi-algebraic with respect to π, that is, it is the image of algebraic subvariety of X along π. More precisely, we will see that totally geodesic subvarieties are Γ-special, that is, sub-locally hermitian spaces of SΓ . From a Hodge theoretical point of view it is natural to consider the image in SΓ of some algebraic subvarieties of D. Namely, let D0 be a Mumford–Tate sub-domain of D. Then any analytic irreducible component of ψ −1 (πZ (D0 ))

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is an algebraic subvariety of SΓ by a famous theorem of Cattani, Deligne and Kaplan [11]. (For the fact that ψ −1 (πZ (D0 )) contains finitely many connected components, see also the work of Bakker, Klingler and Tsimerman [6].) We refer to such irreducible algebraic subvarieties ψ −1 (πZ (D0 ))0 as Z-special subvarieties. We remark here that the definition of Z-special subvarieties is non-trivial even for zero dimensional subvarieties, whereas every point of SΓ could be regarded as both a totally geodesic and a Γ-special subvariety. We will come back to this in Section 7. Until then it is understood that totally geodesic, Z- and Γ-special subvarieties are of dimension > 0. What is the relation between the two special/bi-algebraic structures? Since ψ˜ is just a holomorphic map (rather than, a priori, a totally geodesic immersion), such a question is far from being trivial. To prove Theorems 1.2.1 and 1.2.2 we need the following. Theorem 1.3.2. The totally geodesic subvarieties of SΓ are precisely the Z-special ones, equivalently the Γ-special ones. Theorem 1.3.2 reduces Theorem 1.2.1 to counting the Z-special subvarieties in SΓ , and we will see that this is actually an unlikely intersection problem. (Ssee Section 1.4 for a sketch of the proof of this fact.) A similar strategy appeared also in the work of Wolfart [82], where certain Riemann surfaces C, associated to non-arithmetic lattices, are embedded in Shimura varieties and the André–Oort conjecture is used to deduce that C contains only finitely many CM-points; see also [30], where Wolfart’s programme is completed. Theorem 1.3.2 is crucial to relate our Ax–Schanuel conjecture (Theorem 1.2.2) to the Ax–Schanuel known in Hodge theory by the work of Bakker b Finally, in Section 5.5, we prove and Tsimerman [7], applied to the pair (SΓ , V). some consequences of Theorem 1.3.2 that may be of independent interest. Corollary 1.3.3. Let H be a subgroup of G = PU(1, n) of hermitian type. If Γ ∩ H is Zariski dense in H , then Γ ∩ H is a lattice in H . The second application is about non-arithmetic monodromy groups, as first studied by Nori [62]. After Sarnak [66], a subgroup of a lattice ∆ of a real algebraic group R is called thin subgroup if it has infinite-index in ∆ and it is Zariski dense in R. A link between non-arithmetic lattices and thin subgroups was also noticed by Sarnak; see, e.g., [66, §3.2]. We present a geometric way of constructing many thin subgroups, as long as a non-arithmetic lattice Γ is given. Corollary 1.3.4. Let Γ be a non-arithmetic lattice in G = PU(1, n), n > 1. Let W be an irreducible subvariety of SΓ that is not contained in any of the finitely many maximal totally geodesic subvarieties of SΓ . The image of π1 (W ) → π1 (SΓ ) ∼ =Γ “ “ denotes the Weil restriction from gives rise to a thin subgroup of G(Z), where G K, the (adjoint) trace field of Γ, to Q of G (the K-form of G determined by Γ).

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1.4. Sketch of the proof of Theorem 1.2.1 (in the simplest case). We conclude the introduction presenting a sketch of the proof of Theorem 1.2.1 in the case where Γ is a lattice in PU(1, 2) and the (adjoint) trace field K of Γ has degree two over Q. We hope this section can clarify the relationships between Hodge theory, functional transcendence and o-minimality, which we employ in the sequel. The case of dimension two is particularly interesting since the maximality condition of Theorem 1.2.1 is automatically satisfied, and most of known non-arithmetic lattices were found in PU(1, 2); see, e.g., Section 2.5. Our main theorem reads as follows. Theorem 1.4.1. If Γ ⊂ G = PU(1, 2) is non-arithmetic, then SΓ = Γ\B2 contains only finitely many one dimensional (complex) totally geodesic subvarieties. Sketch of the proof. If SΓ contains no totally geodesic subvarieties, there is nothing to prove. Assume there is a totally geodesic subvariety W of SΓ , say associated to a triplet (H = PU(1, 1), XH = B1 , ΓH = Γ ∩ H), such that ΓH is a lattice in H. Let K be the field generated by the traces of Γ, under the adjoint representation. Thanks to Theorem 1.3.1, it is a totally real number field and we assume, for simplicity, that [K : Q] = 2. Let G be the K-form of “ for the Weil G determined by Γ (resp. H for the K-form of H), and write G “ Theorem 1.3.2 shows that the totally restriction from K to Q of G (resp. H). geodesic subvariety W is an irreducible component of an intersection 0

“ ψ(W ) = ψ(SΓ ) ∩ H(Z)\D “ , H “ “ where H(Z)\D “ is a subperiod domain of G(Z)\D. The only possibilities H for the dimension of the latter space are 4 or 5, depending on whether it is a Shimura variety or not. Observe that, if Γ is non-arithmetic, by the Mostow–Vinberg arithmeticity criterion (Theorem 2.3.1) D is a homogeneous “ space under G(R) = G × G and “ codimG(Z)\D ψ(SΓ ) = dim G(Z)\D − dim SΓ ≥ 2. “ Moreover we observe that “ codimG(Z)\D H(Z)\D “ “ ≥ 2, H “ G

codimG(Z)\D ψ(W ) ≥ 3. “ “ G

The above computations show that, if Γ is non-arithmetic, then W is an unlikely intersection: “ codimG(Z)\D ψ(W ) < codimG(Z)\D ψ(SΓ ) + codimG(Z)\D H(Z)\D “ “. “ “ H “ G

“ G

“ G

To show that unlikely intersections arise in a finite number, we argue using o-minimality and functional transcendence (in the o-minimal structure Ran,exp ,

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given by expanding the semi-algebraic sets with restricted analytic functions and the real exponential). Indeed here we sketch how to show that unlikely intersections can be parametrised by a countable and definable set; see Proposition 6.1.2 for all details. Let F be a semi-algebraic fundamental set for the action of Γ on X, and consider ˜ “ : Im(ψ(x) “ “gˆ−1 }, Π0 (H) := {(x, gˆ) ∈ F × G : S = ResC ˆH R Gm → G) ⊂ g where ψ˜ : X → DG “ is the lift of the period map constructed in Theorem 1.3.1. “ and we then study We will show that Π0 (H) is a definable subset of F × G, “gˆ−1 : (x, gb) ∈ Π0 (H) for some x ∈ F such that dX (x, gˆ) ≥ 1}, Σ1 = Σ(H)1 := {ˆ gH where Ä ä ˜ ˜ “gˆ−1 .ψ(x) dX (x, gˆ) := dimψ(x) gˆH ∩ ψ(X) . ˜ The set Σ1 parametrises all totally geodesic subvarieties (apart from points, of course) of SΓ , and we want to show that it is finite. The fact that it is definable follows from easy observations, at least once the set Π0 (H) is shown to be definable. To prove that Σ1 is countable we use the codimension computations displayed above. They are indeed essential to invoke the Ax–Schanuel Theorem of Bakker and Tsimerman (Theorem 5.4.2), which is used to show that each “gˆ−1 ∈ Σ1 gˆH “ (rather than just a real subgroup of G). “ Since is naturally a Q-subgroup of G “ has only countably many Q-subgroups, the standard fact that a definable G set in an o-minimal structure has only finitely many connected components gives the finiteness of Σ1 and therefore the finiteness of the totally geodesic subvarieties of SΓ .  1.5. Outline of paper. We first fix some notations and discuss preliminary results about lattices in semisimple Lie groups without compact factors. In Section 3, we extend the theory of Shimura varieties starting from a real Shimura datum (G, X, Γ), generalising many classical results and defining Γ-special (and weakly Γ-special) subvarieties. Section 4 is devoted to Z-Hodge theory and period/Mumford–Tate domains. Here we explain how to realise SΓ inside a period domain for Z-VHS. In Section 5, we compare the Mumford–Tate and the monodromy groups associated to the natural OK -VHS on SΓ and the Z-VHS we have constructed, proving a tight relation between the two worlds, namely, Theorem 1.3.2. We prove the non-arithmetic Ax–Schanuel conjecture in Section 5.4. Building on the results of the previous sections, we prove Theorem 1.2.1 in Section 6. Finally, we discuss arithmetic aspects of the theory, and, in Section 7, we discuss various notions of special points and André–Oort type conjectures.

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1.6. Acknowledgements. The second author thanks Bader for an inspiring lecture on his work with Fisher, Miller and Stover in Orsay, in October 2019. The authors are grateful to Klingler for several discussion on related topics and for his cours de l ’IHES on Tame Geometry and Hodge Theory. It is a pleasure to thank Bader, Fisher, Miller and Stover for having explained us their approach and various conversations around superrigidity. We are also grateful to Esnault, Klingler and Labourie for comments on a previous draft of the paper. Finally we thank the anonymous referees for their comments and suggestions that helped improving the exposition and streamlining some arguments. The first author would like to thank the IHES for two research visits in March 2019 and 2020. His work was supported by the Engineering and Physical Sciences Research Council [EP/ L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. 2. Notation, terminology and recollections In this section we fix some notation and discuss the first properties we need about lattices in semisimple groups without compact factors. In particular, we explain the arithmeticity criteria of Mostow–Vinberg and of Margulis. 2.1. Notation. We make free use of the following standard notation. We denote by G real algebraic groups and by G algebraic groups defined over some number field, which usually is either Q, or a totally real number field. Let K be a real field. The K-forms of SU(1, n) are known, by the work of Weil [77], to be obtained as SU(h) for some Hermitian form h on F r , where F is a division algebra with involution over a quadratic imaginary extension L of K and n + 1 = r degL (F ). Regarding algebraic groups, we have • Let G be a connected real algebraic group. By rank of G we always mean the real rank of the group G, i.e., the dimension of a maximal real split torus of G. • If G is reductive, which for us requires also that G is connected, we write Gad for the adjoint of G, i.e., the quotient of G by its centre. (It is a semisimple group with trivial center.) “ • The Weil restriction of G/K from K to Q is usually denoted by G. Regarding subgroups of real algebraic groups, we have • A discrete subgroup Γ of a locally compact group G is a lattice if Γ\G has a finite invariant measure. • All lattices considered in this paper are also assumed to be torsion free. Selberg’s Lemma asserts that if G is semisimple, then Γ has a torsion-free subgroup of finite index; see, for example, [59, Th. 4.8.2].

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• Given H an algebraic subgroup of G, we write ΓH for Γ ∩ H(R). • A lattice Γ ⊂ G in a connected semisimple group without compact factors is reducible if G admits infinite connected normal subgroups H, H 0 such that HH 0 = G, H ∩H 0 is discrete and Γ/(ΓH ·ΓH 0 ) is finite. A lattice is irreducible if it is not reducible. • A subgroup Γ ⊂ G is arithmetic if there exists a semisimple linear algebraic group G/Q and a surjective morphism with compact kernel p : G(R) → G such that Γ lies in the commensurability class of p(G(Z)). Here we denote by G(Z) the group G(Q) ∩ v −1 (GL(VZ )) for some faithful representation v : G → GL(VQ ), where VQ is a finite dimensional Q-vector space and VZ is a lattice in VQ . Torsion-free arithmetic subgroups are lattices. 2.2. Local rigidity. Let G be a real semisimple algebraic group without compact factors, and let Γ be a subgroup of G. Denote by Ad

Ad : Γ ⊂ G −−→ Aut(g) the adjoint representation in the automorphisms of the Lie algebra g of G. Definition 2.2.1. We define the (adjoint) trace field of Γ as the field generated over Q by the set {tr Ad(γ) : γ ∈ Γ}. If Γ is a lattice, its trace field is a finitely generated field extension of Q, which depends only on the commensurability class of Γ. Indeed it is a well-known fact that lattices are finitely generated (or even finitely presented); see, for example, [59, Th. 4.7.10] and references therein. Definition 2.2.2. An irreducible lattice Γ ⊂ G is locally rigid if there exists a neighbourhood U of the inclusion i : Γ ,→ G in Hom(Γ, G), such that every element of U is conjugated to i. The trace field K of a locally rigid lattice is a real number field (which is not, a priori, totally real). By Borel density theorem, lattices in G are Zariski dense. See, for example, [59, Cor. 4.5.6] for details. In particular, Γ determines a K-form of G, which we denote by G, such that, up to conjugation by an element in G, Γ lies in G(K), and K is minimal with this property. For proofs of such facts, we refer to Vinberg’s paper [76]. See also [51, Ch. VIII, Prop. 3.22] and [22, Prop. 12.2.1]. If n > 1, then lattices in G = PU(1, n) are known to be locally rigid. For completeness, we recall a more general1 result, which builds on the study of lattices initiated by Selberg, Calabi and Weil [80].

1 More general in the sense that it allows us to also consider G = SL2 (C), even if local rigidity for non-cocompact lattices in SL2 (C) can fail.

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Theorem 2.2.3 ([34, Th. 0.11]). If G is not locally isomorphic to SL2 (R), then for every irreducible lattice Γ in G, there exist g ∈ G and a subfield K ⊂ R of finite degree over Q such that gΓg −1 is contained in the set of K-rational points of G. Since lattices are finitely generated, we know that there exist a finite set of finite places Σ of K such that Γ lies in G(OK,Σ ) (once a representation is fixed, and up to conjugation by G), where OK,Σ is the ring of Σ-integers of K. In the next subsection we discuss properties of lattices that are contained in G(OK ). 2.3. Lattices and integral points. Let Γ be a lattice in G. Assume that a totally real number field2 K 0 and a form GK 0 of G over K 0 are given, such that a subgroup of finite index of Γ is contained in GK 0 (OK 0 ), where OK 0 denotes the ring of integers of K 0 . Then the field K = Q{tr Ad γ : γ ∈ Γ} is contained in K 0 , as remarked above. The following is [22, Cor. 12.2.8]; see also [60, Lemma 4.1]. It will be the fundamental criterion to detect arithmeticity in Theorem 1.2.1. Theorem 2.3.1 (Mostow–Vinberg arithmeticity criterion). Let Γ ⊂ G(OK 0 ) be a lattice in G. The lattice Γ is arithmetic if and only if for every embedding σ : K 0 → R, not inducing the identity embedding of K into R, the group GK 0 ⊗K 0 ,σ R is compact. Remark 2.3.2. If Γ ⊂ G(OK ) is a non-arithmetic lattice, where K is the trace field of Γ, then G(OK ) is not discrete in G and Γ has infinite index in G(OK ). Let G be a semisimple algebraic group defined over a totally real number field K ⊂ R, and write G for its real points. Denote by Ω∞ the set of all “ for the Weil restriction from K to Q archimedean places of K, and write G of G, as in [81, §1.3]. It has a natural structure of Q-algebraic group and Y “ G(R) = Gσ , σ∈Ω∞

where Gσ is the real group G ×K,σ R. Proposition 2.3.3 ([59, §5.5]). The subgroup G(OK ) of G embeds as an “ via the natural embedding arithmetic subgroup of G “ g 7→ (σ(g))σ∈Ω . G(OK ) ,→ G, ∞ “ Indeed, G(OK ) becomes G(Z). We also have that, if G is simple, then G(OK ) gives rise to an irreducible lattice. Moreover, if for some σ ∈ Ω∞ , Gσ 2 In Theorem 4.2.2 we will see that the trace field of a complex hyperbolic lattice is indeed totally real.

