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Hodge theory, between algebraicity and transcendence

The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, mainly due to the introduction of o- minimal geometry as a natural framework for Hodge theory.


Introduction
Let X be a smooth connected projective variety over C, and X an its associated compact complex manifold. Classical Hodge theory [H51] states that the Betti (i.e. singular) cohomology group H k B (X an , Z) is a polarizable Z-Hodge structure of weight k: there exists a canonical decomposition (called the Hodge decomposition) of complex vector spaces H k B (X an , Z) ⊗ Z C = p+q=k H p,q (X an ) satisfying H p,q (X an ) = H q,p (X an ) and a (−1) k -symmetric bilinear pairing q k : H k B (X an , Z) × H k B (X an , Z) → Z whose complexification makes the above decomposition orthogonal, and satisfies the positivity condition (the signs are complicated but are imposed to us by geometry) i p−q q k,C (α, α) > 0 for any nonzero α ∈ H p,q (X an ).
Deligne [De71a] vastly generalized Hodge's result, showing that the cohomology H k B (X an , Z) of any complex algebraic variety X is functorially endowed with a slightly more general mixed Z-Hodge structure, that makes, after tensoring with Q, H k B (X an , Q) a successive extension of polarizable Q-Hodge structures, with weights between 0 and 2k. As mixed Q-Hodge structures form a Tannakian category MHS Q , one can conveniently (although rather abstractly) summarise the Hodge-Deligne theory as functorially assigning to any complex algebraic variety X a Q-algebraic group: the Mumford-Tate group MT X of X, defined as the Tannaka group of the Tannakian subcategory H • B (X an , Q) of MHS Q generated by H • B (X an , Q). The knowledge of the group MT X is equivalent to the knowledge of all Hodge tensors for the Hodge structure H • B (X an , Q). These apparently rather innocuous semi-linear algebra statements are anything but trivial. They have become the main tool for analysing the topology, geometry and arithmetic of complex algebraic varieties. Let's illustrate what we mean with regard to topology, which we won't go into later. The existence of the Hodge decomposition for smooth projective complex varieties, which holds more generally for compact Kähler manifolds, imposes many constraints on the cohomology of such spaces, the most obvious being that their odd Betti numbers have to be even. Such constraints are not satisfied even by compact complex manifolds as simple as the Hopf surfaces, quotients of C 2 \ {0} by the action of Z given by multiplication by λ = 0, |λ| = 1, whose first Betti number is one. Characterising the homotopy types of compact Kähler manifolds is an essentially open question, which we won't discuss here.
The mystery of the Hodge-Deligne theory lies in the fact that it is at heart not an algebraic theory, but rather the transcendental comparison of two algebraic structures. For simplicity let X be a smooth connected projective variety over C. The Betti cohomology H • B (X an , Q) defines a Q-structure on the complex vector space of the algebraic de Rham cohomology H • dR (X/C) := H • (X, Ω • X/C ) under the transcendental comparison isomorphism: (1.1) ̟ : where the first canonical isomorphism is the comparison between algebraic and analytic de Rham cohomology provided by GAGA, and the second one is provided by integrating complex C ∞ differential forms over cycles (de Rham's theorem). The Hodge filtration F p on H • B (X an , Q) ⊗ Q C is the image under ̟ of the algebraic filtration F p = Im(H • (X, Ω •≥p X/C ) → H • dR (X/C)) on the left hand side. The surprising power of the Hodge-Deligne theory lies in the fact that, although the comparison between the two algebraic structures is transcendental, this transcendence should be severely constrained, as predicted for instance by the Hodge conjecture and the Grothendieck period conjecture: -For X smooth projective, it is well-known that the cycle class [Z] of any codimension k algebraic cycle on X with Q coefficients is a Hodge class in the Hodge structure H 2k (X an , Q)(k). Hodge [H51] famously conjectured that the converse holds true: any Hodge class in H 2k (X, Q)(k) should be such a cycle class.
-For X smooth and defined over a number field K ⊂ C, its periods are the coefficients of the matrix of Grothendieck's isomorphism (generalising (1.1)) (X an , Q) ⊗ Q C with respect to bases of H • dR (X/K) and H • B (X an , Q). The Grothendieck period conjecture (combined with the Hodge conjecture) predicts that the transcendence degree of the field k X ⊂ C generated by the periods of X coincides with the dimension of MT X .
This tension between algebraicity and transcendence is perhaps best revealed when considering Hodge theory in families, as developed by Griffiths [G70a]. Let f : X → S be a smooth projective morphism of smooth connected quasi-projective varieties over C. Its complex analytic fibers X an s , s ∈ S an , are diffeomorphic, hence their cohomologies V Z,s := H • B (X an s , Z), s ∈ S an are all isomorphic to a fixed abelian group V Z and glue together into a locally constant sheaf V Z := R • f an * Z on S an . However the complex algebraic structure on X s , hence also the Hodge structure on V Z,s , varies with s, making R • f an * Z a variation of Z-Hodge structures (ZVHS) V on S an , which can be naturally polarised. One easily checks that the Mumford-Tate group G s := MT Xs , s ∈ S an , is locally constant equal to the socalled generic Mumford-Tate group G, outside of a meagre set HL(S, f ) ⊂ S an , the Hodge locus of the morphism f , where it shrinks as exceptional Hodge tensors appear in H • B (X an s , Z). The variation V is completely described by its period map Φ : S an → Γ\D.
Here the period domain D classifies all possible Z-Hodge structure on the abelian group V Z , with a fixed polarisation and Mumford-Tate group contained in G; and Φ maps a point s ∈ S an to the point of D parameterizing the polarized Z-Hodge structure on V Z defined by V Z,s (well-defined up to the action of the arithmetic group Γ := G ∩ GL(V Z )).
The transcendence of the comparison isomorphism (1.1) for each fiber X s is embodied in the fact that the Hodge variety Γ\D is, in general, a mere complex analytic variety not admitting any algebraic structure; and that the period map Φ is a mere complex analytic map. On the other hand this transcendence is sufficiently constrained so that the following corollary of the Hodge conjecture [Weil79] holds true, as proven by Cattani-Deligne-Kaplan [CDK95]: the Hodge locus HL(S, f ) is a countable union of algebraic subvarieties of S. Remarkably, their result is in fact valid for any polarized ZVHS V on S an , not necessarily coming from geometry: the Hodge locus HL(S, V ⊗ ) is a countable union of algebraic subvarieties of S.
In this paper we report on recent advances in the understanding of this interplay between algebraicity and transcendence in Hodge theory, our main object of interest being period maps Φ : S an → Γ\D. The paper is written for non-experts: we present the mathematical objects involved, the questions, and the results but give only vague ideas of proofs, if any. It is organised as follows. After the Section 2 presenting the objects of Hodge theory (which the advanced reader will skip to refer to on occasion), we present in Section 3 the main driving force behind the recent advances: although period maps are very rarely complex algebraic, their geometry is tame and does not suffer from any of the many possible pathologies of a general holomorphic map. In model-theoretic terms, period maps are definable in the o-minimal structure R an,exp . In Section 4, we introduce the general format of bialgebraic structures for comparing the algebraic structure on S and the one on (the compact dualĎ of) the period domain D. The heuristic provided by this format, combined with o-minimal geometry, leads to a powerful functional transcendence result: the Ax-Schanuel theorem for polarized ZVHS. It also suggests to interprete variational Hodge theory as a special case of an atypical intersection problem. In Section 5 we describe how this viewpoint leads to a stunning improvement of the result of Cattani, Deligne and Kaplan: in most cases HL(S, V ⊗ ) is not only a countable union of algebraic varieties, but is actually algebraic on the nose (at least if we restrict to its components of positive period dimension). Finally in Section 6 we turn briefly to some arithmetic aspects of the theory.
For the sake of simplicity we focus on the case of pure Hodge structures, only mentioning the references dealing with the mixed case. Let n ∈ Z. Let R = Z, Q or R. An R-Hodge structure V of weight n is a finitely generated R-module V R together with one of the following equivalent data: a bigrading V C (:= V R ⊗ R C) = p+q=n V p,q , called the Hodge decomposition, such that V p,q = V q,p (the numbers (dim V p,q ) p+q=n are called the Hodge numbers of V ); or a decreasing filtration F • of V C , called the Hodge filtration, satisfying F p ⊕ F n+1−p = V C . One goes from one to the other through F p = r≥p V r,n−r and V p,q = F p ∩ F q . The following group-theoretic description will be most useful to us: a Hodge structure is an R-module V R and a real algebraic representation ϕ : S → GL(V R ) whose restriction to G m,R is defined over Q. Here the Deligne torus S denotes the real algebraic group C * of invertible matrices of the forms a −b b a , which contains the diagonal subgroup G m,R . Being of weight n is the requirement that ϕ |G m,R acts via the character z → z −n . The space V p,q is recovered as the eigenspace for the character z → z −p z −q of S(R) ≃ C * . A morphism of Hodge structures is a morphism of R-modules compatible with the bigrading (equivalently, with the Hodge filtration or the S-action).
