On the closure of the Hodge locus of positive period dimension

Given V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}$$\end{document} a polarizable variation of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}$$\end{document}-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for V⊗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}^\otimes $$\end{document} is the set of closed points s of S where the fiber Vs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}_s$$\end{document} has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for V⊗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}^\otimes $$\end{document} is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}$$\end{document}. Under the assumption that the adjoint group of the generic Mumford–Tate group of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}$$\end{document} is simple we prove that the union of the special subvarieties for V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}$$\end{document} whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {A}}}_g$$\end{document} of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {A}}}_g$$\end{document} is either a closed algebraic subvariety of S or is Zariski-dense in S.


Motivation: Hodge loci
Let (V Z , V, F • , ∇) be a polarizable variation of Z-Hodge structure (ZVHS) of arbitrary weight on a smooth connected complex quasi-projective variety S. Thus V Z is a finite rank locally free Z S an -local system on the complex manifold S an associated to S; and (V, F • , ∇) is the unique algebraic regular filtered flat connection on S whose analytification is V⊗ Z S an O S an endowed with its Hodge filtration F • and the holomorphic flat connection ∇ an defined by V, see [23, (4.13)]). From now on we will abbreviate the ZVHS (V Z , V, F • , ∇) simply by V. A typical example is the weight zero polarizable ZVHS "of geometric origin" associated to a smooth projective morphism of smooth irreducible complex quasi-projective varieties f : X → S. In this case the Hodge filtration F • is induced by the stupid filtration on the algebraic De Rham complex • X/S and ∇ is the Gauß-Manin connection.
The Hodge locus HL(S, V) is the set of points s ∈ S an for which the Hodge structure V s admits more Hodge classes than the very general fiber V s (for us a Hodge class of a pure Z-Hodge structure H = (H Z , F • ) is a class in H Z whose image in H C lies in F 0 H C , or equivalently a morphism of Hodge structures Z(0) → H ). It is empty if V contains no non-trivial weight zero factor. More generally let V ⊗ be the countable direct sum of polarizable ZVHSs a,b∈N V ⊗a ⊗ (V ∨ ) ⊗b (where V ∨ denotes the ZVHS dual of V). The Hodge locus HL(S, V ⊗ ) is the subset of points s ∈ S an for which the Hodge structure V s admits more Hodge tensors than the very general fiber V s . It contains HL(S, V), usually strictly.
In the geometric case Weil [28] asked whether HL(S, V) is a countable union of closed algebraic subvarieties of S (he noticed that a positive answer follows easily from the rational Hodge conjecture). In [4] Cattani, Deligne and Kaplan proved the following unconditional celebrated result (see [5], we also refer to [3] for an alternative proof): Theorem 1.1 (Cattani-Deligne-Kaplan) Let S be a smooth connected complex quasi-projective algebraic variety and V be a polarizable ZVHS over S. Then HL(S, V) (thus also HL(S, V ⊗ )) is a countable union of closed irreducible algebraic subvarieties of S.
The locus HL(S, V ⊗ ) is easier to understand than HL(S, V) as it has a grouptheoretical interpretation. Recall that the Mumford-Tate group MT(H ) ⊂ GL(H ) of a Q-Hodge structure H is the Tannakian group of the Tannakian category H ⊗ of Q-Hodge structures tensorially generated by H and its dual H ∨ . Equivalently, the group MT(H ) is the fixator in GL(H ) of the Hodge tensors for H . Given a polarized ZVHS V on S as above and Y → S a closed irreducible algebraic subvariety, a point s of Y an is said to be Hodge-generic in Y for V if MT(V s,Q ) has maximal dimension when s ranges through Y an . Two Hodge-generic points in Y an for V have the same Mumford-Tate group, called the generic Mumford-Tate group MT(Y, V |Y ) of Y for V. The Hodge locus HL(S, V ⊗ ) is also the subset of points of S which are not Hodge-generic in S for V. Definition 1.2 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 MT(Z , V |Z ) = MT(Y, V |Y ).
In particular S is always special for V. Theorem 1.1 for HL(S, V ⊗ ) can be rephrased by saying that the set of special subvarieties of S for V is countable and that HL(S, V ⊗ ) is the (countable) union of the strict special subvarieties of S for V.