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is compact, then G(OK ) gives rise to a cocompact lattice. We recall here that a closed subgroup Γ of G is cocompact if Γ\G is compact. Every cocompact, torsion-free, discrete subgroup of G is a lattice. 2.4. Arithmeticity criteria, after Margulis. The results discussed in this section can be found in [51] and are due to Margulis. Theorem 2.4.1 (Arithmeticity). Let G be a semisimple group without compact factors. If G has rank at least 2 and Γ ⊂ G is an irreducible lattice, then Γ is arithmetic. The proof of Theorem 2.4.1 is based on applying the supperrigidity theorem, which is recalled below, to representations obtained from different embeddings of the trace field of Γ into local fields. Theorem 2.4.2 (Superrigidity). Let G be a semisimple group of rank at least 2, and let Γ ⊂ G be an irreducible lattice. Let E be a local field of characteristic zero, and let G0 be an adjoint, absolutely simple E-group. Then every homomorphism Γ → G0 (E) with Zariski-dense and unbounded image extends uniquely to a continuous homomorphism G → G0 (E). Recall that two subgroups Γ1 , Γ2 of G are said to be commensurable with each other if Γ1 ∩ Γ2 has finite index in both Γ1 and Γ2 . The commensurator Comm(Γ) of Γ is a subgroup of G: Comm(Γ) := {g ∈ G : Γ and gΓg −1 are commensurable with each other}. The following is known as the commensurability criterion for arithmeticity, and it will be reproven, in the special case of complex hyperbolic lattices, in Section 6.2, as an “unlikely intersection phenomenon.” Theorem 2.4.3. Assume G is without compact factors and Γ is an irreducible lattice. Then Γ is arithmetic if and only if it has an infinite index in Comm(Γ). 2.5. Non–arithmetic complex hyperbolic lattices. Regarding commensurability classes of non-arithmetic lattices in PU(1, n), at the time of writing this paper, we have n = 2. By the work of Deligne, Mostow and Deraux, Parker, Paupert, there are 22 known commensurability classes in PU(1, 2). See [26], [27] and references therein. n = 3. By the work of Deligne, Mostow and Deraux, there are two commensurability classes of√non-arithmetic lattices in PU(1, 3). In both cases the trace field is Q( 3) and the lattices are not cocompact. See [25] and references therein.

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For n > 3, non-arithmetic lattices are currently not known to exist. One of the biggest challenges in the study of complex hyperbolic lattices is to understand for each n how many commensurability classes of non-arithmetic lattices exist in PU(1, n). 3. Real Shimura data and general properties We extend the (geometric) theory of Shimura varieties, more precisely as defined by Deligne in terms of Shimura datum [20], [21], to include quotients of Hermitian symmetric spaces by arbitrary lattices. We then discuss a generalisation of the theory of toroidal compactifications to non-arithmetic lattices and various definability results, proving in the most general form all the results we need in the rest of the paper. 3.1. Definitions and recollections. Let G be a real reductive group. In this section we let Γ be a discrete subgroup of G, whose image in the adjoint group of G, denoted by Gad , is a lattice. Recall that a normal subgroup N of G induces a decomposition, up to isogeny, of G. In symbols we write G ∼ G0 × N for some subgroup G0 of G. Definition 3.1.1. A Γ-factor of G is either a one dimensional split torus in the centre of G, or a normal subgroup N of G ∼ G0 × N such that the image of Γ along the projection from G to N is a lattice in N . We call a Γ-factor irreducible if the image of Γ in N is an irreducible lattice. In the above definition of Γ-factor, N is not required to be irreducible and may have compact semisimple factors. Moreover the centre of G is a product of irreducible tori that are Γ-factors. Proposition 3.1.2. The group G can be written as a finite product of Γ-irreducible factors. Proof. Given Γ in G, there is a unique direct product decomposition G = Gi , where Gi is normal in G, such that ΓGi is an irreducible lattice in Gi i∈IQ and i ΓGi has finite index in Γ. See, for example, [65, Th. 5.22, p. 86]. 

Q

From now on, we write S for the Deligne torus. This is the real torus obtained as a Weil restriction from C to R of the group C∗ . Definition 3.1.3. A real Shimura datum is a triplet (G, X, Γ) where G is a real reductive algebraic group, Γ ⊂ G is a discrete subgroup such that its image in Gad is a lattice, and X is a G-orbit in the set of morphisms of algebraic groups Hom(S, G) such that for some (all) x ∈ X, the real Shimura–Deligne axioms are satisfied: RSD0. The image of Γ in each GL2 -irreducible factor is an arithmetic lattice. RSD1. The adjoint representation Lie(G) is of type {(−1, 1), (0, 0), (1, −1)}.

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√ RSD2. The involution x( −1) of Gad is a Cartan involution. RSD3. G has no simple compact Γ-factors. Without the axiom RSD0, the above definition would include every Riemann surface. Remark 3.1.4. Let X be a G-orbit in the set of morphisms of algebraic groups Hom(S, G) satisfying RSD1, RSD2 and RSD3. Let x ∈ X be such that x : S → G factorises trough H for some subgroup H of G. Then the H orbit of x in X satisfies RSD1 and RSD2. See, for example, [13, Prop. 3.2]. We have a notion of morphism of real Shimura data. Definition 3.1.5. Let (G1 , X1 , Γ1 ) and (G2 , X2 , Γ2 ) be real Shimura data. A morphism of real Shimura data (G1 , X1 , Γ1 ) → (G2 , X2 , Γ2 ) is a morphism of real algebraic groups f : G1 → G2 such that for each x ∈ X1 , the composition f ◦ x : S → G2 is in X2 and f (Γ1 ) ⊂ Γ2 . We need to work with a more general definition of real sub-Shimura datum, allowing the axiom RSD0 to fail. Indeed, as long as the Hermitian space X has dimension strictly bigger than one, we want to consider arbitrary sub-Shimura data. For example, in Theorem 1.2.1, we want to consider every totally geodesic subvariety of dimension one. Definition 3.1.6. Let (G, X, Γ) be a real Shimura datum, H a subgroup of G and ΓH a lattice in H. A real sub-Shimura datum is a triplet (H, XH , ΓH ), where XH is a H-orbit in the set Hom(S, G) satisfying RSD1, RSD2 and RSD3 of Definition 3.1.3. Definition 3.1.7. Let (G, X, Γ) be a real Shimura datum. We define its adjoint real Shimura datum, simply denoted by (G, X, Γ)ad , as the triplet (Gad , X ad , Γad ), where X ad is the Gad -conjugacy class of morphisms x : S → G → Gad and Γad is the image of Γ in Gad . It is again a real Shimura datum. Definition 3.1.8. Let (G, X, Γ) be a real Shimura datum. We denote by SΓ the quotient Γ\X, and we refer to it as a Shimura variety. Every sub-Shimura datum (H, XH , ΓH ) ⊂ (G, X, Γ), as in Definition 3.1.6, induces a closed (for the analytic topology) subvariety SΓH = ΓH \XH of SΓ . Proposition 3.1.2 can be rephrased as follows. Proposition 3.1.9. Any real Shimura datum (G, X, Γ) can be uniquely written, up to isogeny, as a product of a finite number of real sub-Shimura data

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(Gi , Xi , Γi ) such that the Γi are irreducible lattices in Gi and Y SΓi → SΓ i

is a finite covering (of complex manifolds). Theorem 3.1.10 ([20, Prop. 1.1.14 and Cor. 1.1.17]). Let (G, X, Γ) be a real Shimura datum. Then X has a unique structure of a complex manifold such that, for every real representation ρ : G → GL(VR ), (VR , ρ ◦ h)h∈X is a holomorphic family of Hodge structures. For this complex structure, each family (VR , ρ ◦ h)h∈X is a variation of Hodge structure, and X is a finite disjoint union of Hermitian symmetric domains. Choosing a connected component X + of X, we can also define connected real Shimura data and connected Shimura varieties. In what follows we often implicitly work over some connected component. Remark 3.1.11. Let (G, X) be a Shimura datum in the sense of Deligne (i.e., G is assumed to be a Q-group). For any faithful representation G → GL(VZ ), the triplet (GR , X, G(Q) ∩ GL(VZ )) defines a real Shimura datum. Given a Shimura datum (G, X) as before and KAf a compact open subgroup of the finite adelic points of G, the triplet (GR , X, G(Q) ∩ KAf ) is a real Shimura datum. We refer to the latter case as congruence Shimura datum. Theorem 3.1.12 (Baily–Borel, Siu-Yau, Mok). Every Shimura variety SΓ has a unique structure of a quasi-projective complex algebraic variety. Proof. If the lattice Γ is arithmetic, it was proven by Baily and Borel [5]. To prove the result, we may and do replace Γ by a finite index subgroup. In particular, thanks to Proposition 3.1.9, we may assume that Γ is irreducible. Thanks to Theorem 2.4.1, if G has rank bigger or equal than two, then Γ is arithmetic. Regarding rank one groups we have that X is isomorphic to the complex unit ball Bn ⊂ Cn for some n ≥ 1. If n = 1, then the axiom RSD0 ensures that Γ is arithmetic. If n > 1, Siu-Yau [69] and Mok [55] proved that SΓ is biholomorphic to a quasi-projective variety. To see that SΓ has a unique algebraic structure, one can apply [24, Th. A] in place of the classical Borel extension theorem to the identity map SΓ an → SΓ an , where the left-hand side is understood with a different algebraic structure from the right-hand side.  Definition 3.1.13. Let W ⊂ SΓ be an irreducible algebraic subvariety. We say that W is Γ-special if it is induced by a real sub-Shimura datum (H, XH , ΓH ) of (G, X, Γ). We say that W is weakly-Γ-special if there exists a real sub-Shimura datum (H, XH , ΓH ) of (G, X, Γ) such that its adjoint splits as a product (H, XH , ΓH )ad = (H1 , X1 , Γ1 ) × (H2 , X2 , Γ2 ),

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and W is the image of X1+ × {x2 } for some x2 ∈ X2 . It is a standard fact that Γ-special subvarieties of SΓ are totally geodesic. Definition 3.1.14. Given a real Shimura datum (G, X, Γ), we say that SΓ is an arithmetic Shimura variety if Γ is arithmetic. If Γ is irreducible and non-arithmetic, we say that SΓ is a non-arithmetic ball quotient. 3.2. Real sub-Shimura data in rank one. Let (G = PU(1, n), X, Γ) be a real Shimura datum for some n > 1. We remark here that the main results of the paper remain true, with the same proofs, if one starts with Γ an irreducible lattice in PU(1, n)×K, where K is some compact factor. Let XH be a Hermitian symmetric sub-domain of X with automorphism group H. Such spaces will be often referred to as real sub-Shimura couples of (G, X). In this section we explicitly describe the map H → G inducing the inclusion of XH in G. For more details regarding totally geodesic subvarieties of the complex ball, we refer to [35, §3.1.11] and also [26, §2.2]. We will see that every H appearing in this way can be assumed to be a semisimple group. Let VC be a n + 1-dimensional complex vector space with ϕ : VC × VC → C a hermitian form of signature (1, n). Fix a C-basis (e1 , . . . , en+1 ) such that the quadratic form associated to ϕ becomes z1 z 1 −

n+1 X

zi z i .

i=2

The group G = PU(1, n) can be defined in GL(VC ) as equivalence classes of matrices M satisfying t

M gM = g, where g = diag(1, −1, . . . , −1). Since g = g, G can be seen in GL(VR ). Let WC be a sub-vector space of VC such that ϕ restricts to a hermitian form WC × WC → C, and write VC as the direct sum of WC and its orthogonal complement. We can arrange a basis of WC in such a way that the quadratic form associated to ϕ|WC corresponds to a matrix of the form diag(1, −1, . . . , −1). Let H ⊂ G ⊂ GL(VR ) be the subgroup stabilizing WC . Since H also has to stabilise the orthogonal of WC , it is isomorphic to PU(1, m) × C, where m is the rank of WC , from some compact subgroup C in G. Moreover every H associated to an XH arises in this way. Indeed, up to conjugation by an element in G, any α : S → G can be written as z 7→ diag(z, z, 1, . . . , 1). Given a subgroup H ⊂ G as before, we may therefore assume that α(S) ⊂ H for some choice of a basis for WC . Notice also that a similar description holds when G = GU(1, n).

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3.3. Toroidal compactification of non-arithmetic ball quotients. For the length of this section, we assume that G has real rank one. A compactification of SΓ = Γ\X as a complex spaces with isolated normal singularities was obtained by Siu and Yau [69]. Hummel and Schroeder [42] and then Mok [55, Th. 1] showed that SΓ admits a unique smooth toroidal compactification, which we denote by SΓ . If the parabolic subgroups of Γ are unipotent, the boundary SΓ − SΓ is a disjoint union of abelian varieties. The minimal compactification SΓBB , which is proven to be projective-algebraic in [55], can be recovered by blowing down the boundary. Given a parabolic subgroup P ⊂ G we write • NP for the unipotent radical of P ; • UP for the centre of NP . We may identify UP with its Lie algebra Lie(UP ) ∼ = R. Definition 3.3.1. A parabolic subgroup P ⊂ G is called Γ-rational if its unipotent radical NP intersects Γ as a lattice. 3.3.1. First properties of the toroidal compactification. From now on, when speaking of toroidal compactifications, we replace Γ by a finite index subgroup and always assume that Γ is torsion free and that the parabolic subgroups of Γ are unipotent. The following is a consequence of [55]. Theorem 3.3.2 ([55, Cor. 7.6, Ch. III]). The toroidal compactification SΓ of SΓ is a smooth compactification with strict normal crossing divisor at infinity B := SΓ − SΓ . Denote by ∆ ⊂ C the complex disk and by ∆∗ the punctured disk. As in the arithmetic case, there exists an open cover {Uα }α of SΓ such that Uα = ∆n and Uα ∩ SΓ = ∆n−1 × ∆∗ . We have the following; see also [55, Th. 7.2, Ch. III]. Proposition 3.3.3. For any Γ-rational parabolic P , the set UP ∩ Γ is isomorphic to Z. The image of the fundamental group of Uα ∩ SΓ = ∆n−1 × ∆∗ in the fundamental group of SΓ lies in ΓUP ∼ = Z. Proof. The first part can be found in [55, §1.3]. For the second part, we work with the local coordinates as in [61, pp. 255–256]. Let X ∨ be the compact dual of X and X ⊂ X ∨ be the Borel embedding. Assume that we are working with a Γ-rational boundary component F = {b} corresponding to a Γ-rational parabolic P , and let V be the quotient of the unipotent radical of P by it centre. It is a real vector space of rank n − 1 (where n = dim SΓ ). Set [ Xb := g · X ⊂ X ∨. g∈UP ⊗C

There exists a canonical holomorphic isomorphism j:X∼ = Cn−1 × C × {b},

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where Cn−1 = VC and the latter copy of C is UP ⊗ C. We can naturally identify the universal cover of UP ∩ SΓ = ∆n−1 × ∆∗ with (3.3.1) D∼ = {(z1 , . . . , zn−1 , zn , b) ∈ Cn−1 × C × F : Im(zn ) ≥ 0}. The group UP ⊗ C acts on D, in these coordinates, by (z1 , . . . , zn−1 , zn , b) 7→ (z1 , . . . , zn−1 , zn + a, b). Observe that we can factorise the map π : X → SΓ as exp