Example 2.1. We write R(n) for the unique R-Hodge structure of weight −2n, called the Tate-Hodge structure of weight −2n, on the rank one free R-module (2πi) n R ⊂ C.
One easily checks that the category of R-Hodge structures is an abelian category (where the kernels and cokernels coincide with the usual kernels and cokernels in the category of R-modules, with the induced Hodge filtrations on their complexifications), with natural tensor products V ⊗ W and internal homs hom(V, W ) (in particular duals V ∨ := hom(V, R(0))). For R = Q or R we obtain a Tannakian category, with an obvious exact faithful R-linear tensor functor ω : (V R , ϕ) → V R . In particular R(n) = R(1) ⊗n . If V is an R-Hodge structure we write V (n) := V ⊗R(n) its n-th Tate twist.
If V = (V R , ϕ) is an R-Hodge structure of weight n, a polarisation for V is a morphism of R-Hodge structures q : V ⊗2 −→ R(−n) such that (2πi) n q(x, ϕ(i)y) is a positive-definite bilinear form on V R , called the Hodge form associated with the polarisation. If there exists a polarisation for V then V is said polarizable. One easily checks that the category of polarizable Q-Hodge structures is semi-simple.
Example 2.2. Let M be a compact complex manifold. If M admits a Kähler metric, the singular cohomology H n B (M, Z) is naturally a Z-Hodge structure of weight n, see [H51], [V02,Ch.6]: where H • dR (M, C) denotes the de Rham cohomology of the complex (A • (M, C), d) of C ∞ differential forms on M , the first equality is the canonical isomorphism obtained by integrating forms on cycles (de Rham theorem), and the complex vector subspace H p,q (M ) of H n dR (M, C) is generated by the d-closed forms of type (p, q), and thus satisfies automatically H p,q (M ) = H q,p (M ). Although the second equality depends only on the complex structure on M , its proof relies on the choice of a Kähler form ω on M through the following sequence of isomorphisms: where H n ∆ω (M ) denotes the vector space of ∆ ω -harmonic differential forms on M and H p,q ∆ω (M ) its subspace of ∆ ω -harmonic (p, q)-forms. The heart of Hodge theory is thus reduced to the statement that the Laplacian ∆ ω of a Kähler metric preserves the type of forms. The choice of a Kähler form ω on M also defines, through the hard Lefschetz theorem [V02,Theor. 6.25], a polarisation of the R-Hodge structure Example 2.3. Suppose moreover that M = X an is the compact complex manifold analytification of a smooth projective variety X over C. In that case H n B (X, Z) is a polarizable Z-Hodge structure. Indeed, the Kähler class [ω] can be chosen as the first Chern class of an ample line bundle on X, giving rise to a rational Lefschetz decomposition and (after clearing denominators by multiplying by a sufficiently large integer) to an integral polarisation. Moreover the Hodge filtration F • on H n B (X an , C) can be defined algebraically: upon identifying H n B (X an , C) with the algebraic de Rham cohomology H n dR (X/C) := H n (X, Ω • X/C ), the Hodge filtration is given by F p = Im(H n (X, Ω •≥p X/C ) → H n B (X an , C)). It follows that if X is defined over a subfield K of C, then the Hodge filtration F • on H n B (X an , C) = H n dR (X/K) ⊗ K C is defined over K.
Example 2.4. The functor which assigns to a complex abelian variety A its H 1 B (A an , Z) defines an equivalence of categories between abelian varieties and polarizable Z-Hodge structures of weight 1 and type (1, 0) and (0, 1).

2.2.
Hodge classes and Mumford-Tate group. Let R = Z or Q and let V be an R-Hodge structure. A Hodge class for V is a vector in V 0,0 ∩ V Q = F 0 V C ∩ V Q . For instance, any morphism of R-Hodge structures f : V → W defines a Hodge class in the internal hom(V, W ). Let T m,n V Q denote the Q-Hodge structure V ⊗m Q ⊗ hom(V, R(0)) ⊗n Q . A Hodge tensor for V is a Hodge class in some T m,n V Q . The main invariant of an R-Hodge structure is its Mumford-Tate group. For any R-Hodge structure V we denote by V the Tannakian subcategory of the category of Q-Hodge structures generated by V Q ; in other words V is the smallest full subcategory containing V , Q(0) and stable under ⊕, ⊗, and taking sub-quotients.
If ω V denotes the restriction of the tensor functor ω to V , the functor Aut ⊗ (ω V ) is representable by some closed Q-algebraic subgroup G V ⊂ GL(V Q ), called the Mumford-Tate group of V , and ω V defines an equivalence of categories V ≃ The Mumford-Tate group G V can also be characterised as the fixator in GL(V Q ) of the Hodge tensors for V , or equivalently, writing V = (V R , ϕ), as the smallest Q-algebraic subgroup of GL(V Q ) whose base change to R contains the image Im ϕ. In particular ϕ factorises as ϕ : S → G V,R . The group G V is thus connected, and reductive if V is polarizable. See [An92,Lemma2].
On the other hand the polarisation q defines a Hodge class in hom(V ⊗2 Q , Q(−1)) thus G A has to be contained in the group GSp(V Q , q) of symplectic similitudes of V Q with respect to the symplectic form q. Finally If A = E is an elliptic curve, it follows readily that either D = Q and G E = GL 2 , or D is an imaginary quadratic field (E has complex multiplication) and G E = T D , the Q-torus defined by T D (S) = (D ⊗ Q S) * for any Q-algebra S.
2.3. Period domains and Hodge data. Let V Z be a finitely generated abelian group V Z of rank r. Fix a positive integer n, a (−1) n -symmetric bilinear form q Z on V Z and a collection of non-negative integers (h p,q ), (p, q ≥ 0, p + q = n), such that h p,q = h q,p and h p,q = r. Associated with (n, q Z , (h p,q )) we want to define a period domain D classifying Z-Hodge structures of weight n on V Z , polarized by q Z , and with Hodge numbers h p,q . Setting f p = r≥p h r,n−r we first define the compact dualĎ parametrising the finite decreasing filtrations F • on V C satisfying (F p ) ⊥q Z = F n+1−p and dim F p = f p . This is a closed algebraic subvariety of the product of Grassmannians p Gr(f p , V C ). The period domain D ⊂Ď an is the open subset where the Hodge form is positive definite. If G := GAut(V Q , q Q ) denotes the group of similitudes of q Q , one easily checks that G(C) acts transitively onĎ an , which is thus a flag variety for G C ; and that the connected component G := G der (R) + of the identity in the derived group G der (R) acts transitively on D, which identifies with an open G-orbit inĎ. If we fix a base point o ∈ D and denote by P and M its stabiliser in G(C) and G respectively, the period domain D is thus the homogeneous space Example 2.7. Let n = 1, suppose that the only non-zero Hodge numbers are h 1,0 = h 0,1 = g, q Z is a symplectic form and D is the subset of Gr(g, V C ) consisting of q C -Lagrangian subspaces F 1 on which iq C (u, u) is positive definite. In this case G = GSp 2g , G = Sp 2g (R), M = SO 2g (R) is a maximal compact subgroup of the connected Lie group G and D = G/M is a bounded symmetric domain naturally biholomorphic to Siegel's upper half space H g of g × g-complex symmetric matrices Z = X + iY with Y positive definite. When g = 1, D is the Poincaré disk, biholomorphic to the Poincaré upper half space H.
More generally let G be a connected reductive Q-algebraic group and let ϕ : S → G R be a real algebraic morphism such that ϕ |G m,R is defined over Q. We assume that G is the Mumford-Tate group of ϕ. The period domain (or Hodge domain) D associated with ϕ : S → G R is the connected component of the G(R)-conjugacy class of ϕ : S → G R in Hom(S, G R ). Again, one easily checks that D is an open elliptic orbit of G := G der (R) + in the compact dual flag varietyĎ an , the G(C)- The following geometric feature ofĎ will be crucial for us. The algebraic tangent bundle TĎ naturally identifies, as a G C -equivariant bundle, with the quotient vector bundleV Ad /F 0V Ad , where Ad : G → GL(g) is the adjoint representation on the Lie algebra g of G. In particular it is naturally filtered by the F i TĎ := F iV Ad /F 0V Ad , i ≤ −1. The subbundle F −1 TĎ is called the horizontal tangent bundle ofĎ.
is the relative holomorphic de Rham complex and ∇ is the Gauss-Manin connection. The Hodge filtration on each H The Hodge filtration is usually not preserved by the connection, but Griffiths [G68] crucially observed that it satisfies the transversality constraint is a Z-Hodge structure; and the flat connection ∇ on V defined by V C satisfies Griffiths' transversality: In the geometric situation, such a polarisation exists if there exists an element η ∈ H 2 (Y, Z) whose restriction to each fiber Y b defines a Kähler class, for instance if f is the analytification of a smooth projective morphism of smooth connected algebraic varieties over C.