Main result
In this paper we investigate the geometry of the Zariski-closure of the Hodge locus HL(S, V ⊗ ). Our methods are variational, hence we only detect the special subvarieties of S for V which are of positive period dimension in the following sense: Definition 1.3 A closed irreducible subvariety Y of S is said to be of positive period dimension for V if the local system V |Y is not isotrivial.
Equivalently, Y is of positive period dimension for V if and only if its algebraic monodromy group H Y for V (see Definition 2.1) is not equal to {1}; or equivalently if the period map S : S an → \D + describing V ⊗ (see Sect. 4) does not contract Y an to a point in the connected Hodge variety \D + . When V satisfies the infinitesimal Torelli condition (i.e. the period map S is an immersion), a closed irreducible subvariety Y of S is of positive period dimension for V if and only if it is positive dimensional. Definition 1. 4 We define the Hodge locus of positive period dimension HL(S, V ⊗ ) pos ⊂ HL(S, V ⊗ ) as the union of the strict special subvarieties of S for V which are of positive period dimension for V.
Our main result describes the Zariski-closure of HL(S, V ⊗ ) pos : Theorem 1.5 Let V be a polarizable ZVHS on a smooth connected complex quasi-projective variety S. Suppose that the adjoint group of the generic Mumford-Tate group MT(S, V) is simple (we will say that MT(S, V) is non-product). Then either HL(S, V ⊗ ) pos is a finite union of strict special subvarieties of S; or it is Zariski-dense in S.
In other words: either the set of strict special subvarieties of S for V which are of positive period dimension for V has finitely many maximal elements (for the inclusion); or the union of such special subvarieties is Zariski-dense in S.

Examples
Theorem 1.5 is new even in the much-studied case where the ZVHS V has weight 1 or 2. Let us warn the reader that these cases, which are simpler to describe, are not representative: in higher weight we expect HL(S, V ⊗ ) pos to be algebraic in general.
of prime degree endowed with an involution of the second kind. Ball quotients of Kottwitz type are the simplest examples.
proof adapts immediately to show that HL(S, V ⊗ |S ) pos is analytically dense in S if S has codimension at most g −1. Generalizing the results of [14] to a general connected Shimura variety Sh 0 K (G, X ), Chai (see [6]) showed the following. Let H ⊂ G be a Hodge subgroup. Let HL(S, V ⊗ , H) ⊂ HL(S, V ⊗ ) denote the subset of points s ∈ S whose Mumford-Tate group MT s (V) is G(Q)conjugated to H. Then there exists an explicit constant c(G, X, H) ∈ N, whose value is g in the example above, which has the property that HL(S, V ⊗ , H), hence also HL(S, V ⊗ ) is analytically dense in S as soon as S has codimension at most c(G, X, H) in Sh K (G, X ). Once more it follows from the analysis of the proof of [6] that HL(S, V ⊗ ) pos is analytically dense in S as soon as S has codimension at most c(G, X, H) − 1.

Example 2: classical Noether-Lefschetz locus
be the open subvariety parametrizing the smooth surfaces of degree d in P 3 C . From now on we suppose d > 3. The classical Noether theorem states that any surface Y ⊂ P 3 C corresponding to a very general point [Y ] ∈ B has Picard group Z: every curve on Y is a complete intersection of Y with another surface in P 3 C . The countable union NL(B) of closed algebraic subvarieties of B corresponding to surfaces with bigger Picard group is called the Noether-Lefchetz locus of B. Let V → B be the ZVHS R 2 f * Z, where f : Y → B denotes the universal family of surfaces of degree d. Clearly NL(B) ⊂ HL(B, V ⊗ ). Green (see [26,Prop.5.20]) proved that NL(B) is analytically dense in B (see also [7] for a weaker result). In particular HL(B, V ⊗ ) is dense in B. Once more the analysis of Green's proof shows that in fact HL(B, V ⊗ ) pos is dense in B. Now Theorem 1.5 implies the following: Corollary 1.9 Let S ⊂ B be a Hodge-generic closed irreducible subvariety. Either S ∩ HL(B, V ⊗ ) pos contains only finitely many maximal positive dimensional closed irreducible subvarieties of S, or the union of such subvarieties is Zariski-dense in S. Remark 1. 10 We don't know if Corollary 1.9 remains true if we replace HL(B, V ⊗ ) pos with NL(B).

Organization of the paper
The next Sect. 2 introduce the basic notation concerning local systems and ZVHS we will need. Section 3 then describes the main ingredients and the general strategy for proving Theorem 1.5. The reader will find at the end of Sect. 3 the organization of the rest of the paper.