F X −−−→ expF (X) → S,

where expF : Cn−1 × C × F → Cn−1 × C∗ × F is simply exp(2πi−) on the C-component and the identity on Cn−1 . Moreover expF (X) is ΓUP \X. To conclude, we observe that the diagram expF (X)

expF (X)∨

SΓ

SΓ

is commutative, since the boundary of expF (X)∨ is mapped onto the boundary of SΓ .  Finally we describe in a more explicit way what we discussed in the proof of Proposition 3.3.3. We follow [70, §1] and [55, eq. 8]. Identify X with the complex ball Bn ⊂ Cn . For any b in the boundary of Bn , let Γb ⊂ Γ be the set of elements fixing b. The Siegel domain D representation of X with b corresponding to infinity is ( ) n−1 X D := (z1 , . . . , zn ) : Cn ∈ Im(zn ) > |zi |2 . i=1

For any N > 0, set ( (3.3.2)

D

(N )

:=

n

(z1 , . . . , zn ) ∈ C : Im(zn ) >

n−1 X

) 2

|zi | + N

.

i=1

Let Wb be the set of parabolic isometries fixing b. If N is large enough, two points x, y ∈ D(N ) are equivalent mod Γ if and only if they are equivalent modulo the lattice ΓWb . Moreover, thanks to the work of Margulis and Gromov (see [69, §2], and references therein), for N sufficiently large, one can take D(N ) to be in the set Ab := {x ∈ Bn : min d(x, γx) < } γ∈Γb

for some  > 0 (which depends on Γ). Consider the set E := {x ∈ Bn : min d(x, γx) ≥ }. γ∈Γ

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Notice that the image of E along π : Bn → SΓ is compact. Finally we can write SΓ as the union of a finite number of π(Ab )s and π(E). 3.3.2. Metric at infinity. Let ω be the (1, 1) form on SΓ inducing the natural Hermitian metric. In this section we explain the behaviour of ω near the boundary, as it appears in To’s paper [70, §2]. Fix an open cover {Uα }α of SΓ such that Uα = ∆n and Uα ∩SΓ = ∆n−1 ×∆∗ . Definition 3.3.4. We say that ω is of Poincaré-growth with respect to the toroidal compactification SΓ if, for any α, ω restricted to Uα is bounded by ! n−1 2 X |dz | n Cα |dzk |2 + . |zn |2 | log zn |2 k=1

The above definition does not depend on the choice of coordinates. Whenever we say that ω is of Poincaré-growth, it will be understood that we refer to the growth of ω with respect to the toroidal compactification SΓ . Theorem 3.3.5 ([55, §1.2] and [70, §2]). The Kähler form ω on SΓ is of Poincaré-growth. Finally we notice that the holomorphic tangent bundle T SΓ extends to a holomorphic vector bundle T SΓ on SΓ as follows: in an open neighbourhood Uα = ∆n of SΓ where Uα ∩ SΓ = ∆n−1 × ∆∗ , a local holomorphic basis of T SΓ on Uα is given by ∂ ∂ ∂ ,..., , zn . ∂z1 ∂zn−1 ∂zn 3.4. Some algebraicity results. We first describe Siegel sets in G and X for the action of Γ. This is needed to define semi-algebraic fundamental sets for the action of Γ on X and for proving that every real sub-Shimura datum of (G, X, Γ) gives an algebraic subvariety of SΓ . As usual, the only case we need to describe is when G has rank one. 3.4.1. Siegel sets. Garland and Raghunathan [34] extended Borel’s reduction theory [9] to arbitrary lattices in rank one groups. Notice that Borel follows a different convention. (In this paper G always acts on X on the left.) Since we use the reverse order of multiplication, compared to [9], [34], the inequalities defining the set At are reversed. Write X ∼ = G/K0 for some fixed compact maximal subgroup K0 ⊂ G, and denote by x0 the base point Id K0 of X. Let P be a minimal Γ-rational parabolic subgroup, and let L be the unique Levi subgroup L of P that is stable under the Cartan involution associated to K0 . Consider the Langlands decomposition P = NP AM,

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where A is the split component of P with respect to the basepoint x0 . That is L = AM , where A is a split torus, L ∩ K is a maximal compact in L, and M is the maximal anisotropic subgroup of the connected centralizer Z(A) of A in P . Since Z(A) is compact, M lies in K. For any real number t > 0, let At := {a ∈ A : aα ≥ t}, where α is the unique positive simple root of G with respect to A and P . A Siegel set for G for the data (K0 , P, A) is a product Σ0t,Ω = Ω · At · K0 ⊂ G, where Ω is a compact neighbourhood of the identity in NP . Finally we consider Siegel sets in X Σt,Ω = Ω · At · x0 ⊂ X, where Σt,Ω denotes the image of Σ0t,Ω in X. It is interesting to compare Siegel sets with the D(N ) s described in equation (3.3.2) in the previous section. The arguments of [47, Prop. 3.2] give the following. Proposition 3.4.1. Let Σ = Σt,Ω be a Siegel set in X for the action of Γ. Then Σ is covered by a finite union of open subsets Θ with the following properties. For each Θ, we have • a Γ-rational boundary component F = {b} (corresponding to a Γ-rational parabolic P ), • a positive integer N large enough, • relatively compact subsets U 0 ⊂ U (P ), Y 0 ⊂ Cn−1 such that Θ is bi-isometrically equivalent to n (z1 , . . . , zn ) ∈ Cn−1 × U (P )C , Re(zn ) ∈ U 0 , (z1 , . . . zn ) ∈ Y 0 : Im(zn ) >

n−1 X

|zi |2 + N

o

⊂ D(N ) .

i=1

Definition 3.4.2. A fundamental set for the action of Γ on X is a connected open subset F of X such that Γ · F = X and the set {γ ∈ Γ : γF ∩ F 6= ∅} is finite. Theorem 3.4.3 (Garland-Raghunathan). For any Siegel set Σt,Ω , the set {γ ∈ Γ : γΣt,Ω ∩ Σt,Ω 6= ∅} is finite. There exist a Siegel set Σt0 ,Ω and a finite subset Ξ of G such that • for all b ∈ Ξ, Γ ∩ bNP b−1 is a lattice in bNP b−1 ; • the set Ξ · Σt0 ,Ω is a fundamental set for the action of Γ on X .

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Furthermore, when Ω is chosen to be semi-algebraic, the associated Siegel sets Σt,Ω are semi-algebraic. In particular, the fundamental set F := Ξ · Σt0 ,Ω is semi-algebraic. Proof. The first part of the statement is [34, Th. 4.10]. It is formulated for admissible discrete subgroups of G (see Definition 0.4 in loc. cit.), a class that includes any lattice Γ ⊂ G, as proved in Theorem 0.7 of loc. cit. When Ω is chosen to be semi-algebraic, the Siegel set for G Ω · At · K0 is semi-algebraic since it is defined as product in G of semi-algebraic sets and therefore its image Σt,Ω in X is again semi-algebraic. Finally, since Ξ is finite, Ξ · Σt0 ,Ω is semi-algebraic.  We discuss some functoriality properties of Siegel sets. Let H be a semisimple group of G intersecting Γ as a lattice. As described in Section 3.2, there is a real sub-Shimura datum (H, XH , ΓH ) associated to H. The next proposition describes the intersection of Siegel sets in X with XH (assuming that the dimension of XH is non-zero). Proposition 3.4.4. Let (H, XH , ΓH ) be a real sub-Shimura datum of (G = PU(1, n), X, Γ). Any Siegel sets of H is contained in a Siegel set for G. Proof. Since we are interested only in Siegel set in XH and X, as recalled in Section 3.2, we may assume that H = PU(1, m) for some m < n. Let P be a minimal Γ-rational parabolic subgroup of P and PH be P ∩ H. As above, consider the decompositions relative to K0 and K0 ∩ H: P = NP AM, PH = NH AH MH . As explained in Section 3.2 we have that NP ∩ PH = NH , A ∩ PH = AH and M ∩ PH = MH . The result then follows by the construction of Siegel sets explained above.  3.4.2. Fundamental sets and o-minimality. We have all the ingredients to prove that the uniformising map, opportunely restricted, is definable, extending [47, Th. 4.1] to cover arbitrary lattices. For a general introduction to o-minimal structures, we refer to [29]. Thanks to Theorem 3.1.12, SΓ has a canonical structure of a complex algebraic manifold, and therefore of an Ran,exp manifold; see also [47, App. A]. The following can be proven as [47, Th. 4.1] replacing all references to [2] with the arguments of Section 3.3. Theorem 3.4.5. Let (G, X, Γ) be a real Shimura datum. There exists a semi–algebraic fundamental set F for the action of Γ in X such that the restriction π : F → SΓ of the uniformising map π : X → SΓ is definable in Ran,exp .

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The following will be implicitly used whenever speaking of Γ-special subvarieties. Corollary 3.4.6. Let (H, XH , ΓH ) be a real sub-Shimura datum of (G = PU(1, n), X, Γ). The induced map SΓH → SΓ is algebraic. Proof. By definition, (H, XH , ΓH ) → (G, X, Γ) induces a commutative diagram XH

SΓH

˜i

i

X

SΓ .

For some x ∈ XH , we can write the graph of ˜i as {h.(x, ˜i(x)), h ∈ H}. Let FH be a fundamental set for the action of ΓH in XH . By Proposition 3.4.4, we have that the graph of ˜i restricted to FH is a definable set. Let F be a fundamental set for Γ in X, containing FH , and consider the restriction of the uniformising map XH × X → SΓH × SΓ to FH × F: π|FH ×F : FH × F → SΓH × SΓ . By Theorem 3.4.5, it is a definable map and therefore π|F ×F (graph(˜i|F )) is H

H

definable. The result then follows from Peterzil–Starchenko’s o-minimal GAGA, which is explained below. We have eventually proven that i : SΓH ,→ SΓ is algebraic.  Theorem 3.4.7 ([64, Th. 4.4 and Cor. 4.5]). Let S be a smooth complex algebraic variety. Let Y ⊂ S be a closed complex analytic subset that is also a definable subset (for some o-minimal structure expanding Ran ). Then Y is an algebraic subvariety of S . 4. Z-Hodge theory on non-arithmetic quotients and rigidity Let (G, X, Γ) be a real Shimura datum and SΓ = Γ\X be the associated complex algebraic variety. Thanks to Theorem 3.1.10, faithful linear real representations of G induce real variations of Hodge structure on both X and SΓ . In this section we investigate how to construct a natural Z-variation of Hodge structure on SΓ . The first step is proving that the traces of the image of any γ ∈ Γ along the adjoint representation lie in the ring of integers of a totally real number field K. Given a complex algebraic variety S, we denote by S an the complex points S(C) with its natural structure of a complex analytic variety and by π1 (S) the topological fundamental group of S an . (Unless it is necessary, we omit the base point in the notation.) Regarding variations of Hodge structure (VHS from now

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on) we consider only polarised and pure VHS. More precisely we consider only Q, K, R, C, Z and OK -VHS, as defined in [37], where K is a number field. Let (G = PU(1, n), X, Γ) be a real Shimura datum (n > 1). Recall that there is an isogeny from λ : SU(1, n) → PU(1, n). We may assume that, up to replacing Γ by a finite index subgroup, Γ is the image along λ of a lattice in SU(1, n). For this section we assume this is indeed the case, and we let VC be a complex vector space of dimension n + 1 and SU(1, n) → GL(VC ) be the standard representation of SU(1, n). Let V be the local system associated to (4.0.1)

ρ : π1 (SΓ ) → GL(VC ).

If Γ is an arithmetic subgroup of G, it is a well-known fact that V is induced by a Z-VHS corresponding to a family of principally polarised abelian varieties. We use a recent work of Esnault and Groechenig to prove that lattices in G, not necessarily cocompact, have integral traces. As noticed in [50, Rem. 7.12], if Γ is cocompact, such integrality follows directly from the work of Esnault and Groechenig [32]. This is the first step to construct a Z-VHS on SΓ closely related to the original one (Theorem 4.2.4). Eventually we construct generalised modular embeddings of SΓ in some period domain; see Section 4.3.2. 4.1. Cohomological rigidity and integral traces. The first step in constructing a Z-VHS is the following, which may be of independent interest. (In Theorem 4.2.2 we will prove that the trace field K is in fact totally real.) Theorem 4.1.1. Let Γ be a lattice in G = PU(1, n) for some n > 1, let K be its trace field and let G be the K-form of G determined by Γ. There exists a finite index subgroup Γ0 ⊂ Γ with integral traces. Equivalently, up to conjugation by G, Γ0 lies in G(OK ), where OK is the ring of integers K . In Section 2.2 we discussed locally rigid lattices. To construct a Z-VHS we need a stronger rigidity, namely cohomological rigidity. Building on a study initiated by Weil [80] in the cocompact case, Garland and Raghunathan proved the following. Theorem 4.1.2 ([34, Th. 1.10]). Let G be a semisimple Lie group, not locally isomorphic to SL2 , nor to SL2 (C). For any lattice Γ in G, the first Eilenberg–MacLane cohomology group of Γ with respect to the adjoint representation is zero. In symbols, H 1 (Γ, Ad) = 0. To see why such a vanishing is related to a rigidity result, observe that the the space of first-order deformations of ρ : Γ ,→ G is naturally identified with H 1 (Γ, Ad), where (4.1.1)

ρ

Ad

Ad : Γ − → G −−→ Aut(g)

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is the adjoint representation. The following is proven by Esnault and Groechenig [32], and its proof relies on Drinfeld’s theorem on the existence of `-adic companions over a finite field. Theorem 4.1.1 is a consequence of such result. Theorem 4.1.3 ([32, Th. 1.1]). Let S be a smooth connected quasiprojective complex variety. Then a complex local system V on S is integral; i.e., it comes as extension of scalars from a local system of projective OL -modules of finite type (for some number field L ⊂ C), whenever it is (1) irreducible; (2) quasi-unipotent local monodromies around the components at infinity of a compactification with normal crossings divisor i : S ,→ S ; (3) ccohomologically rigid — that is, H1 (S, i!∗ End0 (V)) vanishes; (4) of finite determinant. Here i!∗ End0 (V) denotes the intermediate extension seen as a perverse sheaf. See [32, Rem. 2.4] for more details. Moreover H1 (S, i!∗ End0 (V)) is the Zariski tangent space at the moduli point of V of the Betti moduli stack of complex local systems of given rank with prescribed determinant and prescribed local monodromies along the components of the normal crossing divisor S − i(S). Proof of Theorem 4.1.1. In the proof we are free to replace Γ by a finite index subgroup, and so we may and do assume that, as in Section 3.3.1, Γ is torsion free and that the parabolic subgroups of Γ are unipotent. Moreover, as explained before, we assume also that Γ lifts to SU(1, n). Let SΓ be the ball quotient Γ\X. Let V be the local system on SΓ associated to the standard representation of SU(1, n), as in (4.0.1). Thanks to Bass–Serre theory [8] (or [76, Th. 2]), V is integral in the sense of Theorem 4.1.3 if and only if the image of every γ ∈ Γ in GL(VC ) has traces in the ring of integers of some number field E ⊂ C (see also [17, Lemma 7.1]). This implies that the traces of Ad(γ) are in OK , where K is the trace field of Γ (which will be proven to be a totally real number field in the next section). As recalled in Section 3.3, SΓ is a smooth quasi-projective variety that admits a smooth toroidal compactification i : SΓ ,→ SΓ , with smooth boundary.3 By construction the local system End0 (V) corresponds to the adjoint representation described in equation (4.1.1). 3 As recalled in the first paragraph of Section 3.3, the boundary is actually a disjoint union of N abelian varieties. Moreover, since G has rank one, the toroidal compactification of SΓ does not depend on any choices.