2.6. Generic Hodge datum and period map. Let S be a smooth connected quasi-projective variety over C and let V be a polarized ZVHS on S an . Fix a base point o ∈ S an , let p : S an → S an be the corresponding universal cover and In [De72,7.5], Deligne proved that there exists a reductive Q-algebraic subgroup ι : G ֒→ GL(V Q ), called the generic Mumford-Tate group of V, such that, for all pointss ∈ S an , the Mumford-Tate group G (V Z ,F • s ) is contained in G, and is equal to G outside of a meagre set of S an (such pointss are said Hodge generic for V). A closed irreducible subvariety Y ⊂ S is said Hodge generic for V if it contains a Hodge generic point. The setup of Section 2.3 is thus in force. Without loss of generality we can assume that the point o is Hodge generic. Let (G, D) be the Hodge datum (called the generic Hodge datum of S an for V) associated with the polarized Hodge structure (V Z , F • o ). The ZVHS p * V is completely described by a holomorphic mapΦ : S an → D, which is naturally equivariant under the monodromy representation ρ : π 1 (S an , o) → Γ := G∩GL(V Z ), hence descends to a holomorphic map Φ : S an → Γ\D, called the period map of S for V. We thus obtain the following commutative diagram in the category of complex analytic spaces: ) on the Hodge variety Γ\D defined by the inclusion ι : G ֒→ GL(V Q ). Griffiths' transversality condition is equivalent to the statement that Φ is horizontal : dΦ(T S an ) ⊂ T h (Γ\D). By extension we call period map any holomorphic, horizontal, locally liftable map from S an to a Hodge variety Γ\D.
The Hodge locus HL(S, V ⊗ ) of S for V is the subset of points s ∈ S an for which the Mumford-Tate group G s is a strict subgroup of G, or equivalently for which the Hodge structure V s admits more Hodge tensors than the very general fiber V s ′ . Thus where the union is over all strict Hodge subdata and Γ ′ \D ′ is a slight abuse of notation for denoting the projection of D ′ ⊂ D to Γ\D.
Let Y ⊂ S be a closed irreducible algebraic subvariety i : Y ֒→ S. Let (G Y , D Y ) be the generic Hodge datum of the ZVHS V restricted to the smooth locus of Y . The algebraic monodromy group H Y of Y for V is the identity component of the Zariski-closure in GL(V Q ) of the monodromy of the restriction to Y of the local system V Z . It follows from Deligne's (in the geometric case) and Schmid's (in general) "Theorem of the fixed part" and "Semisimplicity Theorem" that H Y is a normal subgroup of the derived group G der Y , see [An92, Theorem 1].

Hodge theory and tame geometry
3.1. Variational Hodge theory between algebraicity and transcendence. Let S be a smooth connected quasi-projective variety over C and let V = (V Z , F • ) be a polarized ZVHS on S an . Let (G, D) be the generic Hodge datum of S for V and let Φ : S an → Γ\D be the period map defined by V.
The fact that Hodge theory is a transcendental theory is reflected in the following facts: -Second, and more importantly, the complex analytic Hodge variety Γ\D is in general not algebraizable (i.e. it is not the analytification of a complex quasiprojective variety). More precisely, let us write D = G/M as in Section 2.3. A classical property of elliptic orbits like D is that there exists a unique maximal compact subgroup K of G containing M [GS69]. Supposing for simplicity that G is a real simple Lie group G, then Γ\D is algebraizable only if G/K is a hermitian symmetric domain and the projection D → G/K is holomorphic or anti-holomorphic, see [GRT14].
On the other hand this transcendence is severely constrained, as shown by the following algebraicity results: -If (G, D) is of Shimura type, then Γ\D = Sh an is the analytification of an algebraic variety, called a Shimura variety Sh [BB66], [De71b], [De79]. In that case Borel [Bor72,Theor. 3.10] proved that the complex analytic period map Φ : S an → Sh an is the analytification of an algebraic map.
-Let S ⊂ S be a log-smooth compactification of S by a simple normal crossing divisor Z. Following Deligne [Del70], the flat holomorphic connection ∇ on V defines a canonical extension V of V to S. Using GAGA for S, this defines an algebraic structure on (V, ∇), for which the connection ∇ is regular. Around any point of Z, the complex manifold S an is locally isomorphic to a product (∆ * ) k ×∆ l of punctured polydisks. Borel showed that the monodromy representation ρ : π 1 (S an , s o ) → Γ ⊂ G(Q) of V is "tame at infinity", that is, its restriction to Z k = π 1 ((∆ * ) k × ∆ l ) is quasi-unipotent, see [Sc73, Lemma (4.5)]. Using this result, Schmid showed that the Hodge filtration F • extends holomorphically to the Deligne extension V. This is the celebrated Nilpotent Orbit theorem [Sc73,(4.12)]. It follows, as noticed by Griffiths [Sc73,(4.13)]), that the Hodge filtration on V comes from an algebraic filtration on the underlying algebraic bundle, whether V is of geometric origin or not.
-More recently, an even stronger evidence came from the study of Hodge loci. Cattani, Deligne and Kaplan proved the following celebrated result (generalized to the mixed case in [BP09a], [BP09b], [BP13], [BPS10]): ). Let S be a smooth connected quasi-projective variety over C and V be a polarized ZVHS over S. Then HL(S, V ⊗ ) is a countable union of closed irreducible algebraic subvarieties of S.
In view of this tension between algebraicity and transcendence, it is natural to ask if there is a framework, less strict than complex algebraic geometry but more constraining than complex analytic geometry, where to analyse period maps and explain its remarkable properties.
3.2. O-minimal geometry. Such a framework was in fact envisioned by Grothendieck in [Gro84,§5] under the name "tame topology", as a way out of the pathologies of general topological spaces. Examples of pathologies are Cantor sets, space-filling curves but also much simpler objects like the graph Γ : which prevents any reasonable stratification theory; and Γ ∩ R is not "of finite type". Tame geometry has been developed by model theorists as o-minimal geometry, which studies structures where every definable set has a finite geometric complexity. Its prototype is real semi-algebraic geometry, but it is much richer. We refer to [vdD98] for a nice survey.
Definition 3.2. A structure S expanding the real field is a collection S = (S n ) n∈N , where S n is a set of subsets of R n such that for every n ∈ N: (1) all algebraic subsets of R n are in S n .
(2) S n is a boolean subalgebra of the power set of R n (i.e. S n is stable by finite union, intersection, and complement).
(4) Let p : R n+1 → R n be a linear projection. If A ∈ S n+1 then p(A) ∈ S n . The elements of S n are called the S-definable sets of R n . A map f : A → B between S-definable sets is said to be S-definable if its graph is S-definable.
A dual point of view starts from the functions, namely considers sets definable in a first-order structure S = R, +, ×, <, (f i ) i∈I where I is a set and the f i : where φ is a first-order formula that can be written using only the quantifiers ∀ and ∃ applied to real variables; logical connectors; algebraic expressions written with the f i ; the order symbol <; and fixed parameters λ i ∈ R. When the set I is empty the S-definable subsets are the semi-algebraic sets. Semi-algebraic subsets are thus always S-definable.
One easily checks that the composite of S-definable functions is S-definable, as are the images and the preimages of S-definable sets under S-definable maps. Using that the euclidean distance is a real-algebraic function, one shows easily that the closure and interior of an S-definable set are again S-definable.
The following o-minimal axiom for a structure S guarantees the possibility of doing geometry using S-definable sets as basic blocks. Definition 3.3. A structure S is said to be o-minimal if S 1 consists precisely of the finite unions of points and intervals (i.e. the semi-algebraic subsets of R).
Example 3.4. The structure R sin := R, +, ×, <, sin is not o-minimal. Indeed the infinite union of points πZ = {x ∈ R | sin x = 0} is a definable subset of R in this structure.
Any o-minimal structure S has the following main tameness property: given finitely many S-definable sets U 1 , · · · , U k ⊂ R n , there exists a definable cylindrical cellular decomposition of R n such that each U i is a finite union of cells. Such a decomposition is defined inductively on n. For n = 1 this is a finite partition of R into cells which are points or open intervals. For n > 1 it is obtained from a definable cylindrical cellular decomposition of R n−1 by fixing, for any cell C ⊂ R n−1 , finitely many definable functions f C,i : kC +1 := +∞, and defining the cells of R n as the graphs The simplest o-minimal structure is the structure R alg consisting of semi-algebraic sets. It is too close to algebraic geometry to be used for studying transcendence phenomena. Luckily much richer o-minimal geometries do exist. A fundamental result of Wilkie, building on the result of Khovanskii [Khov80] that any expo- has finitely many connected components, states: In another direction, let us define where a function f : R n → R is a restricted real analytic function if it is zero outside [0, 1] n and if there exists a real analytic function g on a neighbourhood of [0, 1] n such that f and g are equal on [0, 1] n . Gabrielov's result [Ga68] that the difference of two subanalytic sets is subanalytic implies rather easily that the structure R an is o-minimal. The structure generated by two o-minimal structures is not o-minimal in general, but Van den Dries and Miller [vdDM94] proved that the structure R an,exp generated by R an and R exp is o-minimal. This is the o-minimal structure which will be mainly used in the rest of this text.