Notation for local systems
Let S be a smooth connected complex quasi-projective variety. Let V Z be a finite rank locally free Z-local system on S and (V, ∇) the regular algebraic connection on S [11, Theor. 5.9] associated to V Z .
The local system V Z can be uniquely written asS × ρ V Z , where π :S → S denotes the complex analytic universal cover of S associated to the choice of a point s 0 in S, V Z := H 0 (S, π −1 V Z ) V s 0 ,Z is a free Z-module of finite rank and ρ : π 1 (S, s 0 ) → GL(V Z ) denotes the monodromy representation of the local system V Z . This corresponds to a complex analytic trivialization ofṼ := V × SS as a productS × V , where V := V Z ⊗ Z C. We still let π :S × V → V denote the natural projection. Recall the following classical definition: Definition 2.1 Given a closed irreducible algebraic subvariety i : Y → S, let n : Y nor → Y be its normalisation. The algebraic monodromy group H Y of Y for V Z is the (conjugacy class of the) identity component of the Zariski-closure in GL(V Q ) of the monodromy of the restriction to Y nor of the local system n * V Z .

Definition 2.2 Given
The set V(λ) is naturally a connected closed complex analytic subspace of the étalé space of the complex local system V C := V Z ⊗ Z C. We will always endow V(λ) with its reduced analytic structure. When λ = π(s, λ 0 ) is not a complex multiple of an element of V Z , the orbit of λ 0 in V under the monodromy group ρ(π 1 (S, s 0 )) ⊂ GL(V ) has usually accumulation points, in which case V(λ) is not an analytic subvariety of V.

Notation for ZVHS
All ZVHS are assumed to be polarizable. In particular the algebraic monodromy group H S is semi-simple.

Definition 2.3
Let λ ∈ V and i ∈ Z. The locus of classes of F i -type V i (λ) for λ is the intersection of the flat leaf V(λ) with F i V: The locus of F i -type for λ is the projection Again, V i (λ) is naturally a complex analytic subspace (possibly with infinitely many connected components) of the étalé space of the complex local system V C := V Z ⊗ Z C. When λ is not a complex multiple of an element of V C the complex space V i (λ) is in general not an analytic subspace of V; a fortiori its projection S i (λ) ⊂ S is a priori not a complex analytic subvariety of S. Remark 2.4 For i = 0 and λ ∈ V Q the locus V 0 (λ) is also called the locus of Hodge classes for λ, usually denoted Hdg(λ); and S 0 (λ) is the Hodge locus of λ considered by Weil, namely the locus HL(S, λ) of points of S where some determination of the flat transport of λ becomes a Hodge class.
S i (λ) ≥d ⊂ S, be the union of components of V i (λ), resp. S i (λ), of dimension at least d. Remark 2.6 Notice that for λ ∈ V and z ∈ C * , V i (zλ) = zV i (λ) and S i (zλ) = S i (λ) for any z ∈ C * . Hence, for λ ∈ V not in the zero section, S i (λ) depends only on [λ] ∈ PV .
For λ = π(s, λ 0 ) ∈ V it follows from the theorem of the fixed part (see [23,Cor. 7.23]) that S i (λ) = S if and only if the H S -orbit of λ 0 in V is not reduced to a point, equivalently if and only if the orbit of λ 0 under ρ(π 1 (S, s 0 )) ⊂ GL(V ) is infinite. We denote by V nt Q the direct factor of the local system V Q corresponding to the sum of non-trivial irreducible H S -factors of V Q (it is naturally a sub-QVHS of V Q ). By abuse of notation we write V nt Q − {0} for V nt Q with the zero-section removed.

Definition 2.7
We define the locus of non-trivial F i -classes Thus the locus of non-trivial (rational) Hodge classes for V is Hdg(V) := V Q ∩ V 0 ≥0 and the Hodge locus

Ingredients and strategy for Theorem 1.5
Let us now describe the main ingredients and the strategy for the proof of Theorem 1.5. From now on we do not differentiate a complex algebraic variety X from its associated complex analytic space X an , the meaning being clear from the context.