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Notice that, since Γ is irreducible and Zariski dense in G, also V is irreducible. (This of course depends on our choice of the faithful representation of SU(1, n) in GL(VC ).) To prove the integrality of V we are left to check conditions (2), (3) and (4) of Theorem 4.1.3. The fourth condition is satisfied, since every element of SU(1, n) has finite determinant. Proof of (2). Let B be SΓ − i(SΓ ). Write B as union of its N disjoint S irreducible components: B = i Bi . We notice here that the singular locus of B is empty, and therefore SΓ is equal to what is denoted by U in [32, §2]. Let Pi be Γ-rational parabolic corresponding to Bi . As proven in Proposition 3.3.3 the local monodromy at every Bi corresponds to an element Ti ∈ Γ ∩ UPi , which is certainly unipotent.  Proof of (3). Here we show that V is cohomologically rigid (without boundary conditions). Recall that H 1 (SΓ , End0 (V)) ∼ = H 1 (Γ, Ad) = 0, where the last equality follows from Theorem 4.1.2. To show that H1 (SΓ , i!∗ End0 (V)) = 0, it is enough to observe that it injects in H 1 (SΓ , End0 (V)). This follows from the description of the group H1 (SΓ , i!∗ End0 (V)) appearing in [32, Prop. 2.3]; more precisely, see page 4284 line 8 and Remark 2.4 in op. cit.  Eventually we have checked all the conditions of Theorem 4.1.3, concluding the proof of Theorem 4.1.1.  4.2. Weil restrictions after Deligne and Simpson. Let (G, X, Γ) be an arbitrary real Shimura datum. Up to replacing Γ by a finite index subgroup, we can write (G, X, Γ) as a product of (Gi , Xi , Γi ) such that each Γi is an irreducible lattice in Gi (as in Proposition 3.1.9). As recalled in Theorem 2.4.1, if Gi has rank strictly bigger than one, then Γi is an arithmetic lattice, which implies that Γi has integral traces. If Gi is of rank one and isomorphic to GL2 , RSD0 of Definition 3.1.3 ensures that Γi is also an arithmetic lattice. Therefore we may apply Theorem 4.1.1 to the remaining Gi s of rank one, and conclude that each Γi has integral traces. It follows that for some real number field K, we can write every real Shimura datum as follows: (G, X, Γ ⊂ G(OK )), where Γ lies in G(OK ) possibly up to a finite index subgroup, and a conjugation by some g ∈ G. Thanks to the argument explained above, from now on, we

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assume that G = PU(1, n) and that Γ lifts to SU(1, n). In particular, Γ comes with a standard representation on VC , a n + 1 dimensional vector space. After having established Theorem 4.1.1, to complete the proof of Theorem 1.3.1 we are left to show two things; namely, that the trace field K is totally real (rather than just real), and that we can construct a natural Z-VHS on SΓ . For both proofs, the main observation is that for each embedding σ : K → C (or possibly a larger real number field), the local system Vσ associated to the representation ρσ : π1 (SΓ ) → GL(VC ) underlies a polarised complex VHS. The arguments are fairly standard, and essentially due to Simpson, but we recall here the main steps. They appeared, for example, in [68, Th. 5] and, as stated in [50], the basic idea goes back at least to Deligne [20]. For completeness and related discussions, we refer also to [17, §10] and [50, Prop. 7.1]. We point out here that, among the ones cited so far, the only paper dealing with quasi-projective varieties, rather than projective ones, is [17]. For a complete and more general discussion on this issue, we refer to the monograph [53]. Before stating the results we need, we recall an important theorem of Corlette. Theorem 4.2.1 ([17, Th. 8.1]). Suppose V is local system with quasiunipotent monodromy at infinity. If V is rigid, then it underlines a C-VHS. We are now ready to prove that the field K appearing in Theorem 4.1.1 is totally real. In the proof we actually work with some real number field K 0 containing K. Theorem 4.2.2. Let Γ be a lattice in G = PU(1, n), n > 1. Then the trace field K of Γ is totally real. Proof. Possibly after conjugating, the local system (4.0.1) ρ : π1 (SΓ ) → GL(VC ) takes values in some field K 0 . For each σ : K 0 → C, consider (4.2.1)

ρσ = σρ : π1 (SΓ ) → GL(VC ).

The first step is to prove that the local system Vσ corresponding to ρσ is in fact a VHS (either a K 0 or a C-VHS). Since H 1 (Γ, Ad ◦ρσ ) = H 1 (Γ, Ad) ⊗K 0 ,σ C = 0, the twisted representation ρσ is again cohomologically rigid; see also [17, Lemma 6.6]. Moreover Vσ has also quasi-unipotent monodromy at infinity by construction. Indeed, let Pi be a Γ-rational parabolic corresponding to an irreducible divisor Bi of the boundary, and denote by Ti the local monodromy for V and by Ti σ the local monodromy for Vσ . By construction of Vσ , the element Ti σ is obtained by Ti ∈ ΓUPi ⊂ Γ ⊂ G(K 0 ) by applying σ to its entries.

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Since being unipotent is a geometric condition, it is enough to check that Ti σ ∈ Gσ ⊗ C ∼ = G ⊗ C is unipotent, which holds true because Ti is unipotent. (Here we let Gσ (C) be complex group G ×K 0 ,σ Spec(C).) Eventually we can apply Theorem 4.2.1 to conclude that each Vσ is a C-VHS, as claimed above. The second step is as follows, and it is just a matter of reproducing the argument appearing in [68, end of p. 56 and beginning of p. 57]. For each embedding σ of K 0 , let σ denote the complex conjugated embedding. Since ρ is induced by a polarised C-VHS, we have σρ ∼ = σρ∗ , where ρ∗ denotes the dual representation. As ρ is a unitary representation, and therefore self dual, we observe that σρ ∼ = σρ. By taking the traces, the above equality shows that the subfield of K 0 generated by the traces of ρ(γ), varying γ ∈ Γ, is totally real. Moreover, as recalled in Section 2.2, from the work of Vinberg [76, Th. 2] we know that K 0 contains the adjoint trace field Kρ for the representation Ad(ρ). Recall that K, as in Theorem 4.1.1, denotes the trace field of Γ, where traces are computed after applying the adjoint representation (4.1.1). Since Γ comes from SU(1, n), we see that Kρ = K, and therefore K is naturally a subfield of a totally real number field. In particular, it is a totally real number field, concluding the proof of Theorem 4.2.2.  Remark 4.2.3. For each embedding σ : K 0 → R, let Gσ be the group group G ×K 0 ,σ R. Since the groups Gσ are all isomorphic over C, if G = PU(1, n), the proof of Theorem 4.2.2 shows that we can write Gσ as PU(pσ , qσ ) for some positive non-zero integers pσ , qσ such that pσ +qσ = n+1. See also [68, Lemma 4.5]. The following theorem is essentially due to Simpson [68]. Theorem 4.2.4. Let (G, X, Γ) be a real Shimura datum. There exists a totally real number field K 0 such that • K 0 contains K the (adjoint) trace field of Γ. • Γ ⊂ G(OK 0 ). • For each embedding σ : K 0 → R, if we let Vσ be the K 0 -VHS constructed above, then M b := (4.2.2) V Vσ σ:K 0 →R

has a natural structure of (polarized) Q-VHS. Moreover, by Theorem 4.1.1 and Bass–Serre theory, it has a natural structure of a (polarised) Z-VHS.

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After having established Theorem 4.2.2, the proof of Theorem 4.2.4 is a simple application of [68, Th. 5] (and the more detailed [50, Prop. 7.1]). The proof of Theorem 1.3.1 is eventually completed, and in Section 5.2 we will explain how to get rid of K 0 and work only with K. 4.3. Integral variations of Hodge structure and period maps. In this section we discuss period maps associated to Z-VHS, Mumford–Tate groups and the algebraicity theorem of Cattani, Deligne and Kaplan. 4.3.1. Recap of Mumford–Tate domains. For more details about period domains and Mumford–Tate domains, we refer to [37], [45], [19]. Let (HZ , qZ ) be a polarised Z-Hodge structure. We start by recalling the definition of period domains. Let R be the Q-algebraic group Aut(HQ , qQ ). We have • the space D of qZ -polarised Hodge structures on HZ with specified Hodge numbers is an homogeneous space for R; • choosing a reference Hodge structure we have D = R/M , where M is a subgroup of the compact unitary subgroup R ∩ U (h) with respect to the Hodge form h of the reference Hodge structure; • D is naturally an open sub-manifold of the flag variety D∨ . Let M be the Mumford–Tate group, as recalled below, of (HZ , qZ ), and let h ∈ D be the point corresponding to (HZ , qZ ). The Mumford–Tate domain associated to (HZ , qZ ) is the M(R)-orbit of h in the full period domain of polarized Hodge structures constructed above. Let S be a smooth connected quasi-projective complex variety. By period map S an → R(Z)\D we mean a holomorphic locally liftable Griffiths transverse map. It is equivalent to the datum of a (polarised) Z-VHS on S. For future reference and to complete our brief introduction of variational Hodge theory, we recall here the definition of Mumford–Tate group. Let V = (V, F, Q) be a polarized variation of Q-Hodge structure on S. Let λ : Se → S be the universal cover of S, and fix a trivialisation λ∗ V ∼ = Se × V . Let s ∈ S, the Mumford–Tate group at s, denoted by MT(s) ⊂ GL(Vs ), be the smallest Q-algebraic subgroup M ⊂ GL(Vs ) such that the map hs : S → GL(Vs,R ) describing the Hodge-structure on Vs factors through MR . Choosing a point s˜ ∈ λ−1 (s) ⊂ Se we obtain an injective homomorphism MT(s) ⊂ GL(V ). It is well known that there exists a countable union Σ ( S of proper analytic subspaces of S such that • for s ∈ S − Σ, MT(s) ⊂ GL(V ) does not depend on s, nor on the choice of s˜. We call this group the generic Mumford–Tate group of V;

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• for all s and s˜ as above, with s ∈ Σ, MT(s) is a proper subgroup of G, the generic Mumford–Tate group of V. 4.3.2. Generalised modular embeddings. From the real Shimura datum b on SΓ . By (G, X, Γ ⊂ G(OK )), in Theorem 4.2.4 we produced a Z-VHS V b is a reductive Q-subgroup construction, the generic Mumford–Tate group of V 0 “ 0. of the Weil restriction from K to Q of GK 0 = G ⊗K K 0 , which we denote by G Regarding Weil restrictions, we refer to Section 2.3. We now specialise the definitions of Section 4.3.1 to our setting. As usual, c0 (R)+ . Associated to the pair (SΓ , V), b we have a Mumford–Tate “0 = G we set G 0 0 0 0 0 “ “ “0 domain (G , D = DG “0 = G /M ), where M a compact subgroup of G . As in Section 2.3, we write • r0 for the degree of K 0 over Q; • σi : K 0 → R the real embeddings of K 0 , ordered in such a way that σ1 is simply the identity on K 0 ; “ 0 → G0 for the map of K 0 -group schemes obtained by projecting onto • pri : G “ 0 ⊗ K 0. the ith factor of G b we “ 0 (Z)\D0 be the period map associated to the Z-VHS V Let ψ : SΓ an → G constructed in (4.2.2). We have a commutative diagram ψ˜

X

πZ0

πΓ

SΓ an

D 0 = DG “0

ψ

“ 0 (Z)\D0 . G

By construction, DG “0 is the product of X and a homogeneous space under the Q group i=2,...,r0 Gσi . Given such a decomposition of DG “0 , we can write ˜ ψ(x) = (x, xσ2 , . . . , xσr0 ), where xσi is the Hodge structure obtained by the fibre of Vσi , or more precisely of its lift to X, at x. Finally it is important to observe that ψ˜ is holomorphic and Γ-equivariant, in the sense that for each γ ∈ Γ, ˜ ψ(γx) = (γx, ρσ2 (γ)xσ2 , . . . , ρσr0 (γ)xσr0 ), where ρσi , as introduced in (4.2.1), is obtained by applying σi : K 0 → R to the coefficients of Γ. 4.3.3. Definition of Z-special subvarieties. We specialise the definitions of “ 0 (Z)\D0 associated to the Z-VHS Klingler [45] for the period map ψ : SΓ an → G b when G = PU(1, n) for some n > 1. In [7, §1.2] the following spaces are V called weak Mumford–Tate domain. We omit the word weak because we are

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ultimately interested in subvarieties of SΓ , where there are no possibilities of moving Z-special subvarieties in families. Definition 4.3.1. A Mumford–Tate sub-domain D0 of D0 is an orbit M.x, where x ∈ D0 and M is the real group associated to M a normal algebraic Q-subgroup of MT(x). In fact, D0 is a complex sub-manifold of D0 and “ 0 (Z)\D0 , which we π(D0 ) = M(Z)\M.x is a complex analytic subvariety of G call a Mumford–Tate subvariety. The following is a celebrated result of Cattani, Deligne and Kaplan; see also the recent work of Bakker, Klingler and Tsimerman [6, Th. 1.6]. It will be crucial in proving Corollary 1.3.3. Theorem 4.3.2 ([11, Th. 1.1]). Let D0 ⊂ D0 be a Mumford–Tate subdomain. Then the set ψ −1 (π(D0 )) is an algebraic subvariety of SΓ . Definition 4.3.3. Subvarieties of SΓ of the form ψ −1 (π(D0 ))0 , for some Mumford–Tate sub-domain D0 ⊂ D0 , are called Z-special. (Here (−)0 denotes some irreducible component.) An irreducible subvariety W of SΓ is Z-Hodge generic if it is not contained in any strict Z-special subvariety. By definition, SΓ is always Z-special. In the next section we will see that Γ-special subvarieties (see Definition 3.1.13) give examples of (strict) Z-special subvarieties (in the sense of Definition 4.3.3), and vice versa. Moreover we explicitly describe the Z-special subvarieties associated to the Γ-special ones.