Let us now globalize the notion of definable set using charts: Definition 3.6. A definable topological space X is the data of a Hausdorff topological space X , a finite open covering (U i ) 1≤i≤k of X , and homeomorphisms ψ i : As usual the pairs (U i , ψ i ) are called charts. A morphism of definable topological spaces is a continuous map which is definable when read in the charts. The definable site X of a definable topological space X has for objects definable open subsets U ⊂ X and admissible coverings are the finite ones.
Example 3.7. Let X be an algebraic variety over R. Then X(R) equipped with the euclidean topology carries a natural R alg -definable structure (up to isomorphism): one covers X by finitely many (Zariski) open affine subvarieties X i and take U i := X i (R) which is naturally a semi-algebraic set. One easily check that any two finite open affine covers define isomorphic R alg -structures on X(R). If X is an algebraic variety over C then X(C) = (Res C/R X)(R) carries thus a natural R alg -structure. We call this the R alg -definabilization of X and denote it by X R alg .
In the rest of this section, we fix an o-minimal structure S and write "definable" for S-definable. Given a complex algebraic variety X we write X def for the Sdefinabilization X S .
3.3. O-minimal geometry and algebraization. Why should an algebraic geometer care about o-minimal geometry? Because o-minimal geometry provides strong algebraization results.
3.3.1. Diophantine criterion. The first algebraization result is the celebrated Pila-Wilkie theorem: Theorem 3.8 ( [PW06]). Let Z ⊂ R n be a definable set. We define Z alg as the union of all connected positive-dimensional semi-algebraic subsets of Z. Then, denoting by H : Q n → R the standard height function: In words: if a definable set contains at least polynomially many rational points (with respect to their height), then it contains a positive dimensional semi-algebraic set! For instance, if f : R → R is a real analytic function such that its graph Γ f ∩ [0, 1] × [0, 1] contains at least polynomially many rational points (with respect to their height), then the function f is real algebraic [BoP89]. This algebraization result is a crucial ingredient in the proof of functional transcendence results for period maps, see Section 4.
3.3.2. Definable Chow and definable GAGA. In another direction, algebraicity follows from the meeting of o-minimal geometry with complex geometry. The motto is that o-minimal geometry is incompatible with the many pathologies of complex analysis. As a simple illustration, let f : ∆ * → C be a holomorphic function, and assume that f is definable (where we identify C with R 2 and ∆ * ⊂ R 2 is semi-algebraic). Then f does not have any essential singularity at 0 (i.e. f is meromorphic). Otherwise, by the Big Picard theorem, the boundary Γ f \ Γ f of its graph would contain {0} × C, hence would have the same real dimension (two) as Γ f , contradicting the fact that Γ f is definable.
Let us first define a good notion of a definable topological space "endowed with a complex analytic structure". We identify C n with R 2n by taking real and imaginary parts. Given U ⊂ C n a definable open subset, let O C n (U ) denote the C-algebra of holomorphic definable functions U → C. The assignment U O C n (U ) defines a sheaf O C n on C n whose stalks are local rings. Given a finitely generated ideal I ⊂ O C n (U ), its zero locus V (I) ⊂ U is definable and the restriction O V (I) := (O U /IO U ) |V (I) define a sheaf of local rings on V (I).
Definition 3.9. A definable complex analytic space is a pair (X , O X ) consisting of a definable topological space X and a sheaf O X on X such that there exists a finite covering of X by definable open subsets X i on which (X , O X ) |Xi is isomorphic to some (V (I), O V (I) ).
[BBT18, Theor. 2.16] shows that this is a reasonable definition: the sheaf O X , in analogy with the classical Oka's theorem, is a coherent sheaf of rings. Moreover one has a natural definabilization functor (X, O X ) (X def , O X def ) from the category of separated schemes (or algebraic spaces) of finite type over C to the category of definable complex analytic spaces, which induces a morphism g : (X def , O X def ) → (X, O X ) of locally ringed sites.
Let us now describe the promised algebraization results. The classical Chow's theorem states that a closed complex analytic subset Z of X an for X smooth projective over C is in fact algebraic. This fails dramatically if X is only quasi-projective, as shown by the graph of the complex exponential in (A 2 ) an . However Peterzil and Starchenko, generalising [FL81] in the R alg case, have shown the following: Theorem 3.10 ( [PS09], [PS10]). Let X be a complex quasi-projective variety and let Z ⊂ X an be a closed analytic subvariety. If Z is definable in X def then Z is complex algebraic in X.
Chow's theorem, which deals only with spaces, was extended to sheaves by Serre [Se54]: when X is proper, the analytification functor (·) an : Coh(X) → Coh(X an ) defines an equivalence of categories between the categories of coherent sheaves Coh(X) and Coh(X an ). In the definable world, let X be a separated scheme (or algebraic space) of finite type over C. Associating with a coherent sheaf F on X the coherent sheaf F def := F ⊗ g −1 OX O X def on the S-definabilization X def of X, one obtains a definabilization functor (·) def : Coh(X) → Coh(X def ). Similarly there is an analytification functor X X an from complex definable analytic spaces to complex analytic spaces, that induces a functor (·) an : Coh(X ) → Coh(X an ).
Theorem 3.11 ( [BBT18]). For every separated algebraic space of finite type X, the definabilization functor (·) def : Coh(X) → Coh(X def ) is exact and fully faithful (but it is not necessarily essentially surjective). Its essential image is stable under subobjects and subquotients.
Using Theorem 3.11 and Artin's algebraization theorem for formal modification [Art70], one obtains the following useful algebraization result for definable images of algebraic spaces, which will be used in Section 3.6.2: Theorem 3.12 ( [BBT18]). Let X be a separated algebraic space of finite type and let E be a definable analytic space. Any proper definable analytic map Φ : is a closed immersion of definable analytic spaces.

Definability of Hodge varieties.
Let us now describe the first result establishing that o-minimal geometry is potentially interesting for Hodge theory.
Here "functorial" means that that any morphism (G ′ , D ′ ) → (G, D) of Hodge data induces a definable map (Γ ′ \D ′ ) R alg → (Γ\D) R alg of Hodge varieties. Let us sketch the construction of (Γ\D) R alg . Without loss of generality (replacing G by its adjoint group if necessary) we can assume that G is semi-simple, G = G(R) + . For simplicity let us assume that the arithmetic lattice Γ is torsion free. We choose a base point in D = G/M . Notice that G and G/M ⊂Ď R alg are naturally endowed with a G-equivariant semi-algebraic structure, making the projection G → G/M semi-algebraic. To define an R alg -structure on Γ\(G/M ), it is thus enough to find a semi-algebraic open fundamental set F ⊂ G/M for the action of Γ and to write Γ\G/M = Γ\F , where the right hand side is the quotient of F by the closed etale semi-algebraic equivalence relation induced by the action of Γ on D. Here by fundamental set we mean that the set of γ ∈ Γ such that γF ∩ F = ∅ is finite. We construct the fundamental set F using the reduction theory of arithmetic groups, namely the theory of Siegel sets. Let K be the unique maximal compact subgroup of G containing M . For any Q-parabolic P of G with unipotent radical N, the maximal compact subgroup K of G determines a real Levi L ⊂ G which decomposes as L = AQ where A is the center and Q is semi-simple. A semialgebraic Siegel set of G associated to P and K is then a set of the form S = U (aA >0 )W where U ⊂ N(R), W ⊂ QK are bounded semialgebraic subsets, a ∈ A, and A >0 is the cone corresponding to the positive root chamber. By a Siegel set of G associated to K we mean a semialgebraic Siegel set associated to P and K for some Q-parabolic P of G. Suppose now that Γ ⊂ G is an arithmetic group. A fundamental result of Borel [Bor69] states that there exists finitely many Siegel sets S i ⊂ G, 1 ≤ i ≤ s, associated with K, whose images in Γ\G/K cover the whole space; and such that for any 1 ≤ i = j ≤ s, the set of γ ∈ Γ such that γS i ∩ S j = ∅ is finite. We call the images S i,D := S i /M Siegel sets for D. Noticing that these Siegel sets for D are semi-algebraic in D, we can take F = s i=1 S i,D . It is not difficult to show that the R alg -structure thus constructed is independent of the choice of the base point eM ∈ G/M . The functoriality follows from a non-trivial property of Siegel sets with respect to morphisms of algebraic groups, due to Orr [O18].
3.5. Definability of period maps. Once Theorem 3.13 is in place, the following result shows that o-minimal geometry is a natural framework for Hodge theory: Theorem 3.14 ([BKT20]). Let S be a smooth connected complex quasi-projective variety. Any period map Φ : S an → Γ\D is the analytification of a morphism Φ : S Ran,exp → (Γ\D) Ran,exp of R an,exp -definable complex analytic spaces, where the R an,exp -structures on S(C) and Γ\D extend their natural R alg -structures defined in Example 3.7 and Theorem 3.13 respectively.