On the Zariski-closure of the F i -loci
) ⊂ S be the locus of points of S where some determination of the flat transport of λ at s belongs to F i V, as defined in Sect. 2. When i = 0 and λ ∈ V Q is rational, V 0 (λ) is the locus where the flat transport of λ is a rational Hodge class. The precise version of Theorem 1.1 is that for λ rational, V 0 (λ) is a closed algebraic subvariety of V, finite over the finite union of special subvarieties S 0 (λ).
To study the Zariski-closure of HL(S, V ⊗ ) the first idea of this paper consists in studying the geometry of S i (λ) for a general, not necessarily rational, λ ∈ V C . In this generality the subsets S i (λ) are usually not even complex analytic subvarieties of S, see Sect. 2. However we manage to describe the Zariski-closure of any of their components (see Definition 2.5 for the notion of component of S i (λ)): Theorem 3.1 For any i ∈ Z and any λ ∈ V C , the Zariski-closure of any of the (possibly infinitely many) components of S i (λ) is a weakly special subvariety of S for V.
Here the weakly special subvarieties of S for V are a generalisation, introduced in [15], of the special subvarieties of S for V. See Definition 4.1 for the original definition and Corollary 4.14 for a more geometric description. Theorem 3.1 provides a strong information on components of S i (λ) which are of positive period dimension.

A global algebraicity result for the locus of classes of F i -type
The second ingredient in the proof of Theorem 1.5 is a global algebraicity statement for the union of the F i -loci of dimension bounded below. Precisely, for any integer d ≥ 0, let V i ≥d ⊂ F i V be the locus of classes λ ∈ F i V whose orbit under monodromy is infinite and such that

Theorem 3.2 Let V be a polarized ZVHS on a smooth quasi-projective variety S. For any i ∈ Z and any d
In words: the property of a point λ ∈ V of having a flat leaf intersecting F i V in dimension at least d > 0 is closed in the Zariski-topology. Theorem 3.2 is in fact a special case of a more general result on algebraic flat connections, see Theorem 7.1. It uses in a crucial way the properties of parallel transport.

Strategy for the proof of Theorem 1.5
Let us indicate how Theorem 1.5 follows from Theorem 3.1 and Theorem 3.2.
First, using a finiteness result of Deligne, we are reduced to showing that for V a polarizable ZVHS with non-product generic Mumford-Tate group, the Hodge locus of positive period dimension HL(S, V) pos is either a finite union of strict special subvarieties of S for V or is Zariski-dense in S.
Let us assume for simplicity that the period map S for V is an immersion.

In that case the locus of exceptional rational Hodge classes in
≥0 ); and the Hodge locus of positive period dimension HL(S, V) pos is the projection  is non-product imply that each Y x is contained in a strict special subvariety S x of S for V. As such an S x is contained in HL(S, V) pos it follows that HL(S, V) pos Zar = HL(S, V) pos . But then HL(S, V) pos is a finite union of special subvarieties. The general case where S is not a submersion is dealt with similarly using stratifications and the geometry of S 0 d (V) for all d ≥ 1.

A converse to Theorem 1.1
Recall that for λ ∈ V Q the precise version of Theorem 1.1 states that V 0 (λ) is a closed algebraic subvariety of V, finite over the finite union of special subvarieties S 0 (λ). As a preliminary to Theorem 3.1, Theorem 3.2 and Theorem 1.5, we also provide for the convenience of the reader the following kind of converse to Theorem 1.1, which might be well-known to experts but which does not seem to have appeared before.

Organization of the remaining sections
Section 4 studies the geometric properties of the weakly special subvarieties of S for V. In particular we prove that they are closed algebraic subvarieties, obtain a key geometric description (Corollary 4.14), prove that they coincide in fact with the bi-algebraic subvarieties of S for the natural bi-algebraic structure on S defined by V (see Proposition 4.20, a result stated in [15,Prop.7.4] without proof), and state the Ax-Lindemann Theorem 4.21 for them. The following sections provide the proofs of Proposition 3.3, Theorem 3.1, Theorem 3.2 and Theorem 1.5 successively.

Weakly special subvarieties and bi-algebraic geometry for (S, V)
In this section we recall the definition of the weakly special subvarieties of S for V given in [15], study their geometry and prove their bi-algebraic characterisation (stated in [15] without proof). We recall below the definitions of Hodge theory we need and introduced in [15] (inspired by [21] and [22]), and refer to [15] for more details.
Let G be the generic Mumford-Tate group of S for V. Any Hodge generic point s ∈ S defines a morphism of real algebraic groups h s : C * → G R . All such morphisms belong to the same connected component of a G(R)conjugacy class D in Hom(C * , G R ), which has a natural structure of complex analytic space (see [15,Prop.3.1]). The space D + is a so-called Mumford-Tate domain, a refinement of the classical period domain for V defined by Griffiths. The pair (G, D + ) is a connected (pure) Hodge datum in the sense of [15, Section 3.1], called the generic Hodge datum of V. The ZVHS V is entirely described by its period map is a finite index subgroup and Hod 0 (S, V) := \D + is the associated connected Hodge variety (see [15,Def. 3.18 and below]). We denote by˜ S :S → D + the lift of S .