5. Comparison between the Γ-special and Z-special structures Let (G = PU(1, n), X, Γ ⊂ G(OK )) be a real Shimura datum. In this section we introduce and compare two notions of monodromy groups for subvarieties of SΓ = Γ\X: on the one hand with respect to the natural OK -VHS, and, b of Theorem 4.2.4. Thanks to the Deligne–André on the other, to the Z-VHS V Theorem, such comparison translates to a computation of the Mumford–Tate b The main result of the section is Theorem 1.3.2, which will be group of V. used to prove Theorem 1.2.1 in Section 6. We also prove the non-arithmetic Ax–Schanuel conjecture in Section 5.4. Finally several corollaries, which may be of independent interest, are discussed. 5.1. Computing the monodromy for the Z-VHS. Recall that Γ is assumed to be torsion free, and that it defines a K-form G of G, where K denotes the (adjoint) trace field of Γ. Let K 0 be the totally real field of Theorem 4.2.4 (which contains K), and denote the Weil restriction from K 0 to Q of GK 0 = G ⊗K K 0 “ 0 . It naturally contains G, “ the Weil restriction from K to Q of G. by G

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b which, We consider the real representation of Γ ∼ = π1 (SΓ ) associated to V, by construction, is given by Y “0 = (5.1.1) ρVb : Γ → G Gσ , γ → 7 (σ1 (γ) = γ, . . . , σr0 (γ)), σ∈Ω∞

where, as in Section 4.3.2, σi are the r0 real embeddings of K 0 . In this section b we compute the algebraic monodromy of subvarieties of SΓ for the Z-VHS V. 0 “⊂G “. Among other things, we prove that ρVb has Zariski dense image in G Let W be an irreducible (closed) algebraic subvariety of SΓ of positive f be an analytic component of π −1 (W ), where π : X → SΓ dimension, and let W denotes the quotient map. As in Section 4, we omit the base point in the notation of the fundamental group and we replace W by its smooth locus. As follows, we denote the Γ-monodromy of W , that is, the monodromy of W with respect to the K-local system V: Γ

Zar,0

MW := Im(π1 (W ) → π1 (SΓ ) = Γ)

⊂ G,

that is, the identity component of the Zariski closure in G of the image of π1 (W ). Lemma 5.1.1. The K-algebraic group Γ MW is non-trivial, and its real points are not compact. Proof. Heading for a contradiction suppose that the intersection between f is Γ and Γ MW is finite. By Riemann existence theorem, this implies that W algebraic and therefore does not admit bounded holomorphic functions. This, however, contradicts the fact that X = Bn admits a bounded realisation. Finally, since Γ ∩ Γ MW is a discrete and infinite subgroup of Γ MW , Γ MW cannot be compact.  The group Γ MW is defined over K, the adjoint trace field of Γ, but it could happen that it is defined over a smaller field. In the next definition we look at the field of definition of the adjoint of Γ MW , in the sense of Section 2.2. Definition 5.1.2. The field generated by the traces of the elements in Γ ∩ MW , under the adjoint representation (of MW ), is called the trace field of W and is denoted by KW . Γ

Remark 5.1.3. The field KW is naturally a subfield of K. Indeed, by construction, the field L := Q{tr ad|MW (γ) : γ ∈ Γ ∩ Γ MW } is a subfield of K, where ad|MW denotes the restriction of the adjoint representation of G to MW . By Vinberg’s theorem [76], as explained in Section 2.2, we have that the adjoint trace field of Γ∩ Γ MW is a subfield of L, and therefore of K.

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We define the Z-monodromy of W , denoted by Z MW , as the monodromy b we constructed in Theorem 4.2.4. That is, of W with respect to the Z-VHS V Zar,0

Z

“ 0 (Z)) MW := Im(π1 (W ) → G

“ 0. ⊂G

“ the Weil restriction from K to Q of G, It is a connected Q-subgroup of G, “ 0. which is seen diagonally as a subgroup of G We record here two simple observations relating the two “special structures,” relying only on the functoriality of the fundamental group: • if W is contained in a Γ-special subvariety (as introduced in Definition 3.1.13) associated to (H, XH , ΓH ⊂ H(OK )), then the Γ-monodromy of W is contained in H; • if W is contained in a Z-special subvariety (as introduced in Definition 4.3.3) associated to (R, DR , R(Z)), then the Z-monodromy of W is contained in R. We define Γ\ M as the Weil restriction from K to Q of the adjoint of W

W

Γ

MW . It is a Q-algebraic group whose real points can be identified as the
product of the Γ Mad , varying σ : KW → R. Each factor has a structure of a W σ KW -algebraic group. Since KW is contained in K, as explained in Remark 5.1.3, the factors also have a structure of K-group scheme. The following will be crucial for the future arguments, and it is the first step towards the proof of Theorem 1.3.2. Theorem 5.1.4. The Z-monodromy of W is the Weil restriction from KW to Q of the Γ-monodromy of W . In symbols, we write \ Γ M = Z Mad . W

W

The representation π1 (W ) → Γ Mad W, b from SΓ to (the smooth locus of) associated to the restriction of the Z-VHS V W is obtained, by equation (5.1.1), using each embedding σ : K → R: π (W ) → Γ Mad → Γ\ M . 1

W

W

In particular, the adjoint of Z MW is contained in Γ\ MW . Proof. If KW = Q, there is nothing to prove, so we may assume that [KW : Q] > 1. For each σ : KW → R, denote by Mσ the image of the adjoint of Z MW along the projection “ → Gσ . prσ : G
We first prove that Mσ is the adjoint of Γ MW σ . By equation (5.1.1), we have that Mσ contains the image of the monodromy representation associated to the VHS Vσ|W , which is Ä ä gW : π1 (W ) → Γ MW . σ

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Since the image of π1 (W ) is Zariski dense in Γ MW , we have that the image of
Γ gW is contained in Mσ and it is Zariski dense in MW σ . It follows that Ä äad Γ (5.1.2) MW = Mσ . σ

For any two of the rW archimedean places σi , σj of KW , we prove that Z MW surjects onto Mσi × Mσj . This is enough to conclude when rW = 2, and then we argue by induction. Let H be the intersection of Z MW with Mσi × Mσj . It is an algebraic group whose minimal field of definition is KW . For ` ∈ {i, j}, we have a short exact sequences (in the category of algebraic KW -groups) 1 → K` → H → Mσ` → 1. Since we are working with semisimple group, the only possibilities for K` is to be either finite or equal to Mσ`0 for `0 such that {`, `0 } = {i, j}. If K` = Mσ`0 , then H = Mσi × Mσj , proving the claim. If both kernels are finite, we argue using Goursat’s Lemma, as recalled below. Lemma 5.1.5 (Goursat’s Lemma). Let G1 , G2 be algebraic groups and H be a closed subgroup of G1 × G2 such that the two projections from H to G1 and G2 are surjective. For i = 1, 2, let Ki be the kernel of H → Gi . The image of H in G1 /K2 × G2 /K1 is the graph of an isomorphism G1 /K2 → G2 /K1 . Assume that both Ki and Kj are finite groups. By applying Lemma 5.1.5 we can write H as the graph of an isomorphism of KW group schemes f : Mσi /Kj → Mσj /Ki . In particular, up to a quotient by a finite group, we obtain a commutative square Mσi (KW )

fKW

σj

σi

Mσi (R)

Mσj (KW )

fR

Mσj (R).

The diagram induces an equivalence between the two places σi , σj . Indeed, since KW is the field generated by the traces of the γ ∈ Γ ∩ Γ MW , and Q{tr(ad(γ)) : γ ∈ σi (Γ ∩ Γ MW )} = Q{tr(ad(γ)) : γ ∈ σj (Γ ∩ Γ MW )}

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we obtain that σi (KW ) = σj (KW ). This is impossible because σi , σj are places of KW , the smallest field of definition of Γ MW . To finish the proof we argue by induction. Assume that Z MW surjects onto any products of m factors for some m < rW . We have to prove that it surjects onto any products of m + 1 factors. Let S be a set of m + 1 places of KW , and write S = S 0 ∪ {σ` } for some σ` ∈ S. Set Y Y MS 0 := Mσ , MS := Mσ , σ∈S 0

σ∈S

and consider the exact sequence 1 → KS 0 → H → MS 0 → 1. The kernel KS 0 is either finite, or equal to Mσ` . In the latter case, we conclude that H = MS . In the former case, consider the short exact sequence 1 → K` → H → Mσ` → 1. By applying Lemma 5.1.5, we can have an isomorphism of K-group schemes Mσ` ∼ = MS 0 /K` . Since MS 0 is a product of simple groups, the right-hand side has to be isomorphic to (a finite quotient of) Mσ`0 for some `0 ∈ S 0 . We therefore obtain the same commutative square as in the case where S had cardinality two, concluding the proof of Theorem 5.1.4.  5.2. Monodromy and Mumford–Tate groups. We are ready to compute the b restricted to subvarieties of SΓ , in the sense generic Mumford–Tate groups for V of Section 4.3.1. We first recall the following important theorem, which relates the monodromy and the Mumford–Tate groups. It can be found in [1, Th. 1] and [19] (see also [48, Th. 4.10]). Theorem 5.2.1 (Deligne, André). Let W a smooth quasi-projective variety supporting a Z-VHS, and let s ∈ W be a Hodge generic point. Let Ms be the algebraic monodromy of W (at s). Then Ms is a normal subgroup of the derived subgroup MT(s)der of the Mumford–Tate group at s. In symbols we simply write Ms / MT(s)der . Combining Theorems 5.2.1 and 5.1.4, we obtain two important corollaries. The former will be used to prove that Γ-special subvarieties are Z-special, and the latter to establish the converse implication. “ i.e., Γ has Zariski Corollary 5.2.2. The Z-monodromy group of SΓ is G; “ dense image in G. Moreover the derived subgroup of the generic Mumford–Tate b is G. “ group of V

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Proof. By definition, the Γ-monodromy of SΓ is given by G. Theorem 5.1.4 “ since KW is just K, applied to W = SΓ shows that the Z-monodromy of SΓ is G, the trace field of Γ. By Theorem 5.2.1, we conclude that generic Mumford–Tate b is G. “ group of V  Remark 5.2.3. As a direct consequence of Corollary 5.2.2, we have the following. Let SΓH be a Γ-special subvariety of SΓ associated to a triplet b (H, XH , ΓH ⊂ H(OK )) ⊂ (G, X, Γ ⊂ G(OK )). The Mumford–Tate group of V restricted to the (smooth locus of) SΓH is the Weil restriction from KH to Q of H, where KH is the subfield of K given by the trace field of the lattice ΓH . b Now that we have finally computed the generic Mumford–Tate group of V, we can upgrade the diagram presented in Section 4.3.2 to one targeted in “ D “) (rather than its decorated version). After having explained this, K 0 (G, G “ 0 will no longer appear in the paper. First notice that and G Y 0 “0 = G Gσ [K :K] . σ:K→R

Write

Ñ b= V

é

M

M

σ:K→R

`:K 0 →R, `|σ

V`

,

and notice that the original K-VHS V is one of the factors appearing above. We work at the places ` above “the identity” σ1 : K → R. The lift of the period map ψ˜ induces a holomorphic map ψ˜Id from X to a homogeneous space under 0 G[K :K] , which we denote by DId . The image of ψ˜Id lies in a ∆(G)-orbit, where 0

∆ : G → G[K :K] is the diagonal embedding. Since, in the coordinates of D0 , we have a component of ψ˜ corresponding to the identity, we have that for any g ∈ G, ψ˜Id (gx) = (gx, gx` , . . . , gx` 0 ). 2

[K :K]

As a consequence of the above G-equivariance, we observe that ψ˜Id : X → DId ∼ = X × · · · × X, where on the right-hand side we have [K 0 : K]-factors. In such coordinates we can indeed write ψ˜Id (x) = (x, g`1 x, . . . , g`[K 0 :K] x). 0 By choosing the opportune biholomorphism DId ∼ = X [K :K] , ψ˜Id becomes just the “ 0 will no longer diagonal embedding. This explains why, from now on, K 0 and G appear. We therefore refine the notation of Section 4.3.2 by “undecorating it”: • we let r be the degree of K over Q; • we let σi : K → R be the real embeddings of K, ordered in such a way that σ1 is simply the identity on K;

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b “ and we let ψ : SΓ an → G(Z)\D be the period map associated to the Z-VHS V we constructed in (4.2.2). We have a commutative diagram ψ˜

X

D = DG “ πZ

πΓ

SΓ an

ψ

“ G(Z)\D

∼ X × D>1 , where D>1 is a homogeneous space under in such a way that D = Q ˜ i>1 Gσi , and the first component of ψ is simply the identity. Q Remark 5.2.4. We note here that, by Theorem 2.3.1, the group i=2,...,r Gσi is compact if and only if Γ is an arithmetic lattice. This will be a fundamental step in Section 6, whose starting point is the following: Γ is non-arithmetic if and only if codimG(Z)\D (ψ(SΓ an )) > 0. “ Corollary 5.2.5. Let H/Q be the the generic Mumford–Tate group of a subvariety (of positive dimension) W ⊂ SΓ . Then H/Q is a Weil restriction from KW to Q of some subgroup F of G. Proof. Let W be a (smooth, irreducible, closed) subvariety of SΓ . Applying Theorem 5.2.1 to W , we see that Z

“ MW / MT(w)der ⊆ G,

where w ∈ W is a point whose Mumford–Tate group is H, the generic Mumford– Tate of W . This is a Hodge generic point in the sense of Section 4.3.1. Theorem 5.1.4 shows that the left-hand side is the Weil restriction, from KW to Q, of some subgroup F ⊂ G. Since the centraliser of FR in GR is finite (being G of rank one), we conclude that b = Z MW = MT(w)der , F 

as desired.

5.3. Proof of Theorem 1.3.2. We first prove one of the two implications of Theorem 1.3.2. (The fact that Γ-special subvarieties are the same as the totally geodesic ones was already discussed in Section 3.) Proposition 5.3.1 (Γ-special ⇒ Z-special). Let (H, XH , ΓH ⊂ H(OK )) “ the Weil be a real sub-Shimura datum of (G, X, Γ ⊂ G(OK )). Denote by H restriction from K to Q of H. The subvariety SΓH of SΓ can be written as an irreducible component of the preimage along the period map “ ψ : SΓ an → G(Z)\D “ G

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“ “ of the Mumford–Tate sub-domain H(Z)\D “ ⊂ G(Z)\DG “. H Proof. As explained in Theorem 3.1.12 and Corollary 3.4.6, SΓH is an b from SΓ to irreducible algebraic subvariety of SΓ . We can restrict the Z-VHS V 4 (the smooth locus of ) SΓH . From Corollary 5.2.2 (see, e.g., Remark 5.2.3), we know that b |S )der = H “ der . (5.3.1) MT(V ΓH

“ denotes the Weil restriction from K to Q of H ⊂ G/K. To be more Here H precise we should restrict from the KH , the trace field of ΓH , but as will be explained in Remark 5.3.2, this difference plays no role in the sequel. “ From (5.3.1), we see that H(Z)\D “ is, as claimed in the statement of the H “ proposition, indeed a Mumford–Tate sub-domain of G(Z)\D “. By construction G and the definition of generic Mumford–Tate group recalled in Section 4.3, “ “ H(Z)\D “ is indeed the smallest sub-domain of G(Z)\DG “ containing ψ(SΓH ). H We therefore have a commutative diagram ψ˜|XH

XH

DH “ ⊂ DG “ π

SΓH an

ψ|SΓ

an

H

“ “ H(Z)\D “ ⊂ G(Z)\DG “. H

Observe that ψ˜−1 (DH “ ) = XH . This is enough to see that 0 “ SΓH ⊆ ψ −1 (H(Z)\D “) . H

The right-hand side is a closed irreducible analytical subvariety of SΓ an . Since both objects have the same dimension, being locally isomorphic to XH , we conclude that SΓH is of the desired shape.  Remark 5.3.2. It could happen that H is defined over a smaller field (but not over a bigger one, as explained in Remark 5.1.3). The trace field of ΓH , which we denote by KH = KSΓH , may be strictly contained in K. (It could even happen that KH = Q. In this case we can consider the Weil restriction H ResK Q H and the associated period domain ˜ := D KH D ⊂ DG “. Res H(R) Q

The variety ΓH \XH is smooth (at least when Γ is torsion free), but it may be that its image in SΓ is not smooth. 4

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˜ 0 . Indeed D ˜ is We remark here that SΓH can also be written as ψ −1 π(D) “ ˜ contained in DH “ and dim ψ(SΓH ) ∩ π(D) = dim ψ(SΓH ) ∩ H(Z)\DH “ , since, as in the proof of Proposition 5.3.1, both are locally isomorphic to XH . The following is the converse of Proposition 5.3.1, and their combination gives Theorem 1.3.2. An important input in the proof is Corollary 5.2.5. Proposition 5.3.3 (Z-special ⇒ Γ-special). Let (G = PU(1, n), X, Γ) be a real Shimura datum, and let SΓ be the associated Shimura variety. Any Z-special subvariety W of SΓ is Γ-special. Proof. Let W ⊂ SΓ be a Z-special subvariety, as usual assumed to be of f be such that π(w) is smooth and MT(w) = positive dimension. Let w ∈ W MT(W ). By Corollary 5.2.5, we can assume that MT(w) is the Weil restriction of some K-subgroup of G. To keep the notation more compact, we write M c for its Weil for the associated K-subgroup of G, M for its real points, and M restriction to Q. Being W a Z-special subvariety with generic Mumford–Tate c means that group M (5.3.2)

0 c W = ψ −1 (M(Z)\D c) . M

c denotes the associated real group and (−)0 some analytic component. Here M ˜ c Notice also that DM c is the M -orbit of ψ(w) . Our goal is to show that W is associated to some sub-Shimura datum of (G = PU(1, n), X, Γ). The natural candidate appears to be (M, XM , ΓM ), where XM = ψ˜−1 (DM c ) = M.w ⊂ X is an Hermitian symmetric subspace of X (since w : S → G factorises through M ). The only problem is that, a priori, we do not know that ΓM = Γ ∩ M has finite covolume in M . Equation (5.3.2) is enough to show that the first of the following equalities holds true: W = π(M.w) = ΓM \XM . As W is an algebraic subvariety of SΓ , by the work of Cattani, Deligne and Kaplan (recalled before as Theorem 4.3.2), we can apply Lemma 5.3.4 below to conclude that ΓM is a lattice in M . Therefore we proved that the Z-special subvariety W is Γ-special and associated to (M, XM , ΓM ).  To conclude the proof of Proposition 5.3.3, we need the following. Lemma 5.3.4. Let (G = PU(1, n), X, Γ) be a real Shimura datum. Let XH be a Hermitian symmetric subspace of X , associated to a subgroup H ⊂ G, such that π(XH ) is an algebraic subvariety of SΓ . Then (H, XH , ΓH ) is a real sub-Shimura datum of (G, X, Γ); that is, ΓH is a lattice in H .