In down to earth terms, this means that we can cover S by finitely many open affine charts S i such that Φ restricted to (Res C/R S i )(R) = S i (C) and read in a chart of Γ\D defined by a Siegel set of D, can be written using only real polynomials, the real exponential function, and restricted real analytic functions! This statement is already non-trivial when S = Sh is a Shimura variety and Φ an : S an → Γ\D is the identity map coming from the uniformization π : D → S an of S an by the hermitian symmetric domain D = G/K. In that case the R alg -definable varieties Sh R alg and (Γ\D) R alg are not isomorphic but Theorem 3.14 claims that their R an,expextensions Sh Ran,exp and (Γ\D) Ran,exp are. This is equivalent to showing that the restriction π |SD : S D → S Ran,exp to a Siegel set for D can be written using only real polynomials, the real exponential function, and restricted real analytic functions. This is a nice exercise on the j-function when Sh is a modular curve, was done in [PS13] and [PT14] for Sh = A g , and [KUY16] in general.
Let us sketch the proof of Theorem 3.14. We choose a log-smooth compactification of S, hence providing us with a definable cover of S Ran by punctured polydisks (∆ * ) k × ∆ l . We are reduced to showing that the restriction of Φ to such a punctured polydisk is R an,exp -definable. This is clear if k = 0, as in this case ϕ : ∆ k+l → Γ\D is even R an -definable. For k > 0, let e : exp(2πi·) : H → ∆ * be the universal covering map. Its restriction to a sufficiently large bounded vertical strip V := [a, b]×]0, +∞[⊂ H = {x + iy, y > 0} is R an,exp -definable. Considering the following commutative diagram: Let the coordinates of (∆ * ) k × ∆ l be t i , 1 ≤ i ≤ k + l, those of H k be z i , 1 ≤ i ≤ k, so that e(z i ) = t i . Let T i be the monodromy at infinity of Φ around the hyperplane (z i = 0), boundary component of S \ S. By Borel's theorem T i is quasiunipotent. Replacing S by a finiteétale cover we can without loss of generality assume that each T i = exp(N i ), with N i ∈ g nilpotent. The Nilpotent Orbit Theorem of Schmid is equivalent to saying thatΦ : V k × ∆ l → D can be written as Φ(z 1 , . . . , z k , t k+1 , . . . , t k+l ) = exp( k i=1 z i N i ) · Ψ(t 1 , . . . , t k+l ) for Ψ : ∆ k × ∆ l → D an a holomorphic map. On the one hand Ψ is R an -definable as a function of the variables t i , hence R an,exp -definable as a function of the variables z i , 1 ≤ i ≤ k, and the variables t j , k + 1 ≤ j ≤ k + l . On the other hand exp( k i=1 z i N i ) ∈ G(C) is polynomial in the variables z i , as the monodromies N i are nilpotent and commute pairwise. As the action of G(C) onĎ is algebraic, it follows thatΦ : V k × ∆ l → D is R an,exp -definable. The proof of Theorem 3.14 is thus reduced to the following, proven by Schmid when k = 1, l = 0 [Sc73, 5.29]: Remark 3.16. Theorem 3.13 and Theorem 3.14 have been extended to the mixed case in [BBKT20].
3.6.1. About the Cattani-Deligne-Kaplan theorem. As a corollary of Theorem 3.14 and Theorem 3.10 one obtains the following, which, in view of (2.3), implies immediately Theorem 3.1: Theorem 3.17 ([BKT20]). Let S be a smooth quasi-projective complex variety. Let V be a polarized ZVHS on S an with period map Φ : S an → Γ\D. For any special subvariety Γ ′ \D ′ ⊂ Γ\D its preimage Φ −1 (Γ ′ \D ′ ) is a finite union of irreducible algebraic subvarieties of S.
Indeed it follows from Theorem 3.13 that Γ ′ \D ′ is definable in (Γ\D) R alg . By Theorem 3.14 its preimage Φ −1 (Γ ′ \D ′ ) is definable in S Ran,exp . As Φ is holomorphic and Γ ′ \D ′ ⊂ Γ\D is a closed complex analytic subvariety, Φ −1 (Γ ′ \D ′ ) is also a closed complex analytic subvariety of S an . By Theorem 3.10 it is thus algebraic in S.
Let Y ⊂ S be a closed irreducible algebraic subvariety. Let (G Y , D Y ) ⊂ (G, D) be the generic Hodge datum of V restricted to the smooth locus of of Y . There exist a smallest Hodge subvariety Γ Y \D Y of Γ\D containing Φ(Y an ). The following terminology will be convenient: Definition 3.19. Let S be a smooth quasi-projective complex variety. Let V be a polarized ZVHS on S an with period map Φ : S an → Γ\D. A closed irreducible subvariety Y ⊂ S is called a special subvariety of S for V if it coincides with an irreducible component of the preimage Equivalently, a special subvariety of S for V is a closed irreducible algebraic subvariety Y ⊂ S maximal among the closed irreducible algebraic subvarieties Z of S such that the generic Mumford-Tate group G Z of V |Z equals G Y .
3.6.2. A conjecture of Griffiths. Combining Theorem 3.14 this time with Theorem 3.12 leads to a proof of an old conjecture of Griffiths [G70b], claiming that the image of any period map has a natural structure of quasi-projective variety (Griffiths proved it when the target Hodge variety is compact): Theorem 3.20 ( [BBT18]). Let S be a smooth connected quasi-projective complex variety and let Φ : S an → Γ\D be a period map. There exists a unique dominant morphism of complex algebraic varieties f : S → T , with T quasi-projective, and a closed complex analytic immersion ι : T an ֒→ Γ\D such that Φ = ι • f an .
Let us sketch the proof. As before, let S ⊂ S be a log-smooth compactification by a simple normal crossing divisor Z. It follows from a result of Griffiths [G70a,Prop.9.11i)] that Φ extends to a proper period map over the components of Z around which the monodromy is finite. Hence, without loss of generality, we can assume that Φ is proper. The existence of f in the category of algebraic spaces then follows immediately from Theorem 3.14 and Theorem 3.12 (for S = R an,exp ). The proof that T is in fact quasi-projective exploits a crucial observation of Griffiths that Γ\D carries a positively curved Q-line bundle L := p det(F p ). This line bundle is naturally definable on (Γ\D) def . Using the definable GAGA Theorem 3.11, one shows that its restriction to T def comes from an algebraic Q-line bundle L T on T , which one manages to show to be ample.

Functional transcendence
4.1. Bi-algebraic geometry. As we saw, Hodge theory, which compares the Hodge filtration on H • dR (X/C) with the rational structure on H • B (X an , C), gives rise to variational Hodge theory, whose fundamental diagram (2.2) compares the algebraic structure of S with the algebraic structure on the dual period domainĎ. As such, it is a partial answer to one of the most classical problem of complex algebraic geometry: the transcendental nature of the topological universal cover of complex algebraic varieties. If S is a connected complex algebraic variety, the universal cover S an has usually no algebraic structure as soon as the topological fundamental group π 1 (S an ) is infinite. As an aside, let us mention an interesting conjecture of Kóllar and Pardon [KP12], predicting that if X is a normal projective irreducible complex variety whose universal cover X an is biholomorphic to a semialgebraic open subset of an algebraic variety then X an is biholomorphic to C n × D × F an , where D is a bounded symmetric domain and F is a normal, projective, irreducible, topologically simply connected, complex algebraic variety. We want to think of variational Hodge theory as an attempt to provide a partial algebraic uniformization: the period map emulates an algebraic structure on S an , modeled on the flag varietyĎ. The remaining task is then to describe the transcendence properties of the complex analytic uniformization map p : S an → S an with respects to the emulated algebraic structure on S an and the algebraic structure S on S an . A few years ago, the author [K17], together with Ullmo and Yafaev [KUY18], introduced a convenient format for studying such questions, which encompasses many classical transcendence problems and provides a powerful heuristic.
Definition 4.1. A bi-algebraic structure on a connected quasi-projective variety S over C is a pair where Z denotes an algebraic variety (called the algebraic model of S an ), Aut(Z) is its group of algebraic automorphisms, ρ is a group morphism (called the monodromy representation) and f is a ρ-equivariant holomorphic map (called the developing map).
An irreducible analytic subvariety Y ⊂ S an is said to be an algebraic subvariety of S an for the bi-algebraic structure (f, ρ) if Y is an analytic irreducible component irreducible algebraic subvariety Y ⊂ S an , resp. W ⊂ S, is said to be bi-algebraic if p(Y ) is an algebraic subvariety of S, resp. any (equivalently one) analytic irreducible component of p −1 (W ) is an irreducible algebraic subvariety of S an . The bi-algebraic subvarieties of S are precisely the ones where the emulated algebraic structure on S an and the one on S interact non-trivially. and f is the identity morphism of C n . An irreducible algebraic subvariety Y ⊂ C n (resp. W ⊂ (C * ) n ) is bi-algebraic if and only if Y is a translate of a rational linear subspace of C n = Q n ⊗ Q C (resp. W is a translate of a subtorus of (C * ) n ).