Weakly special subvarieties
The weakly special subvarieties of S for V are defined in terms of the weakly special subvarieties of the connected Hodge variety Hod 0 (S, V), which we first recall.

Weakly special subvarieties of Hodge varieties
Let (G, D + ) be a connected Hodge datum and Y = \D + an associated connected Hodge variety. Hence Y is an arithmetic quotient in the sense of [3, Section 1] endowed with a natural complex analytic structure (which is not algebraic in general). Recall that a Hodge morphism between connected Hodge varieties is the complex analytic map deduced from a morphism of the corresponding Hodge data (see [15,Lemma 3.9]). The special and weakly special subvarieties of Y are irreducible analytic subvarieties of Y defined as follows (see [15,Def.7 → Y between (possibly mixed) connected Hodge varieties and any point r ∈ R. Then any irreducible component of i(π −1 (r )) is called a weakly special subvariety of Y . When Y is pure, i.e. G is a reductive group, one easily checks that this definition reduces to Definition 4.1(2) above.

Remark 4.3
Considering the connected Hodge variety T 2 = {t 2 } associated to the trivial algebraic group, any special subvariety of Y is a weakly special subvariety of Y .

Remark 4.4
As noticed in [22,Rem. 4.8] in the case of Shimura varieties, any irreducible component of an intersection of special (resp. weakly special) subvarieties of the Hodge variety Y is a special (resp. a weakly special) subvariety of Y . The proof is easy and the details are left to the reader. Any irreducible complex analytic component of −1 where Y is a special (resp. weakly special) subvariety of the connected mixed Hodge variety Hod 0 (S, V), is called a special (resp. weekly special) subvariety of S for V. It is said to be strict if it is distinct from S.
Notice that an irreducible component of an intersection of special (resp. weakly special) subvarieties of S for V is not anymore necessarily a special (resp. a weakly special) subvariety of S for V: it might happen that for Y ⊂ Hod 0 (S, V) a special (resp. weakly special) subvariety the preimage −1 S (Y ) decomposes as a union Z 1 ∪ Z 2 with Z i , i = 1, 2 irreducible; in which case Z 1 and Z 2 are special (resp. weakly special) subvarieties in S but an irreducible component of Z 1 ∩ Z 2 is not. To take this minor inconvenience into account we define more generally: Definition 4.6 Let Y ⊂ Hod 0 (S, V) be a special (resp. weakly special) subvariety. An irreducible component of the intersection of some irreducible components of −1 S (Y ) is called a special (resp. weakly special) intersection in S for V.
The following follows immediately from Remark 4.4: Lemma 4.7 An irreducible component of an intersection of special (resp. weakly special) intersections for V is a special (resp. weakly special) intersection for V.

Algebraicity of weakly special subvarieties of S
The very definition of the Hodge locus HL(S, V ⊗ ) implies that special subvarieties of S for V in the sense of Definition 4.5 coincide with the ones defined in Definition 1.2. In particular, in view of Theorem 1.1, any special subvariety of S (hence any special intersection in S) is a closed irreducible algebraic subvariety of S. An alternative proof of Theorem 1.1 using o-minimal geometry was provided in [3, Theor. 1.6]. The approach of [3] gives immediately the following more general algebraicity result, which is implicit in the discussion of [15, Section 7]:

Special and weakly special closure
One deduces immediately from Lemma 4.7 the following Corollary 4.9 Any irreducible algebraic subvariety i : W → S is contained in a smallest weakly special (resp. special) intersection W ws (resp. W s ) of S for V, called the weakly special (resp. special) closure of W in S for V.  The description of the weakly special closure W ws is a bit more involved but similar to the one obtained by Moonen [19,Section 3]  Let H W be the algebraic monodromy group of W for V. Thus H W is the identity component of the Zariski-closure of ( S • n) * (π 1 (W nor )) ⊂ in GL(V ). As W nor is normal the open immersion j : W nor,0 → W nor of the smooth locus W nor,0 of W nor defines a surjection j * : π 1 (W nor,0 ) → π 1 (W ).
In particular H W is also the algebraic monodromy group of the restriction of n * V Z to W nor,0 . It thus follows from [1, Theor.1] that H W is a normal subgroup of the derived group G der W . As G W is reductive there exists a normal subgroup G W ⊂ G W such that G W is an almost direct product of H W and G W . In this way we obtain a decomposition of the adjoint Hodge datum (G ad W , D + W ) into a product inducing a decomposition of connected Hodge varieties By the very definition of the algebraic monodromy group H W the group λ(π 1 (W nor )) ⊂ GL(H Z ) is finite. Applying the theorem of the fixed part (see [23,Cor. 7.23]) to the corresponding étale cover of W nor we deduce that the period map p 2 • W nor is constant.