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Proof. Let Ω ⊂ Cm be a bounded domain, and let ΓΩ be a discrete subgroup of the automorphism group of Ω such that ΓΩ \Ω has the structure of a quasiprojective variety. Recall that Griffiths [38, Prop. 8.12] proved that ΓΩ \Ω has finite Kobayashi–Eisenman volume [31]. (See also the discussion after [38, Question 8.13].) Applying such a result to Ω = XH = Bm (for some m < n) and ΓΩ = ΓH we see that ΓH \XH has finite Kobayashi–Eisenman volume. Thanks to [31, Prop. 2.4, p. 57], the Kobayashi–Eisenman volume form is equal to the volume associated to the Kähler form on Bm . Therefore ΓH \Bm has finite volume with respect to the volume form associated to the standard Kähler form on Bm ; that is, ΓH is a lattice in H.  After having established Theorem 1.3.2, which says that Γ-special subvarieties are the same thing as the Z-special ones, we give the following. Definition 5.3.5. A smooth irreducible closed algebraic subvariety W ⊂ SΓ (of dimension > 0) is special if it is Γ-special or equivalently Z-special or equivalently a totally geodesic subvariety. We define the special closure of W as the smallest special subvariety of SΓ containing W . We say that W is Hodge generic if it is not contained in any strict special subvariety of SΓ . The reader interested only in the proof of Theorem 1.2.1 may skip the next two subsections and go directly to Section 6. Notice, however, that in Proposition 6.1.4 we will use Theorem 5.4.2, which is recalled below. 5.4. Proof of the non-arithmetic Ax–Schanuel conjecture. In this section we prove Theorem 1.2.2, which, thanks to Theorem 1.3.2 (proven in Section 5.3), becomes the following. Theorem 5.4.1 (Non-arithmetic Ax–Schanuel Conjecture). Let (G, X, Γ) be an irreducible Shimura datum. Let W ⊂ X × SΓ be an algebraic subvariety and Π ⊂ X × SΓ be the graph of π : X → SΓ . Let U be an irreducible component of W ∩ Π such that codim U < codim W + codim Π, the codimension being in X × SΓ or, equivalently, dim W < dim U + dim SΓ . If the projection of U to SΓ is not zero dimensional, then it is contained in a strict special subvariety of SΓ . Before starting the proof of Theorem 5.4.1, we recall the Ax–Schanuel for b constructed in Theorem 1.3.1. The next theorem follows from the the Z-VHS V work of Bakker and Tsimerman [7, Th. 1.1], Theorem 1.3.2 and the fact that G has rank one. We refer to it as the Z-Ax–Schanuel (or simply as Z-AS).

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c ⊂ D “ × SΓ be an algebraic subvariety. Let U “ be Theorem 5.4.2. Let W G Ä ä c ∩ D“ ד an irreducible component of W S ⊂ DG “ × SΓ such that G G(Z)\D Γ “ < codim W c + codim D “ × “ codim U S , G G(Z)\D Γ “ the codimension being in DG “ × SΓ . Then the projection of U to SΓ is contained in a strict special subvariety of SΓ , unless it is zero dimensional. Proof of Theorem 5.4.1. If Γ is arithmetic, the result is a special case of [57, Th. 1.1], so we may and do assume that Γ is non-arithmetic. Thanks to Theorem 1.3.1 and the discussion after Corollary 5.2.2, we have the following commutative diagram X

ψ˜

D = DG “ πZ

πΓ

SΓ an

ψ

“ G(Z)\D.

In the proof we actually use that the above diagram is cartesian. Finally we can Q write D ∼ = X × D>1 for some homogeneous space D>1 under ri>1 Gσi , where r = [K : Q]. Let W ⊂ X × SΓ be an algebraic subvariety, and consider the algebraic subvariety of D × SΓ c := W × D>1 W (by applying the natural isomorphism X × SΓ ∼ = SΓ × X). We claim that W ∩ Π c can be identified with W ∩ (D ×G(Z)\D SΓ ) ⊂ D × SΓ . To see this, notice that “ W ∩ Π = {w = (w0 , w1 ) ∈ W ⊂ X × SΓ : πΓ (w1 ) = w0 } and ˜ c ∩ (D × “ c ∩ {(πΓ (x), ψ(x)) W S )∼W : x ∈ X}. G(Z)\D Γ = The natural map identifying the two complex analytic varieties is given by (5.4.1)

˜ 1 )) ∈ W c ∩ (D “ × “ W ∩ Π 3 w 7→ (w0 , ψ(w S ). G G(Z)\D Γ

Let U be an irreducible component of W ∩ Π such that (5.4.2)

codimX×SΓ U < codimX×SΓ W + codimX×SΓ Π.

Thanks to the identification appearing in (5.4.1), we see that U corresponds “, an irreducible component of W c ∩ (D × “ to U S ). From (5.4.2), we obtain G(Z)\D Γ that “ < codim W c + codim D × “ codim U S , G(Z)\D Γ

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the codimension being in D × SΓ ∼ = D>1 × X × SΓ . Eventually we may apply Theorem 5.4.2 to conclude.  5.5. Some corollaries. Let (G = PU(1, n), X, Γ) be a real Shimura datum and π = πΓ : X → SΓ = Γ\X be the projection map. Let Y ⊂ X be a positive dimensional algebraic subvariety. Applying the non-arithmetic Ax–Schanuel Conjecture, Theorem 5.4.1, to Zar

W := Y × π(Y )

,

we obtain the following, which recovers Mok’s result [56, Main Theorem]. Corollary 5.5.1 (Non-arithmetic Ax–Lindemann–Weierstrass). Let (G = PU(1, n), X, Γ) be a real Shimura datum and Y ⊂ X be an algebraic subvariety. Then any irreducible component of the Zariski closure of π(Y ) in SΓ is special. Another direct consequence of the Ax–Lindemann–Weierstrass theorem is the following characterisation of special subvarieties of SΓ . Corollary 5.5.2. Let W ⊂ SΓ be an irreducible algebraic subvariety (of dimension > 0). The following are equivalent: (1) W is totally geodesic; (2) W is bi-algebraic — i.e., some (equivalently any) analytic component of the preimage of W along π : X → SΓ is algebraic; (3) W is a special subvariety (in the sense of Definition 5.3.5); (4) W = ψ −1 (πZ (Y ))0 for some algebraic subvariety Y of DG “. “ The fact that a non-arithmetic Γ gives rise to a thin subgroup of G(Z) follows already from Corollary 5.2.2. As announced in the introduction, we can prove Corollary 1.3.4 (at least up to the finiteness of maximal special subvarieties, Theorem 1.2.1, which is proven in the next section). Corollary 5.5.3. Let (G = PU(1, n), X, Γ ⊂ G(OK )) be a real Shimura datum. Let W be a Hodge generic subvariety of SΓ (in the sense of Definition 5.3.5). If Γ is non-arithmetic, the image of π1 (W ) → Γ “ gives rise to a thin subgroup of G(Z). Proof. As in Remark 2.3.2, since Γ is non-arithmetic, Γ has infinite index “ in G(OK ) and so the same holds for its image in G(Z). Because of the Hodge genericity assumption, the André–Deligne monodromy theorem (Theorem 5.2.1) “ “ and implies that the image of π1 (W ) → Γ → G(Z) is Zariski dense in G “ therefore a thin subgroup of G(Z). 

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Finally the following was announced in the introduction as Corollary 1.3.3. The proof is similar to that of Proposition 5.3.3. Corollary 5.5.4. Let (F, XF ) be a real sub-Shimura couple of (G = PU(1, n), X) (in the sense of Section 3.2). If ΓF is Zariski dense in F , then ΓF is a lattice in F . Proof. That (F, XF ) is a real sub-Shimura couple of (G = PU(1, n), X) implies that there exists some x ∈ X such that the map x:S→G

(5.5.1)

factorises trough F . Since ΓF ⊂ F is Zariski dense in F , it determines the b be the Weil restriction from K to K-form of F , which we denote by F. Let F Q of F. Consider the sub-pair ˜ b D “ = F(R). b “ D = D “). (F, ψ(x)) ⊂ (G, F G b D “) is a sub-Mumford–Tate domain of Because (5.5.1) factorises trough F , (F, F “ D = D “). (G, G As usual we argue using the fundamental cartesian diagram (coming from Theorem 1.3.1 and Corollary 5.2.2): X

ψ˜

D = DG “ πZ

πΓ

SΓ an

ψ

“ G(Z)\D.

We observe that ψ˜−1 (DF“) = XF , and so 0 b ΓF \XF = πΓ (XF ) = ψ −1 (F(Z)\D “) F 0 −1 b b for the component ψ −1 (F(Z)\D “) of ψ (F(Z)\DF “) containing ΓF \XF . TheF orem 4.3.2 implies that the right-hand side is algebraic. Eventually Lemma 5.3.4 applies and shows that ΓF is a lattice in F , as desired. 

6. Finiteness of special subvarieties Let n be an integer > 1. Let (G = PU(1, n), X, Γ) be a real Shimura datum, as in Definition 3.1.3, and let SΓ = Γ\X be the associated Shimura variety (not necessarily of arithmetic type). Recall that, thanks to Theorem 1.3.1, the lattice Γ ⊂ G determines a totally real number field K, and a K-form G of the real group G for which, up to G-conjugation, Γ lies in G(OK ).

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Here we prove Theorem 1.2.1, which, thanks to Definition 5.3.5 and Theorem 1.3.2, reads as follows. Moreover, in Section 6.2, we explain how a similar strategy can be used to reprove the Margulis commensurability criterion for arithmeticity for lattices in PU(1, n), which was recalled as Theorem 2.4.3. Theorem 6.0.1. If SΓ is non-arithmetic, then it contains only finitely many special subvarieties. A special geodesic subvariety S 0 ⊂ SΓ is called maximal if the only special subvariety of SΓ strictly containing S 0 is SΓ itself. Recall that, from Definition 5.3.5, (positive dimensional) special subvarieties of SΓ are the same as totally geodesic subvarieties, Γ-special and Z-special subvarieties. The maximality condition cannot be dropped in the statement, unless n = 2, where it is always satisfied. For example, SΓ could contain a special subvariety S 0 ( SΓ associated to a real Shimura datum (H, XH , ΓH ) where ΓH is arithmetic, and S 0 could contain countably many special subvarieties. An equivalent way of phrasing the above theorem is the following. Theorem 6.0.2. Let (Yi )i∈N be a family of special subvarieties of SΓ . Then every irreducible component of the Zariski closure of the union [ Yi i∈N

is either one of the Yi s, or it is a special and arithmetic subvariety of SΓ . To establish the equivalence of the two statements we only need to show that Theorem 6.0.1 implies Theorem 6.0.2. Let S 0 be an irreducible component S Zar of i∈N Yi . If S 0 is not one of the Yi , we can consider the smallest special subvariety S 00 of SΓ containing S 0 . By construction, S 00 contains infinitely many maximal totally geodesic subvarieties, and therefore, by Theorem 6.0.1, it is arithmetic. To see that S 0 = S 00 , just notice that S 0 is a subvariety of an arithmetic Shimura variety containing a Zariski dense set of sub-Shimura varieties. It follows that S 0 is special [73, Th. 4.1] and, by minimality of S 00 , that S 0 = S 00 . We summarise here the strategy of the proof of Theorem 6.0.1, explaining what we are left to do. • In Section 4, we constructed a polarised integral variation of Hodge structure b on SΓ , corresponding to a period map (Z-VHS) V “ ψ : SΓ an → G(Z)\D “, G “ denotes the Weil restriction from K to Q of G, and G “ Q R. See “ = G× where G Section 4.3 for the Hodge theoretic preliminaries needed, and Corollary 5.2.2 b “ is the generic Mumford–Tate group of V; for the proof of the fact that G

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• In Section 5, we proved that the totally geodesic subvarieties of SΓ are Z-special, that is they can be described as intersections between SΓ and some “ explicit sub-period domains of G(Z)\D “ (see Section 5.3); G • Here we show that such Z-special subvarieties are actually unlikely intersections, and prove, using tools from functional transcendence for Z-VHS, that the maximal ones appear in a finite number, unless ψ is an isomorphism (which happens if and only if Γ is arithmetic). See also the sketch of the proof presented in Section 1.4. The last item, which is inspired by the arguments of [73], [41], [18], is proven in Section 6.1. We will show that maximal Z-special subvarieties are parametrised by a countable and definable set in some o-minimal structure (as introduced in Section 3.4.2). This is enough to conclude that they form a finite set, by the very definition of o-minimal structure. A key input to show b as recently the countability is the Ax–Schanuel conjecture for the pair (SΓ , V), proven by Bakker and Tsimerman [7] (recalled before as Theorem 5.4.2). From now on we refer simply to special subvarieties of SΓ , bearing in mind that we want to conclude something about the Γ-special subvarieties, by arguing on the Z-special ones. This is where we employ Theorem 1.3.2, which was proven in Section 5.3. 6.1. Proof of Theorem 1.2.1. If SΓ contains no special subvariety, there is nothing to prove. Assume there is a special subvariety W of SΓ of maximal dimension among special subvarieties. To describe W , we will use Theorem 1.3.2 several times. First we see that W is associated to (H, XH , ΓH ⊂ H(OK )), a real sub-Shimura datum of (G, X, Γ), where H is a strict semisimple K-subgroup “ the Q-subgroup of of G, since W is, in particular, Γ-special. Denote by H “ “ the G obtained as a Weil restriction, from K to Q, of H, and denote by H 5 associated real group. 6.1.1. Preliminaries. Before moving on with the argument, we need the following. Lemma 6.1.1. Let (G = PU(1, n), X, Γ) be a real Shimura datum. There exists a compact subset C of SΓ such that every special subvariety S 0 ⊂ SΓ has a non-empty intersection with C . Proof. If Γ is arithmetic, this appears as [13, Lemma 4.5]; for the general case, see [49, Prop. 3.2]. Here it is important that G has rank one, and therefore 5 Here we could consider the Weil restriction from the trace field KH of ΓH , rather than from K, of H, but, as explained in Remark 5.3.2, both groups describe the same Γ-special subvariety. The argument of Section 6.1.2 will, moreover, show that the smaller KH is, the more atypical W is.