(b) abelian varieties: S = A is a complex abelian variety of dimension n. Let p : Lie A ≃ C n → A be the uniformizing map of a complex abelian variety A of dimension n. Once more S an = C n and f is the identity morphism. One checks easily that an irreducible algebraic subvariety W ⊂ A is bi-algebraic if and only if W is the translate of an abelian subvariety of A. What are its bi-algebraic subvarieties? To answer this question, we need to define the weakly special subvarieties of Γ\D, as either a special subvariety or a subvariety of the form is a Hodge subdatum of (G ad , D) and {t} is a Hodge generic point in Γ L \D L . Generalising Theorem 3.17, the preimage under Φ of any weakly special subvariety of Γ\D is an algebraic subvariety of S, [KO21]. An irreducible component of such a preimage is called a weakly special subvariety of S for V (or Φ).

Theorem 4.3 ([KO21]
). Let Φ : S an → Γ\D be a period map. The bi-algebraic subvarieties of S for the bi-algebraic structure defined by Φ are precisely the weakly special subvarieties of S for Φ. In analogy with Definition 3.19, they are also the closed irreducible algebraic subvarieties Y ⊂ S maximal among the closed irreducible algebraic subvarieties Z of S whose algebraic monodromy group H Z equals H Y .
When S = Sh is a Shimura variety, these results are due to Moonen [Moo98] and [UY11]. In that case the weakly special subvarieties are also the irreducible algebraic subvarieties of Sh whose smooth locus is totally geodesic in Sh an for the canonical Kähler-Einstein metric on Sh an = Γ\D coming from the Bergman metric on D, see [Moo98].
To study not only functional transcendence but also arithmetic transcendence, we enrich bi-algebraic structures over Q. A Q-bi-algebraic structure on a quasiprojective variety S defined over Q is a bi-algebraic structure (f : S an → Z an , h : π 1 (S an ) → Aut(Z)) such that Z is defined over Q and the homomorphism h takes values in Aut Q Z. An algebraic subvariety Y ⊂ S an is said to be defined over Q

if its model f (Y )
Zar ⊂ Z is. A Q-bi-algebraic subvariety W ⊂ S is an algebraic subvariety of S defined over Q and such that any (equivalently one) of the analytic irreducible components of p −1 (W ) is an algebraic subvariety of S an defined over Q.
A Q-bi-algebraic point s ∈ S(Q) is also called an arithmetic point. Example 4.2a) is naturally defined over Q, with arithmetic points the torsion points of (C * ) n . In Example 4.2b) the bi-algebraic structure can be defined over Q if the abelian variety A has CM, and its arithmetic points are its torsion points. Example 4.2c) is naturally a Q-bi-algebraic structure, with arithmetic points the special points of the Shimura variety (namely the special subvarieties of dimension zero), at least when the pure part of the Shimura variety is of Abelian type, see [ShWo95]. In all these cases it is interesting to notice that the Q-bi-algebraic subvarieties are the bi-algebraic subvarieties containing one arithmetic point (in Example 4.2c) these are the special subvarieties of the Shimura variety). The bi-algebraic structure associated with a period map Φ : S an → Γ\D is defined over Q as soon as S is. In this case, we expect the Q-bi-algebraic subvarieties to be precisely the special subvarieties, see [K17, 2.6 and 3.4].

The Ax-Schanuel theorem for period maps.
The geometry of bialgebraic structures is controlled by the following functional transcendence heuristic, whose idea was introduced by Pila in the case of Shimura varieties, see [Pil14], [Pil15] : Ax-Schanuel principle: Let S be an irreducible algebraic variety endowed with a non-trivial bi-algebraic structure. Let U ⊂ S an × S an be an algebraic subvariety (for the product bi-algebraic structure) and let W be an analytic irreducible component of U ∩∆, where ∆ denotes the graph of p : S an → S an . Then where W bi denotes the smallest bi-algebraic subvariety of S containing p(W ).
When applied to a subvariety U ⊂ S an × S an of the form Y × p(Y ) Zar for Y ⊂ S an algebraic, the Ax-Schanuel principle specializes to the following: Ax-Lindemann principle: Let S be an irreducible algebraic variety endowed with a non-trivial bi-algebraic structure. Let Y ⊂ S an be an algebraic subvariety. Then p(Y ) Zar is a bi-algebraic subvariety of S.
Ax [Ax71], [Ax72] showed that the abstract Ax-Schanuel principle holds true for Example 4.2a) and Example 4.2b) above, using differential algebra. Notice that the Ax-Lindemann principle in Example 4.2a) is the functional analog of the classical Lindemann theorem stating that if α 1 , . . . , α n are Q-linearly independent algebraic numbers then e α1 , . . . , e αn are algebraically independent over Q. This explains the terminology. The Ax-Lindemann principle in Example 4.2c) was proven by Pila [Pil11] when S is a product Y (1) n × (C * ) k , by Ullmo-Yafaev [UY14b] for projective Shimura varieties, by Pila-Tsimerman [PT14] for A g , and by Klingler-Ullmo-Yafaev [KUY16] for any pure Shimura variety. The full Ax-Schanuel principle was proven by Mok-Pila-Tsimerman for pure Shimura varieties [MPT19].
We conjectured in [K17,Conj. 7.5] that the Ax-Schanuel principle holds true for the bi-algebraic structure associated to a (graded-)polarized variation of (mixed) ZHS on an arbitrary quasi-projective variety S. Bakker and Tsimerman proved this conjecture in the pure case:  Then the projection of U to S is contained in a strict weakly special subvariety of S for Φ.
The proof of Theorem 4.4 follows a strategy started in [KUY16] and fully developed in [MPT19] in the Shimura case, see [Tsi18b] for an introduction. It does not use Theorem 3.14 but only a week version equivalent to the Nilpotent Orbit Theorem, and relies crucially on the definable Chow Theorem 3.10, the Pila-Wilkie Theorem 3.8 and the proof that the volume (for the natural metric on Γ\D) of the intersection of a ball of radius R in Γ\D with the horizontal complex analytic subvariety Φ(S an ) grows exponentially with R (a negative curvature property of the horizontal tangent bundle).

4.3.
On the distribution of the Hodge locus. Theorem 4.4 is most useful, even in its simplest version of the Ax-Lindemann theorem. After Theorem 3.1 one would like to understand the distribution in S of the special subvarieties for V. For instance, are there any geometric constraints on the Zariski closure of HL(S, V ⊗ )? To approach this question, let us decompose the adjoint group G ad into a product G 1 ×· · ·×G r of its simple factors. It gives rise (after passing to a finiteétale covering if necessary) to a decomposition of the Hodge variety Γ\D into a product of Hodge varieties Γ 1 \D 1 × · · · × Γ r \D r . A special subvariety Z of S for V is said of positive period dimension if dim C Φ(Z an ) > 0; and of factorwise positive period dimension if moreover the projection of Φ(Z an ) on each factor Γ i \D i has positive dimension. The Hodge locus of factorwise positive period dimension HL(S, V ⊗ ) fpos is the union of the strict special subvarieties of positive period dimension, it is contained in the Hodge locus of positive period dimension HL(S, V ⊗ ) pos union of the strict special subvarieties of positive period dimension, and the two coincide if G ad is simple.
Using the Ax-Lindemann theorem special case of Theorem 4.4 and a global algebraicity result in the total bundle of V, Otwinowska and the author proved the following: Theorem 4.6 ([KO21]). Let V be a polarized ZVHS on a smooth connected complex quasi-projective variety S. Then either HL(S, V ⊗ ) fpos is Zariski-dense in S; or it is an algebraic subvariety of S (i.e., the set of strict special subvarieties of S for V of factorwise positive period dimension has only finitely many maximal elements for the inclusion).
Example 4.7. The simplest example of Theorem 4.6 is the following. Let S ⊂ A g be a Hodge-generic closed irreducible subvariety. Either the set of positive dimensional closed irreducible subvarieties of S which are not Hodge generic has finitely many maximal elements (for the inclusion), or their union is Zariski-dense in S. In the same way that the Ax-Schanuel principle controls the geometry of bi-algebraic structures, the diophantine geometry of Q-bi-algebraic structures is controlled by the following heuristic: Atypical intersection principle: Let S be an irreducible algebraic Q-variety endowed with a Q-bi-algebraic structure. Then the union S atyp of atypical Q-bi-algebraic subvarieties of S is an algebraic subvariety of S (i.e. it contains only finitely many atypical Q-bi-algebraic subvarieties maximal for the inclusion).
Here a Q-bi-algebraic subvariety Y ⊂ S is said to be atypical for the given bialgebraic structure on S if it is obtained as an excess intersection of f ( S an ) with its model f (Ỹ ) Zar ⊂ Z; and S atyp denotes the union of all atypical subvarieties of S. As a particular case of the atypical intersection principle: Sparsity of arithmetic points principle: Let S be an irreducible algebraic Q-variety endowed with a Q-bi-algebraic structure. Then any irreducible algebraic subvariety of S containing a Zariski-dense set of atypical arithmetic points is a Q-bi-algebraic subvariety.