Conversely, as any irreducible component of an intersection of weakly special subvarieties of W \D +
W is still weakly special, one easily checks that any weakly special subvariety Y : It then follows immediately: Corollary 4.14 The weakly special subvarieties of S for V (see Definition 4.5) are precisely the closed irreducible algebraic subvarieties Y ⊂ S maximal among the closed irreducible algebraic subvarieties Z of S whose algebraic monodromy group H Z with respect to V equals H Y . Remark 4.15 The reader will notice that the characterisation of the weakly special subvarieties given above is strictly analogous to the characterisation Definition 1.2 of the special subvarieties, replacing the generic Mumford-Tate group by the algebraic monodromy group.

Definition 4.16 A bi-algebraic structure on a connected complex algebraic variety S is a pair
where π :S → S denotes the universal cover of S, X is a complex algebraic variety, Aut(X ) its group of algebraic automorphisms, ρ : π 1 (S) → Aut(X ) is a group morphism (called the holonomy representation) and D is a ρequivariant holomorphic map (called the developing map).
The datum of a bi-algebraic structure on S tries to emulate an algebraic structure on the universal coverS of S: Definition 4.17 Let S be a connected complex algebraic variety endowed with a bi-algebraic structure (D, ρ).
(i) An irreducible analytic subvariety Z ⊂S is said to be a closed irreducible algebraic subvariety ofS if Z is an analytic irreducible component of is a closed algebraic subvariety of S, resp. any (equivalently one) analytic irreducible component of π −1 (W ) is a closed irreducible algebraic subvariety ofS.
As in Sect. 4.1.2 an irreducible component of an intersection of closed algebraic subvarieties ofS is not necessarily algebraic in the sense above, as the map D is not assumed to be injective.

Definition 4.18 An algebraic intersection inS is an irreducible analytic component of an intersection of closed algebraic subvarieties ofS.
An algebraic intersection Z ⊂S, resp. a closed irreducible algebraic subvariety W ⊂ S, is called a bi-algebraic intersection if π(Z ) is a closed algebraic subvariety of S, resp. any (equivalently one) analytic irreducible component of π −1 (W ) is an algebraic intersection inS.
Let V be a polarized ZVHS on S. It canonically defines a bi-algebraic structure on S as follows. Letˆ

Definition 4.19
Let p : V → S be a polarized ZVHS on a quasi-projective complex manifold S. The bi-algebraic structure on S defined by V is the pair (ˆ S :S →D, ρ S := ( S ) * : π 1 (S) → ⊂ G(C)).
The following proposition, stated in [15,Prop. 7.4] without proof, characterizes the weekly special subvarieties of S for V in bi-algebraic terms. It was proven by Ullmo-Yafaev [24] in the case where S is a Shimura variety, and in some special cases by Friedman and Laza [13]. (S, V) be a ZVHS. The weakly special subvarieties (resp. the weakly special intersections) of S for V are the bi-algebraic subvarieties (resp. the bi-algebraic intersections) of S for the bi-algebraic structure on S defined by V.

Proposition 4.20 Let
Proof The proof is similar to the proof of [24,Theor.4.1], we provide it for completeness.
Notice that the statement for the weakly special intersections follows immediately from the statement for the weakly special subvarieties. Hence we are reduced to prove that the weakly special subvarieties of S coincide with the bi-algebraic subvarieties of S.
That a weakly special subvariety of S is bi-algebraic follows from the fact that a Hodge morphism of Hodge varieties ϕ : T → Y is defined at the level of the universal cover by a closed analytic embedding D + T → D + Y restriction of a closed algebraic immersionD T →D Y .
Conversely let W be a bi-algebraic subvariety of S. With the notations of Proposition 4.13 the period map S |W : W → Hod 0 (S, V) factorises trough the weakly special subvariety ϕ(( H W \D + H W ) × {t }) of Hod 0 (S, V). Let Z be an irreducible component of the preimage of W inS and consider the lifting |Z : Z → D + H W of S |W to Z . As W is bi-algebraic the Zariski-closure of |Z (Z ) inD H W has to be stable under the monodromy group H W (C), hence equal toD H W . Thus Z = (˜ |Z ) −1 (D + H W ) and W is weakly special.
We will need the following result, proven for Shimura varieties in [16], conjectured in general in [15,Conj.7.6] as a special case of [15,Conj.7.5] Zar is a bi-algebraic subvariety of S, i.e. a weakly special subvariety of S for V.