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that there is no difference between special (or totally geodesic) subvarieties and what is called in [13] strongly special subvarieties; see Section 4.1 in loc. cit..  Let F ⊂ X be a fundamental set for the action of Γ as in Section 3.4.1, and set C := π −1 (C) ∩ F, where π : X → SΓ is the canonical projection map. Lemma 6.1.2. The set (6.1.1)

˜ “ : Im(ψ(x) “ ⊂ gˆH “gˆ−1 } Π0 (H) := {(x, gˆ) ∈ C × G : S → G)

is definable in an o-minimal structure. To unpack the definition of Π0 (H), recall that ψ˜ : X → D denotes the lift of the period map “ ψ : SΓ an → G(Z)\D “. G By the definition of period domains and Mumford–Tate domains, recalled in Section 4.3.1, every element in D corresponds to a Hodge structure (whose “ In particular, associated Mumford–Tate group is a reductive Q-subgroup of G). ˜ given x ∈ X, ψ(x) corresponds to a map of real algebraic groups ˜ “ ψ(x) : S → G, where S, as in Section 3.1, is the Deligne torus ResC R Gm . From now on, by definable we always mean definable in the o-minimal structure Ran,exp (which was introduced in Section 3.4.2). Proof of Lemma 6.1.2. Thanks to the description of Siegel sets of Section 3.4.3, we can assume that C is the closure of an open semi-algebraic subset of F. Note that the restriction of ψ˜ : X → DG “ to C is holomorphic and therefore ˜ definable in Ran . Eventually the condition on the image of ψ(x) is a definable condition.  Remark 6.1.3. The reason why we work with the compact C, rather than with the fundamental set F, is to avoid discussing the definability of (opportune ˜ However it should be possible to apply the arguments of [6, restrictions of) ψ. Th. 1.5] to prove that ψ˜|F is definable. It would be a consequence of the fact that ψ˜ restricted to D(N ) , where D(N ) ⊂ X was defined in 3.3.2, is definable, at least for N large enough, and that the complement of a compact subset of F is covered by a finite union of D(N ) s. ˜ The condition on the image of ψ(x) appearing in (6.1.1) ensures that we “ by elements gˆ ∈ G “ such that are only considering conjugated of H ˜ “gˆ−1 .ψ(x) gˆH

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˜ “gˆ−1 (C).ψ(x) has a complex structure. In particular, gˆH is an algebraic subvariety ˜ “gˆ−1 .ψ(x). of the flag variety of DG whose intersection with D ˆH That “ “ is g G −1 ˜ “gˆ .ψ(x) is, gˆH is an algebraic subvariety of DG “ in the sense of [47, App. B]. Associated to (x, gˆ) ∈ Π0 (H) there is 0 ˜ “gˆ−1 .ψ(x)) Sx,ˆg := ψ −1 π(ˆ gH ⊂ SΓ an ,

which could happen to be a special subvariety of SΓ , or not. Indeed Sx,ˆg is a “gˆ−1 has a structure of a special subvariety of SΓ only when the real group gˆH “ Q-subgroup of G. In what follows, by dimension we always mean the complex dimension. For ˜ (x, gˆ) ∈ Π0 (H), consider the function computing the local dimension at ψ(x): Ä ä ˜ ˜ “gˆ−1 .ψ(x) dX (x, gˆ) := dimψ(x) gˆH ∩ ψ(X) . ˜ Finally, given 0 < j < n = dim SΓ , we consider the sets (6.1.2)

Πj1 (H) := {(x, gˆ) ∈ Π0 (H) : dX (x, gˆ) ≥ j}, “gˆ−1 : (x, gb) ∈ Πj (H) for some x ∈ C}. Σj = Σ(H)j := {ˆ gH 1

6.1.2. Main proposition and the Z-Ax–Schanuel. Let W be a special subvariety of SΓ of maximal dimension among special subvarieties, say associated to (H, XH , ΓH ⊂ H(OK )). Proposition 6.1.4. Let m = dim W be the maximum of the dimensions of the strict special subvarieties of SΓ . The set Σm = Σ(H)m is finite. Proof of Proposition 6.1.4. It is enough to prove that Σm is definable and countable. First observe that, since the local dimension of a definable set at a point is a definable function, Πm 1 (H) is a definable set. Consider the map “ “(H), “ Πm 1 (H) → G/NG

“ (x, gb) 7→ gb · NG “(H),

“ “ “ where NG “(H) denotes the normaliser of H in G. Such a map is definable, and therefore its image, which is in bijection with Σm , is definable. “gˆ−1 is To prove that Σm is countable, it is enough to show that each gˆH m defined over the rational numbers. We show that such a (x, gˆ) ∈ Σ gives rise to a special subvariety of SΓ ; that is, ˜ ˜ “gˆ−1 .ψ(x) Sx,ˆg = gˆH ∩ ψ(X) is a special subvariety of SΓ . For simplicity, just write D = DG “. Consider the ∨ ∨ algebraic subset of SΓ × D , where D denotes the flag variety associated to D, Ä ä ˜ c := SΓ × gˆH “gˆ−1 .ψ(x) W . ˜ “ be a component at ψ(x) Let U of the intersection c ∩ SΓ × “ W D, G(Z)\D

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“ to SΓ contains Sx,ˆg . We claim that, if Γ is such that the projection of U “ is an unlikely/atypical intersection. That is, non-arithmetic, U Ä ä “ < codimS ×D W c + codimS ×D SΓ × “ (6.1.3) codimSΓ ×D U D . Γ Γ G(Z)\D Assuming the claim, the Z-Ax–Schanuel, which was introduced before as Theo“ to SΓ is contained in a strict special rem 5.4.2, asserts that the projection of U subvariety S 0 of SΓ . We have Sx,ˆg ⊂ S 0 ( SΓ . By definition of Σm , dim Sx,ˆg ≥ m and, since m is the maximum of the dimensions of the strict special subvarieties of SΓ , dim S 0 ≤ m. It follows that “gˆ−1 is defined Sx,ˆg = S 0 . That is, Sx,ˆg is a special subvariety of SΓ and so gˆH over Q. This concludes the proof, assuming equation (6.1.3). To prove the validity of equation (6.1.3), for each embedding σ1 , σ2 , . . . , σr : K → R (ordered in such a way that σ1 is just the identity on K), we set di := dim DGσi , dHi := dim DHσi for any i = 1, . . . , r. Observe that each dHi is smaller than or equal to di . We have r X c dim W = d1 + dHi i=1

and, thanks to the definition of the set Πm 1 (H), we also have “ = dX (x, gˆ) ≥ dH . dim U 1 Putting everything together, we obtain that (6.1.4)

Ä ä “ = codimS ×D W c + codimS ×D SΓ × “ codimSΓ ×D U D Γ Γ G(Z)\D

if and only if “ = dim D − dim D “ + dim D ≤ d1 + dim D − dH , codim U 1 H or simply if and only if, for all i 6= 1, di = dHi = 0. This happens if and only if each Hσi and Gσi are compact groups. By the Mostow–Vinberg arithmeticity criterion, Theorem 2.3.1, it is equivalent to Γ being an arithmetic lattice in G. This concludes the proof of Proposition 6.1.4. 

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6.1.3. End of the proof and induction. We have all the tools to finally prove Theorem 1.2.1, arguing by induction as in the proof of [73, Th. 4.1]. Proof of Theorem 1.2.1. Let m be the maximum of the dimensions of the strict special subvarieties of SΓ . For any j ≤ m, consider the set Ej := {maximal special subvarieties of SΓ of dimension j}. “ there are only a finite number of semisimple Up to conjugation by elements of G, “ Therefore, Proposition 6.1.4 implies that there are only a finite subgroups of G. number of special subvarieties of SΓ of maximal dimension m. That is, the set Em is finite. We argue by induction (downward) on the js for which Ej = 6 ∅. Let F be a semisimple subgroup of G for which there exists a special subvariety associated to (F, XF , ΓF ⊂ F(K)) lying in Ej . Consider the set “gˆ−1 : (x, gb) ∈ Σ(F)j , Sx,bg ∈ Σ2 (F)j := {ˆ gF / Ej 0 , ∀j 0 > j}, where Σ(F)j is defined as in (6.1.2). The inductive assumption implies that the second condition is definable (each Ej 0 being finite). The proof of Proposition 6.1.4 shows that Σ2 (F)j is countable. Since, up to conjugation by elements “ there are only a finite number of possibilities for F, we proved that Ej is of G, finite. This concludes the proof of Theorem 1.2.1.  Remark 6.1.5. Recently Klingler [45] has proposed a generalisation of the Zilber–Pink Conjecture for arbitrary irreducible smooth quasi-projective complex varieties supporting a Z-VHS. It is interesting to notice that his conjecture for b implies Theorem 1.2.1. Given an irreducible subvariety W ⊂ SΓ , the pair (SΓ , V) Klingler introduced a notion of Hodge codimension 6 of W and a corresponding notion of Hodge optimality. Conjecture 1.9 in loc. cit. asserts that there are only finitely many Hodge optimal subvarieties of SΓ . Now (6.1.3), appearing in Section 6.1.2, indeed shows that if Γ is non-arithmetic, then the maximal special subvarieties (of positive dimension) are Hodge optimal, and therefore Klingler’s conjecture predicts Theorem 1.2.1. 6.2. Commensurability criterion for arithmeticity of complex hyperbolic lattice. We conclude this chapter interpreting the commensurability criterion for arithmeticity of Margulis as an unlikely intersection phenomenon. We offer a new proof Theorem 2.4.3, in the complex hyperbolic case, following the strategy used in the proof of Theorem 1.2.1, therefore using no equidistribution/superrigidity techniques. Let (G = PU(1, n), X, Γ ⊂ G(OK )) be a real Shimura datum and SΓ be the associated Shimura variety. 6 Namely, the rank of the horizontal part of the Lie algebra of the generic Mumford–Tate group of W minus the dimension of ψ(W ).

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Definition 6.2.1. A Γ-special correspondence for SΓ is a Γ-special subvariety of SΓ × SΓ whose projections π1 , π2 : SΓ × SΓ → SΓ are finite and surjective. Theorem 6.2.2. Let (G = PU(1, n), X, Γ) be a real Shimura datum. The lattice Γ is arithmetic if and only if SΓ admits infinitely many Γ-special correspondences. Remark 6.2.3. If n = 1, the axiom RSD0 of real Shimura datum implies that Γ is arithmetic and therefore that SΓ admits infinitely many Γ-special correspondences. The proof of Theorem 6.2.2 carries on assuming only that Γ is (cohomologically7 ) rigid. However, we are not able to recover Theorem 2.4.3 for all lattices in PU(1, 1). The diagonal embedding ∆ : SΓ → SΓ × SΓ realises SΓ as a Γ-special correspondence associated to the real sub-Shimura datum (∆(G), ∆(X), ∆(Γ)) ⊂ (G × G, X × X, Γ × Γ). Notice also that ∆(SΓ ) is a strict maximal Γ-special subvariety of SΓ × SΓ . Given g ∈ G, we consider the maps (6.2.1)

ig : G → G × G, f 7→ (f, gf g −1 ),

(6.2.2)

ig : X → X × X, x 7→ (x, g.x).

Let π : X → SΓ be the quotient map. The subvariety π × π(ig (X)) ⊂ SΓ × SΓ is a Γ-special subvariety, in the sense of Definition 3.1.6, if and only if g lies in Comm(Γ). In this case π × π(ig (X)) is indeed the image of ig : SΓ∩ig (Γ) → SΓ × SΓ . If Γ is arithmetic, then it has infinite index in its commensurator, and therefore, in the proof of Theorem 6.2.2 we may assume that Γ is non-arithmetic. Using the map “ “ ψ × ψ : SΓ an × SΓ an → G(Z)\D “ × G(Z)\DG “, G we also have a definition of Z-special correspondences and a similar description. Definition 6.2.4. A Z-special correspondence for SΓ is a Z-special subvariety of SΓ × SΓ whose projections πi : SΓ × SΓ → SΓ are finite and surjective. The arguments appearing in the proof of Theorem 1.3.2, namely in Section 5.3, also show the following. Proposition 6.2.5. The Γ-special correspondences are precisely the Zspecial ones. 7 Over a one dimensional base, all rigid systems are known to be cohomologically rigid [43, Th. 1.1.2, Cor. 1.2.4].

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From now on we just speak of special correspondences for SΓ . We are ready to prove Theorem 6.2.2. 6.2.1. Proof of Theorem 6.2.2. As in Section 6.1, let C ⊂ F ⊂ X be a definable compact set such that XH ∩ C = 6 ∅ for any real sub-Shimura datum “ we write (H, XH , ΓH ) ⊂ (G, X, Γ). Given gˆ ∈ G, Y gˆ = (g1 , . . . , gr ) ∈ Gσi , i=1,...,r

where r = [K : Q] and σi : K → R are ordered in such a way that σ1 induces the identity on K. Consider the map “→ G “ × G, “ igˆ : G

ˆ gˆfˆgˆ−1 ), fˆ 7→ (f,

“gˆ for its image. Consider the set and write ∆ (6.2.3) ˜ “ : g1 .x ∈ C, Im((ψ˜ × ψ)(x, “ × G) “ ⊂∆ “gˆ}. Π0 := {(x, gˆ) ∈ C × G g1 .x) : S → G By Lemma 6.1.2, Π0 is a definable set. Associated to (x, gˆ) ∈ Π0 there is Ä ä ˜ ˜ 1 .x)) 0 ⊂ SΓ an × SΓ an , “gˆ.(ψ(x), Cx,ˆg := ψ −1 × ψ −1 π × π(∆ ψ(g which could happen to be a special correspondence for SΓ , or not. Moreover any special correspondence for SΓ arises as a Cx,ˆg for some (x, gˆ) ∈ Π0 . Define the local dimension Ä ä ˜ ˜ ˜ “ ˜ c(x, gˆ) := dim(ψ(x), ˜ ˜ 1 .x)) ∆gˆ .(ψ(x), ψ(g1 .x)) ∩ ψ × ψ(X × X) . ψ(g The set “gˆ : (x, gb) ∈ Π0 for some x ∈ C, c(x, gˆ) = n} Σ := {∆ is definable. To show that Σ is countable, we argue as in the proof of Proposi“gˆ comes tion 6.1.4. More precisely, we use Z-AS, Theorem 5.4.2, to prove that ∆ “ × G. “ from a Q-subgroup of G Assume that Γ is non-arithmetic, and consider the algebraic subset of SΓ × SΓ × DG “ × DG “, Ä ä ˜ ˜ 1 .x)) . c := SΓ × SΓ × ∆ “gˆ.(ψ(x), W ψ(g ˜ ˜ 1 .x)) of the intersection “ be a component at (ψ(x), Let U ψ(g ä Ä c ∩ SΓ × SΓ × “ W D × D , “ “ “ G G G(Z)\D×G(Z)\D “ to SΓ × SΓ contains Cx,ˆg . A direct computation such that the projection of U shows that, if Γ is non-arithmetic, such an intersection is unlikely and the Z-Ax– Schanuel, namely Theorem 5.4.2, implies that Cx,ˆg for (x, gˆ) ∈ Σ is a special “gˆ is defined over the rational numbers. correspondence of SΓ . It follows that ∆