This principle that arithmetic points are sparse is a theorem of Mann [Ma65] in Example 4.2a). For abelian varieties over Q (Example 4.2b)) this is the Manin-Mumford conjecture proven first by Raynaud [Ray88], saying that an irreducible subvariety of an abelian variety over Q containing a Zariski-dense set of torsion point is the translate of an abelian subvariety by a torsion point. For Shimura varieties of abelian type (Example 4.2c)) this is the classical André-Oort conjecture [An89], [Oort94] stating that an irreducible subvariety of a Shimura variety containing a Zariski-dense set of special points is special. It has been proven in this case using tame geometry and following the strategy proposed by Pila-Zannier [PiZa08] (let us mention [Pil11], [U14], [PT14], [KUY16], [AGHM], [YuZh], [Tsi18a]; and [Gao16] in the mixed case; see [KUY18] for a survey). Recently the André-Oort conjecture in full generality has been obtained in [PST21], reducing to the case of abelian type using ingredients from p-adic Hodge theory. We refer to [Za12] for many examples of atypical intersection problems.
For a general polarized ZVHS V with period map Φ : S an → Γ\D, which we can assume to be proper without loss of generality, we already mentioned that even the geometric characterisation of the Q-bi-algebraic subvarieties as the special subvarieties is unknown. Replacing the Q-bi-algebraic subvarieties of S by the special ones, we define: is contained in the singular locus of the complex analytic variety Φ(S an )), or if Φ(S an ) and Γ Z \D Z do not intersect generically along Φ(Z): Otherwise it is said to be typical.
Defining the atypical Hodge locus HL(S, V ⊗ ) atyp ⊂ HL(S, V ⊗ ) as the union of the atypical special subvarieties of S for V, we obtain the following precise atypical intersection principle for ZVHS, first proposed in [K17] in a more restrictive form: Conjecture 5.2 (Zilber-Pink conjecture for ZVHS, [K17], [BKU21]). Let V be a polarizable ZVHS on an irreducible smooth quasi-projective variety S. The atypical Hodge locus HL(S, V ⊗ ) atyp is a finite union of atypical special subvarieties of S for V. Equivalently: the set of atypical special subvarieties of S for V has finitely many maximal elements for the inclusion.
Notice that this conjecture is in some sense more general than the above atypical intersection principle, as we don't assume that S is defined over Q; this has to be compared to the fact that the Manin-Mumford conjecture holds true for every complex abelian variety, not necessarily defined over Q.
Example 5.3. Recently Baldi and Ullmo [BU20] proved a special case of Conjecture 5.2 of much interest. Margulis' arithmeticity theorem states that any lattice in a simple real Lie group G of real rank at least 2 is arithmetic: it is commensurable with a group G(Z), for G a Q-algebraic group such that G(R) = G up to a compact factor. On the other hand the structure of lattices in a simple real Lie group of rank 1, like the group P U (n, 1) of holomorphic isometries of the complex unit ball B n C endowed with its Bergman metric, is an essentially open question. In particular there exist non-arithmetic lattices in P U (n, 1), n = 2, 3. Let ι : Λ ֒→ P U (n, 1) be a lattice. The ball quotient S an := Λ\B n C is the analytification of a complex algebraic variety S. By results of Simpson and Esnault-Groechenig, there exists a ZVHS Φ : S an → Γ\(B n C × D ′ ) with monodromy representation ρ : Λ → P U (n, 1) × G ′ whose first factor Λ → P U (n, 1) is the rigid representation ι. The special subvarieties of S for V are the totally geodesic complex subvarieties of S an . When Λ is nonarithmetic, they are automatically atypical. In accordance with Conjecture 5.2 in this case, Baldi and Ullmo prove that if Λ is non-arithmetic, then S an contains only finitely many maximal totally geodesic subvarieties. This result has been proved independently by Bader, Fisher, Miller, and Stover [BFMS20], using completely different methods from homogeneous dynamics.
Among the special points for a ZVHS V, the CM-points (i.e. those for which the Mumford-Tate group is a torus) are always atypical except if the generic Hodge datum (G, D) is of Shimura type and the period map Φ is dominant. Hence, as explained in [K17, Section 5.2], Conjecture 5.2 implies the following: Conjecture 5.4 (André-Oort conjecture for ZVHS, [K17]). Let V be a polarizable ZVHS on an irreducible smooth quasi-projective variety S. If S contains a Zariskidense set of CM-points then the generic Hodge datum (G, D) of V is a Shimura datum, and the period map Φ : S an → Γ\D is an algebraic map, dominant on the Shimura variety Γ\D.
Example 5.5. Consider the Calabi-Yau Hodge structure V of weight 3 with Hodge numbers h 3,0 = h 2,1 = 1 given by the mirror dual quintic. Its universal deformation space S is the projective line minus 3 points, which carries a ZVHS V of the same type. This gives a non-trivial period map Φ : S an → Γ\D, where D = Sp(4, R)/U (1) × U (1) is a 4-dimensional period domain. This period map is known not to factorize through a Shimura subvariety (its algebraic monodromy group is Sp 4 ). Conjecture 5.4 in that case predicts that S contains only finitely many points CM-points s. A version of this prediction already appears in [GuVa04]. The more general Conjecture 5.2 also predicts that S contains only finitely many points s where V s splits as a direct sum of two (Tate twisted) weight one Hodge structures (V 2,1 s ⊕ V 1,2 s ) and its orthogonal for the Hodge metric (V 3,0 s ⊕ V 0,3 s ) (the so-called "rank two attractors" points, see [Moore98]).
Conjecture 5.2 about the atypical Hodge locus takes all its meaning if we compare it to the expected behavior of its complement, the typical Hodge locus HL(S, V ⊗ ) typ := HL(S, V ⊗ ) \ HL(S, V ⊗ atyp ): Conjecture 5.6 (Density of the typical Hodge locus, [BKU21]). If HL(S, V ⊗ ) typ is not empty then it is dense (for the analytic topology) in S an .
Conjecture 5.2 and Conjecture 5.6 imply immediately the following, which clarifies the possible alternatives in Theorem 4.6: Conjecture 5.7 ( [BKU21]). Let V be a polarizable ZVHS on an irreducible smooth quasi-projective variety S. If HL(S, V ⊗ ) typ is empty then HL(S, V ⊗ ) is algebraic; otherwise HL(S, V ⊗ ) is analytically dense in S an .
In [BKU21] Baldi, Ullmo and I establish the geometric part of Conjecture 5.2: the maximal atypical special subvarieties of positive period dimension arise in a finite number of families whose geometry is well-understood. We can't say anything on the atypical locus of zero period dimension (for which different ideas are certainly needed): such that Z contains a Zariski-dense set of atypical special subvarieties for Φ ′′ of zero period dimension. Moreover Z is Hodge generic in the special subvariety Φ −1 (Γ GZ \D GZ ) 0 of S for Φ, which is typical.
Conjecture 5.2, which also takes into account the atypical special subvarieties of zero period dimension, predicts that the branch (b) of the alternative in the conclusion of Theorem 5.8 never occurs. Theorem 5.8 is proven using properties of definable sets and the Ax-Schanuel Theorem 4.4, following an idea originating in [U14].
As an application of Theorem 5.8, let us consider the Shimura locus of S for V, namely the union of the special subvarieties of S for V which are of Shimura type (but not necessarily with dominant period maps). In [K17], I asked (generalizing the André-Oort conjecture for ZVHS) whether a polarizable ZVHS V on S whose Shimura locus in Zariski-dense in S is necessarily of Shimura type. As a corollary of Theorem 5.8 we obtain: Theorem 5.9 ([BKU21]). Let V be a polarizable ZVHS on a smooth irreducible complex quasi-projective variety S, with generic Hodge datum (G, D). Suppose that the Shimura locus of S for V of positive period dimension is Zariski-dense in S. If G ad is simple then V is of Shimura type.

5.3.
On the algebraicity of the Hodge locus. In view of Conjecture 5.7, it is natural to ask if there a simple combinatorial criterion on (G, D) for deciding whether HL(S, V) typ is empty. Intuitively, one expects that the more "complicated" the Hodge structure is, the smaller the typical Hodge locus should be, due to the constraint imposed by Griffiths' transversality. Let us measure the complexity of V by its level: when G ad is simple, it is the greatest integer k such that g k,−k = 0 in the Hodge decomposition of the Lie algebra g of G; in general one takes the minimum of these integers obtained for each simple Q-factor of G ad . While strict typical special subvarieties usually abound for ZVHSs of level one (e.g. families of abelian varieties, see Example 4.7; or families of K3 surfaces) and can occur in level two (see Example 4.8), they do not exist in level at least three! Theorem 5.10 ( [BKU21]). Let V be a polarizable ZVHS on a smooth connected complex quasi-projective variety S. If V is of level at least 3 then HL(S, V ⊗ ) typ = ∅ (and thus HL(S, V ⊗ ) = HL(S, V ⊗ ) atyp ).