A converse to Theorem 1.1: proof of Proposition 3.3
Let f : S → S be a finite étale cover and let V := f * V. By abuse of notation let f still denote the natural map V → V. The reader will immediately check the following (where, with the notations of Sect. 1, we naturally identify V with V ): (b) the f -image of a special subvariety of S for V is a special subvariety of S for V; conversely the f -preimage of a special subvariety of S for V is a finite union of special subvarieties of S for V .
Hence proving Proposition 3.3 for V is equivalent to proving it for V . As any finitely generated linear group admits a torsion-free finite index subgroup (Selberg's lemma) we can thus assume without loss of generality by replacing S by a finite étale cover if necessary that the monodromy ρ(π 1 (S, ) ⊂ PV is also algebraic. As the projection p : PV → S is a proper morphism, it follows that the set S i (λ) := p(V i ([λ]) is an algebraic subvariety of S.
Let n : S → S i (λ) be the smooth locus of the normalisation of one irreducible component of S i (λ). Hence S is connected. Let π :S → S be its universal cover and let ρ : π 1 (S , s 0 ) → GL(V ) be the monodromy of As V i (λ) ⊂S × {λ} and p : V i (λ) →S is surjective, it follows that V is an algebraic subvariety of V . On the other hand the set ρ(π 1 (S , s 0 )) · λ is countable. Thus V i (λ)∩V is a finite set of points, in particular p : V i (λ) → S is finite étale. It follows that the smallest Q-sub-local system W Q ⊂ V Q whose complexification W ⊂ V contains V i (λ) has finite monodromy. As the monodromy ρ(π 1 (S )) is a subgroup of ρ(π 1 (S) which is assumed to be torsion-free, it follows that the local system W Q is trivial. By the theorem of the fixed part (see [23,Cor. 7.23]) W Q is a constant sub-QVHS of V . It follows easily that n(S ) is the smooth locus of an irreducible component of the Hodge locus in S defined by the fiber W Q ⊂ V Q of W . This finishes the proof that S i (λ) is a union of special subvarieties of S and that p : Let A ≥d,∞ := n∈N A ≥d,n . As the A ≥d,n are algebraic subvarieties of V, so is A ≥d,∞ .
The result then follows from Lemma 7.2 below.

Lemma 7.2
The equality A ≥d = A ≥d,∞ holds.
Proof The inclusion A ≥d ⊂ A ≥d,∞ is equivalent to the inclusions A ≥d ⊂ A ≥d,n for all n ∈ N, which we show by induction on n. By definition A ≥d ⊂ F = A ≥d,0 . Assume that A ≥d ⊂ A ≥d,n for some n ∈ N. By definition of A ≥d , for any x ∈ A ≥d the variety A ≥d contains an irreducible component N of N F,x through x of dimension at least d. Hence hence x ∈ A ≥d,n+1 . This shows A ≥d ⊂ A ≥d,n+1 and finishes the proof that A ≥d ⊂ A ≥d,∞ .
Conversely let us prove that A ≥d,∞ ⊂ A ≥d . Let h :Ṽ → V denote the composition (where the first isomorphism is provided by the flat trivialisation). For x ∈ V andx ∈ π −1 (x) ⊂Ṽ S × V let Nx be the union of the irreducible components passing throughx of the complex analytic subvariety h −1 (h(x))∩ π −1 (F) ofṼ. Thus the local biholomorphism π :Ṽ → V identifies Nx locally atx with N x locally at x.
By noetherianity there exists an n ∈ N such that A ≥d,n = A ≥d,n+1 = A ≥d,∞ . Hence for any

Applications to Q-local systems
The following saturation result will be crucial in the proof of Theorem 1.5: For anyx ∈Ũ we have on the other hand Hence for any x (for the usual topology) and x∈U N 0 x ⊂ W . As this holds for any irreducible component W of A ≥d , the result follows.