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We proved that Σ is definable and countable, therefore finite. That is, if Γ is non-arithmetic, then SΓ contains only finitely many special correspondences. 7. Special points and their Zariski closure Let (G = PU(1, n), X, Γ ⊂ G(OK )) be a real Shimura datum and SΓ = Γ\X be the associated Shimura variety. We end the paper presenting several notions of special point of SΓ , inspired by the distribution of special points of arithmetic Shimura varieties. We propose four different definitions of special points, and we formulate four André–Oort-like conjectures relating special points to special subvarieties in the sense of Definition 5.3.5. (These were studied in details in the previous sections.) For an introduction to the André–Oort conjecture for (arithmetic) Shimura varieties, we refer, for example, to [48]. 7.1. Γ-special points. We first introduce the notion of zero dimensional Γ-special subvariety of SΓ . Here we denote by K the totally real field associated to Γ and by G the K-form of G, given by Theorem 1.3.1. Definition 7.1.1. A point x ∈ X is pre-Γ-special if the smallest K-subgroup of G whose extension to the real numbers contains the image of x : S → GR is commutative. A point s ∈ SΓ is Γ-special if it is the image along π : X → SΓ of a pre-Γ-special point x ∈ X. The following can be proven as in the classical case of arithmetic Shimura varieties. From now on, we fix an algebraic closure of the field of rational numbers, denoted by Q. Proposition 7.1.2. Every Γ-special subvariety contains an analytically dense set of Γ-special points. Pre-Γ-special point are defined over Q, with respect to the Q-structure given by looking at X inside its associated flag variety X ∨ . Proof. Let SΓH be a Γ-special subvariety associated to (H, XH , ΓH ). First notice that if one Γ-special point exists in SΓH , then there is a dense set of Γ-special points. Indeed each point in π(H(K).x) ⊂ SΓH is again Γ-special and the set H(K).x is dense by the usual approximation property. For the existence of a Γ-special point, we argue as in [20, §5]. Let x ∈ XH and T ⊂ G be a maximal torus containing the image of x : S → H. Write T as the centraliser of some regular element λ of the Lie algebra of T . Choose λ0 ∈ H(K) sufficiently close to λ. It is still a regular element and its centralizer T0 in G is a maximal torus in G. Since there are only finitely many

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conjugacy classes of maximal real tori in G, we may assume that T 0 and T are conjugated by some h ∈ H. The point π(hx) is a Γ-special point in π(XH ). Finally if x ∈ X is pre-Γ-special, we can factorise its associated Hodge co-character as hx : C∗ → (Tx )C , where Tx is a maximal K-torus. The above factorisation has to hold even over Q. Therefore the point associated to x in the flag variety is fixed by Tx (Q). This is enough to conclude that x is defined over Q. For a complete discussion, we refer to [75, Prop. 3.7].  As in the case of arithmetic Shimura varieties, the Γ-Ax–Schanuel proven in Section 5.4, or more precisely Corollary 5.5.1, could be helpful in proving the following. Conjecture 7.1.3 (Γ-André–Oort). An irreducible subvariety W ⊂ SΓ is Γ-special if it contains a Zariski dense set of Γ-special points. 7.2. Z-special points. Another natural possibility is to look at zero dimensional intersections between ψ(SΓ an ) and sub-domains of the Mumford–Tate “ domain G(Z)\D “ constructed in Theorem 1.3.1; see also Remark 6.1.5. That G b where V b is is the zero dimensional components of the Hodge locus of (S, V), the Z-VHS introduced in Theorem 1.3.1. Definition 7.2.1. A point s ∈ SΓ is Z-special if it is a zero dimensional Z-special subvariety. A first relation between a Z-special point and a Γ-special point is given by the following. Proposition 7.2.2. Let s = π(x) ∈ SΓ be a Z-special point. Then s is Γ-special (in the sense of Definition 7.1.1). “ inducing a Mumford–Tate sub-domain Proof. Let R be a Q-subgroup of G an DR of DG “. If R(Z)\DR intersects ψ(SΓ ) in a finite number of points, then “ R ∩ Gσ1 is a compact subgroup of G(R). Let x ∈ ψ˜−1 (DR ) ⊂ X. It corresponds to a map x:S→G and, by construction, x(S) is contained in R ∩ Gσ1 . Since G is of adjoint type, x(S) is contained in the centre of R ∩ Gσ1 . However R ∩ Gσ1 has a natural structure of a K-subgroup of G, which we denote by Rσ . It follows that x(S) is contained in a commutative K-subgroup of G. That is, MT(x) is commutative and x is pre-Γ-special. 

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In rank one, the Γ-AO Conjecture, thanks to Proposition 7.2.2, predicts that the irreducible components of the Zariski closure of a set of Z-special points are Z-special subvarieties of SΓ . Proposition 7.2.3. If Γ is non-arithmetic, Z-special points of SΓ are an “ atypical intersections in G(Z)\D “ between ψ(SΓ ) and Mumford–Tate subG domains of DG “. “ be as in the proof of Proposition 7.2.2. The intersection Proof. Let R ⊂ G between R(Z)\DR and ψ(SΓ an ) is typical when codimG(Z)\D {s} = codimG(Z)\D SΓ + codimG(Z)\D R(Z)\DR , “ “ “ “ G

“ G

“ G

an

where s is one of the finite points lying in R(Z)\DR ∩ ψ(SΓ ). Let r = [K : Q]. Equivalently, ! r r r X X X di = di + d1 + di − dRi , i=1

i=2

i=2

where di and dRi are as defined in the proof of Proposition 6.1.4. That is, s is typical if and only if di = dRi for all i ≥ 2, which is impossible unless “ R = G.  As usual we expect that a subvariety of SΓ having a dense set of atypical points is atypical. Conjecture 7.2.4. Let (G, X, Γ) be a real Shimura datum and W ⊂ SΓ an irreducible algebraic subvariety. Then W contains a Zariski dense set of Z-special points if and only if it is special and arithmetic. We can also observe that the intersection of Γ-special subvarieties gives rise to Z-special, and therefore Γ-special, points. Proposition 7.2.5. Let SΓ1 , SΓ2 be Γ-special subvarieties of positive dimension in SΓ intersecting in a finite number of points. If s ∈ SΓ1 ∩ SΓ2 , then s is Z-special. Proof. Since SΓ1 and SΓ2 are also Z-special varieties, every point in the intersection can be written as the the preimage along the period map of the intersection of two Mumford–Tate sub-domains. The point s is therefore Zspecial.  7.3. Complex multiplication points. Recall that, as in Theorem 1.3.1, we b for the Z-VHS on SΓ induced by V. Another interesting class of points write V is given by the following. Definition 7.3.1. A point s ∈ SΓ is called a CM-point if the Mumford–Tate b at s is commutative. group of V

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Such points are both Z and Γ-special, but they are even more special. The following is a special case of [45, Conj. 5.6], and it is indeed predicted by, the more difficult, Conjecture 7.2.4. Conjecture 7.3.2. Let (G, X, Γ) be a real Shimura datum and W ⊂ SΓ an irreducible algebraic subvariety. Then W contains a Zariski dense set of CM-points if and only if it is special and arithmetic. If Γ is an arithmetic lattice, this is the classical André-Oort conjecture. If Γ “ is non-arithmetic, but G(Z)\D “ is a Shimura variety, then the above conjecture G follows from the André–Oort conjecture for Shimura varieties of abelian type, which is now a theorem [71]. 7.4. Bi-arithmetic points. In the classical case of arithmetic Shimura varieties, another option is to look at bi-arithmetic points and bi-arithmetic subvarieties; that is, points x ∈ X(Q) such that π(x) ∈ SΓ (Q) and subvarieties W ⊂ SΓ that are defined over Q and such that some component of π −1 (W ) is algebraic and defined over some number field. See also [48, §4.2]. Let (G = PU(1, n), X, Γ) be a real Shimura datum.8 We briefly show how to prove that SΓ and its positive dimensional special subvarieties S 0 ⊂ SΓ have a model over a number field. This follows again from the rigidity of lattices. 7.4.1. Models. Let S be a smooth complex quasi-projective variety. We say that S admits a Q-model if there exists Y /Q such that Y ×Q C ∼ = S, with respect 0 to some embedding Q ,→ C. Two Q-models Y, Y are certainly isomorphic, both over C and Q. We say that S has a unique Q-structure if S admits a Q-model and any two Q-models of S are uniquely isomorphic. For example, if Γ is cocompact, then [67, Th. 1] and [10, Th. 1] imply the following.9 Theorem 7.4.1 (Shimura, Calabi–Vesentini). Let (G = PU(1, n), X, Γ) be a real Shimura datum such that SΓ is projective. Let Θ be the sheaf of germs of holomorphic sections of the tangent bundle of SΓ . If the dimension of SΓ is greater or equal to two, then for i = 0, 1, H i (SΓ , Θ) = 0. It follows that SΓ has a unique Q-structure. If Γ is arithmetic, the above proof can be generalised to cover the case when SΓ is not compact by using Mumford’s theory [61]; see, for example, [33], [63]. 8 From axiom RSD0, a real Shimura datum (G = PU(1, 1), X, Γ) corresponds to a Shimura curve (for the standard definition). 9 Actually Shimura does not discuss the uniqueness of the Q-structure.

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With the tools described in Section 3.3 it should be possible to generalise such arguments to the case of non-arithmetic lattices. For length reasons, we prefer to give a different argument that uses only the fact that all lattices in PU(1, n), for n > 1, have entries in some number field, as explained in Section 2.2, even if such a point of view does not address the uniqueness of the Q-structure. 7.4.2. The action of the automorphisms of C on a Shimura variety. Theorem 7.4.2. Let (G, X, Γ) be a real Shimura datum, and let σ ∈ Aut(C). There exists a real Shimura datum (σ G, σ X, σ Γ) such that σ

SΓ ∼ = Sσ Γ .

Moreover, • G has rank one if and only if σ G has rank one; • Γ is arithmetic if and only if σ Γ is arithmetic. Proof. In the arithmetic case, the result follows from the work of Borovoi and Kazdhan [44]. To prove the result, we may and do replace Γ by a finite index subgroup. In the non-arithmetic case, thanks to the work of Margulis (Theorem 2.4.1), the first part of the above statement reduces to the case of ball quotients (of dimension strictly bigger than one). The result follows from fact that the Chern numbers of SΓ satisfy the equality from Yau’s solution of the Calabi conjecture (in the cocompact case, and by Tsuji [72] in the general case), which characterise ball quotients by their Chern numbers. See, for example, [58, proof of the reduction, p. 257] and references therein. Indeed, let S be a smooth complex projective algebraic of dimension n > 1, and let D be a smooth divisor on S. Assume that (1) KS +(1−)D is ample for every sufficiently small positive rational number ; (2) KS +D is numerically trivial on D and it is ample modulo D and semi-ample (see [72, Def. 1.2]). Tsuji [72, Th. 1] proved that the inequality holds, (7.4.1)

cn1 (Ω1S (log D)) ≤

2n + 2 n−2 1 c1 (ΩS (log D))c2 (Ω1S (log D)), n

and the equality holds if and only if S − D is an unramified quotient of the unit ball Bn in Cn ; that is, if and only if S − D is a ball quotient of the form SΓ0 for some lattice Γ0 in PU(1, n). Let S be the toroidal compactification of SΓ , as described in Section 3.3, and σ let σ ∈ Aut(C). We have that S satisfies (1) and (2) above, since such properties were satisfied by S and are preserved by the action of Aut(C). Since the Chern σ numbers of S are also preserved under the conjugate of a Galois action, S also satisfies the equality in (7.4.1). Therefore it is the toroidal compactification of another ball quotient Sσ Γ , associated to (σ G = G, σ X = X = Bn , σ Γ) for some

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other lattice σ Γ in G. This proves the first part of Theorem 7.4.2, and the first bullet of the second part (since G has rank one if and only if X is a complex ball). We are left to check that Γ is arithmetic if and only if σ Γ is arithmetic. Notice that elements in the commensurator of Γ, denoted by Comm(Γ), are in bijection with special algebraic correspondences of SΓ × SΓ , in the sense of Section 6.2. Special correspondences are preserved by the action of Aut(C), as observed in the proof of [28, Prop. 1.2]. Therefore σ −1 induces an isomorphism Comm(Γ)/Γ ∼ = Comm(σ Γ)/σ Γ. By the Margulis commensurability criterion (as recalled in Sections 2.4.3 and 6.2), we conclude that Γ is arithmetic if and only if σ Γ is arithmetic. Theorem 7.4.2 is eventually proven.  Remark 7.4.3. To complete our discussion of the action of Aut(C), we should explain what happens to special subvarieties of positive dimension of SΓ (i.e., in the sense of Definition 5.3.5). Indeed, it is a well-known fact that, in the theory of arithmetic Shimura varieties, Aut(C) maps special subvarieties of SΓ to special subvarieties of Sσ Γ . Thanks to the description of special subvarieties of SΓ as the Hodge locus of a Z-VHS, as we established in Corollary 5.5.2, we can see that σ ∈ Aut(C) maps Z-special subvarieties of dimension > 0 of SΓ to Z-special subvarieties of Sσ Γ as a special case of the main result of [46]. Theorem 7.4.4. Let (G, X, Γ) be a real Shimura datum. Then SΓ admits a model over a number field. Proof. If Γ is arithmetic, the result follows, for example, from Faltings [33]. Therefore we may and do assume that G = PU(1, n) for some n > 1. Let SΓ BB be the Baily–Borel compactification of SΓ , as discussed in Section 3.3. From [36, Criterion 1, [p. 3], the following are known to be equivalent: (a) SΓ BB can be defined over Q; (b) the set {σ SΓ BB : σ ∈ Aut(C/Q)} contains only finitely many isomorphism classes of complex projective varieties; (c) the set {σ SΓ BB : σ ∈ Aut(C/Q)} contains only countably many isomorphism classes of complex projective varieties. Since the action of Aut(C) on the Baily–Borel compactification of SΓ preserves the smooth locus, it is enough to check that {σ SΓ : σ ∈ Aut(C/Q)} contains countably many isomorphism classes. By Theorem 7.4.2, it is enough to show that there are most countably many lattices in G = PU(1, n) (for some n > 1 fixed), up to conjugation by elements in G. But this follows from the fact that complex hyperbolic lattices enjoy local rigidity, as discussed in Section 2.2. 

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Finally, Remark 7.4.3 shows that the positive dimensional special subvarieties of SΓ are also defined over some number field. 7.4.3. Zariski closure of bi-arithmetic points. The last conjecture we propose is as follows. Conjecture 7.4.5. An irreducible subvariety W ⊂ SΓ contains a dense set of bi-arithmetic points if and only if it is bi-arithmetic. This is known to be equivalent to the André–Oort conjecture only for arithmetic Shimura varieties of abelian type (thanks to Wüstholz’ analytic subgroup theorem). See [48, Ex. 4.17] and references therein. Finally it would be interesting to show that CM-points in SΓ , in the sense of Definition 7.3.1, are bi-arithmetic.

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(Received: July 17, 2020) (Revised: February 2, 2022) I.H.E.S., Université Paris-Saclay, CNRS, Laboratoire Alexandre Grothendieck. 35 Route de Chartres, 91440 Bures-sur-Yvette, France E-mail : [email protected] I.H.E.S., Université Paris-Saclay, CNRS, Laboratoire Alexandre Grothendieck. 35 Route de Chartres, 91440 Bures-sur-Yvette, France E-mail : [email protected]