The proof of Theorem 5.10 is purely Lie theoretic. Let (G, D) be the generic Hodge datum of V and Φ : S an → Γ\D its period map. Suppose that Y ⊂ S is a typical special subvariety, with generic Hodge datum (G Y , D Y ). The typicality condition and the horizontality of the period map Φ imply that g −i,i Y = g −i,i for all i ≥ 2 (for the Hodge structures on the Lie algebras g Y and g defined by some point of D Y ). Under the assumption that V has level at least 3, we show that this is enough to ensure that g Y = g, hence Y = S. Hence there are no strict typical special subvariety.
Notice that Conjecture 5.2 and Theorem 5.10 imply: Conjecture 5.11 (Algebraicity of the Hodge locus in level at least 3, [BKU21]). Let V be a polarizable ZVHS on a smooth connected complex quasi-projective variety S. If V is of level at least 3 then HL(S, V ⊗ ) is algebraic.
The main result of [BKU21], which follows immediately from Theorem 5.8 and Theorem 5.10, is the following stunning geometric reinforcement of Theorem 3.1 and Theorem 4.6: Theorem 5.12 ( [BKU21]). If V is of level at least 3 then HL(S, V ⊗ ) fpos is algebraic.
As a simple geometric illustration of Theorem 5.12, we prove the following, to be contrasted with the n = 2 case (see Example 4.8): Corollary 5.13. Let P N (n,d) C be the projective space parametrising the hypersurfaces X of P n+1 be the Zariski-open subset parametrising the smooth hypersurfaces X and let V → U n,d be the ZVHS corresponding to the primitive cohomology H n (X, Z) prim . If n ≥ 3 and d > 5 then HL(U n,d , V ⊗ ) pos ⊂ U n,d is algebraic.

5.4.
On the typical Hodge locus in level one and two. In the direction of Conjecture 5.6, we obtain: Theorem 5.14 (Density of the typical locus, [BKU21]). Let V be a polarized ZVHS on a smooth connected complex quasi-projective variety S. If the typical Hodge locus HL(S, V ⊗ ) typ is non-empty (hence the level of V V is one or two by Theorem 5.10) then HL(S, V ⊗ ) is analytically (hence Zariski) dense in S.
Notice that, in Theorem 5.14, we also treat the typical Hodge locus of zero period dimension. Theorem 5.14 is new even for S a subvariety of a Shimura variety. Its proof is inspired by the arguments of Chai [Chai98] in that case.
It remains to find a criterion for deciding whether, in level one or two, the typical Hodge locus HL(S, V ⊗ ) typ is empty or not. We refer to [KOU,Theor. 2.15] and [Ta20], [TaTh21] for results in this direction.

Arithmetic aspects
We turn briefly to some arithmetic aspects of period maps. 6.1. Field of definition of special subvarieties. Once more the geometric case provides us with a motivation and a heuristic. Let f : X → S be a smooth projective morphism of connected algebraic varieties defined over a number field L ⊂ C and let V be the natural polarizable ZVHS on S an with underlying local system R • f an * Z. In that case, the Hodge conjecture implies that each special subvariety Y of S for V is defined over Q and that each of the Gal(Q/L)-conjugates of Y is again a special subvariety of S for V. More generally, let us say that a polarized ZVHS V = (V Z , (V, F • , ∇), q) on S an is defined over a number field L ⊂ C if S, V, F • and ∇ are defined over L (with the obvious compatibilities).
Conjecture 6.1. Let V be a ZVHS defined over a number field L ⊂ C. Then any special subvariety of S for V is defined over Q, and any of its finitely many Gal(Q/L)-conjugates is a special subvariety of S for V.
There are only few results in that direction: see [V07, Theor. 0.6] for a proof under a strong geometric assumption; and [SaSc16], where it is shown that when S (not necessarily V) is defined over Q, then a special subvariety of S for V is defined over Q if and only if it contains a Q-point of S. In [KOU] Otwinowska, Urbanik and I provide a simple geometric criterion for a special subvariety of S for V to satisfy Conjecture 6.1. In particular we obtain: Theorem 6.2 ( [KOU]). Let V be a polarized ZVHS on a smooth connected complex quasi-projective variety S. Suppose that the adjoint generic Mumford-Tate group G ad of V is simple. If S is defined over a number field L, then any maximal (strict) special subvariety Y ⊂ S of positive period dimension is defined over Q. If moreover V is defined over L then the finitely many Gal(Q/L)-translates of Y are special subvarieties of S for V.
As a corollary of Theorem 5.12 and Theorem 6.2, one obtains the following, which applies for instance in the situation of Corollary 5.13. Corollary 6.3. Let V be a polarized variation of Z-Hodge structure on a smooth connected quasi-projective variety S. Suppose that V is of level at least 3, and that it is defined over Q. Then HL(S, V ⊗ ) fpos is an algebraic subvariety of S, defined over Q.
It is interesting to notice that Conjecture 5.11, which is stronger than Theorem 5.12, predicts the existence of a Hodge generic Q-point in S for V in the situation of Corollary 6.3.
As the criterion given in [KOU] is purely geometric, it says nothing about fields of definitions of special points. It is however strong enough to reduce the first part of Conjecture 6.1 to this particular case: Theorem 6.4. Special subvarieties for ZVHSs defined over Q are defined over Q if and only if it holds true for special points.

Absolute Hodge locus.
Interestingly, Conjecture 6.1 in the geometric case follows from an a priori much weaker conjecture than the Hodge conjecture. Let f : X → S be a smooth projective morphism of smooth connected complex quasi-projective varieties. For any automorphism σ ∈ Aut (C/Q) we can consider the algebraic family f σ : X σ → S σ , where σ −1 : S σ = S × C,σ C ∼ − → S is the natural isomorphism of abstract schemes; and the attached polarizable ZVSH V σ = (V σ Z , V σ , F •σ , ∇ σ ) with underlying local system V σ Z = Rf σan * Z on (S σ ) an . The algebraic construction of the algebraic de Rham cohomology provides compatible canonical comparison isomorphisms ι σ : (V σ , F •σ , ∇ σ ) ∼ − → σ −1 * (V, F • , ∇) of the associated algebraic filtered vector bundles with connection. More generally a collection of ZVHS (V σ ) σ with such compatible comparison isomorphisms is called a (de Rham) motivic variation of Hodge structures on S, in which case we write V := V Id . Following Deligne (see [CS14] for a nice exposition), an absolute Hodge tensor for such a collection is a Hodge tensor α for V s such that the conjugates σ −1 * α dR of the De Rham component of α defines a Hodge tensor in V σ σ(s) for all σ. The generic absolute Mumford-Tate group for (V σ ) σ is defined in terms of the absolute Hodge tensors as the generic Mumford-Tate group is defined in terms of the Hodge tensors. Thus G ⊂ G AH . In view of Definition 3.19 the following is natural: Definition 6.5. Let (V σ ) σ be a (de Rham) motivic variation of Hodge structure on a smooth connected complex quasi-projective variety S. A closed irreducible algebraic subvariety Y of S is called absolutely special if it is maximal among the closed irreducible algebraic subvarieties Z of S satisfying G AH Z = G AH Y . In the geometric case, the Hodge conjecture implies, since any automorphism σ ∈ Aut (C/Q) maps algebraic cycles in X to algebraic cycles on X σ , the following conjecture of Deligne: Conjecture 6.6 ( [DMOS82]). Let (V σ ) σ be a (de Rham) motivic variation of Hodge structure on S. Then all Hodge tensors are absolute Hodge tensors, i.e. G = G AH .
This conjecture immediately implies: Conjecture 6.7. Let (V σ ) σ be a (de Rham) motivic variation of Hodge structure on S. Then any special subvariety of S for V is absolutely special for (V σ ) σ .
Let us say that a (de Rham) motivic variation (V σ ) σ is defined over Q if V σ = V for all σ ∈ Aut(C/Q). In the geometric case any morphism f : X → S defined over Q defines such a (de Rham) motivic variation (V σ ) σ over Q. Notice that the absolutely special subvarieties of S for (V σ ) σ are then by their very definition defined over Q, and their Galois conjugates are also special. In particular Conjecture 6.7 implies Conjecture 6.1 in the geometric case. As proven in [V07], Deligne's conjecture is actually equivalent to a much stronger version of Conjecture 6.1, where one replaces the special subvarieties of S (components of the Hodge locus) with the special subvarieties in the total bundle of V ⊗ (components of the locus of Hodge tensors).
Recently T.Kreutz, using the same geometric argument as in [KOU], justified Theorem 6.2 by proving: Theorem 6.8 ( [Kr21a]). Let (V σ ) σ be a (de Rham) motivic variation of Hodge structure on S. Suppose that the adjoint generic Mumford-Tate group G ad is simple. Then any strict maximal special subvariety Y ⊂ S of positive period dimension for V is absolutely special.