Application to ZVHS: proof of Theorem 3.2 and corollary for Hodge loci
Suppose now that V is a ZVHS and F = F i V.
Consider now the Zariski-closure p(V Q ∩ V 0 ≥d ) Zar . It coincides with the ). For λ ∈ F i V the projection p(V i,0 (λ)) is a component of dimension at least d of S i ( p(λ)). By Theorem 3.1 the Zariski-closure of any such components is a weakly special subvariety of S of dimension at least d. We thus obtain Following Deligne (see [27,Theor. 4.14] and the comment above it), there exists a bound on the tensors one has to consider for defining HL(S, V ⊗ ). Thus HL(S, V ⊗ ) = n i=1 HL(S, V i ) for finitely many irreducible weight zero ZVHS V i ⊂ V ⊗ . It follows that HL(S, V ⊗ ) pos = n i=1 HL(S, V i ) pos . Hence, replacing V by V ⊕ n i=1 V i if necessary (this does not change the generic Mumford-Tate group, the period map, or the special subvarieties), we are reduced without loss of generality to showing that for V a polarizable ZVHS the Hodge locus of positive period dimension HL(S, V) pos is either a finite union of special subvarieties of S for V or Zariski-dense in S.
To make the proof of Theorem 1.5 more transparent we deal first with special cases. Either there exists x ∈ U such that W x = S, in which case HL(S, V) pos Zar = S. Or for all x ∈ U the weakly special subvariety W x of S is strict. In this case the assumption that MT(S, V) is non-product and the description of weakly special subvarieties given in Sect. 4.1 implies that each W x is contained in a unique strict special subvariety S x of positive period dimension for V. As S x belongs by definition to HL(S, V) pos , it follows in this case that HL(S, V) pos Zar = HL(S, V) pos is a finite union of strict special subvarieties of S, hence the result. Case 2: the period map S has constant relative dimension d. The proof is the same as in the first case, replacing (V i ) 0 ≥1 and "of positive period dimension" by (V i ) 0 ≥d and "of period dimension at least (d + 1)". General case: As the period map S is definable in the o-minimal structure R an,exp (see [3]), it follows from the trivialization theorem [25,Theor. (1.2) p.142] that the locus S d ⊂ S where the fibers of S are of complex dimension at least d is an R an,exp -definable subset of S. As S d is also a closed complex analytic subset of S, if follows from the o-minimal Chow theorem [20,Theor.4.4 and Cor. 4.5] of Peterzil-Starchenko that S d is a closed algebraic subvariety of S. Finally we obtain an algebraic filtration S = S d 0 S d 1 · · · S d k S d k+1 = ∅. Suppose that HL(S, V) pos is not algebraic. Let i ∈ {0, · · · , k} be the smallest integer such that (S d i −S d i+1 )∩HL(S, V) pos is not a closed algebraic subvariety of S d i − S d i+1 . As HL(S − S d i+1 , V ⊗ |S−S d i+1 ) pos = HL(S, V) pos ∩ (S − S d i+1 ), to prove that HL(S, V) pos is Zariski-dense in S we can and will assume without loss of generality that i = k (replacing S by S − S d i+1 if necessary).
Without loss of generality we can assume that HL(S, V) pos is contained in S d i : this is clear if i = 0, as S = S d 0 in this case; if i > 0 there are only finitely many maximal special subvarieties of positive period dimension Z 1 , . . . , Z m of S for V intersecting S d i−1 − S d i and we can without loss of generality replace S by S − (Z 1 ∪ · · · ∪ Z m ).
Thus HL(S, V) pos coincide with p((V) 0 ≥d i +1 ∩V Q ). Applying Corollary 7.5 with d = d i + 1, it follows that the union Z of irreducible components of HL(S, V) pos Zar contains a Zariski-open dense set U such that for every point x ∈ U there exists a weakly special subvariety W x of S for V of dimension at least d i + 1 passing through x and contained in Z . If i > 0 the weakly special subvariety W x ⊂ Z ⊂ S d i is strict, and we conclude as above: each W x is contained in a unique strict special subvariety S x of positive period dimension for V, thus Z = HL(S, V) pos , which contradicts the assumption that HL(S, V) pos is not an algebraic subvariety of S.
Thus i = 0. Hence we are in Case 2 above and we conclude that HL(S, V) pos is Zariski-dense in S d 0 = S. This finishes the proof of Theorem 1.5.