The following saturation result will be crucial in the proof of Theorem1.5:
Proposition 7.3 In the situation of Theorem 7.1suppose moreover that the local systemV=VQ⊗QCis defined overQ. Let AF,≥d,Q:= AF,≥d∩VQbe the locus of rational classes x whose flat transport meets F at x in dimension
≥ d and let AF,≥d := AF,≥d,QZar
be its Zariski-closure inV. There exist a Zariski-open dense subset U of AF,≥d and, for each x ∈ U , a component NF0,xof NF,x of dimension at least d such that U ⊂
x∈U NF0,x ⊂ AF,≥d. Proof As in the proof of Theorem7.1we remove from now on the reference toF in our notations.
First notice that A≥d,Q⊂ A≥d, henceA≥d ⊂ A≥d, asA≥d is algebraic by Theorem7.1.
LetW be an irreducible component of A≥d. Since A≥d+1 ⊂ A≥d, pro-ceeding by decreasing induction on d, we may and shall assume that d is the largest integer such thatW ⊂ A≥d. HenceW∩ A≥d+1 is a strict closed algebraic subvariety ofW. LetU ⊂ W be the Zariski-open dense subset of all x ∈ W −(W ∩ A≥d+1) such that the variety A≥d is smooth at x and
As this holds for any irreducible componentW of A≥d, the result follows.
7.3 Application toZVHS: proof of Theorem3.2and corollary for
Hodge loci
Suppose now that V is a ZVHS and F = FiV. Then AF,≥d = Vi≥d and Theorem7.1in this case is Theorem3.2.
Moreover AF,≥d,Q=Vi≥d ∩VQand Proposition7.3reads:
Proposition 7.4 Let S be a smooth complex quasi-projective algebraic variety andVbe a polarizedZVHS over S. Let i ∈Zand d∈N. There exist a com-ponent of dimension at leastdofSi(p(λ)). By Theorem3.1the Zariski-closure of any such components is a weakly special subvariety of Sof dimension at leastd. We thus obtain
Corollary 7.5 Let S be a smooth complex quasi-projective algebraic variety andVbe a polarizedZVHS over S. Let d∈N. Then p(VQ∩V0≥d)Zarcontains a Zariski-open dense set U with the following property: for each point x ∈U there exists a weakly special subvariety Yx ⊂ p(VQ∩V0≥d)Zarof dimension at least d passing through x.
8 Proof of Theorem1.5
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⊗). generic Mumford–Tate group, the period map, or the special subvarieties), we are reduced without loss of generality to showing that forVa polarizable ZVHS the Hodge locus of positive period dimension HL(S,V)posis either a finite union of special subvarieties ofSforVor Zariski-dense inS.
To make the proof of Theorem1.5more transparent we deal first with special cases.
Case 1: the period mapS is an immersion.In that case HL(S,V)pos= p((VQ∩(V)0≥1) .
Applying Corollary7.5ford =1 toVit follows that HL(S,V)pos
Zar con-tains a Zariski-open dense subset U with the following property: for each point x ∈ U there exists a weakly special subvariety Wx of positive period dimension forVpassing throughx and contained in HL(S,V)pos
Zar. Either there existsx ∈Usuch thatWx =S, in which case HL(S,V)pos
Zar= S. Or for all x ∈ U the weakly special subvariety Wx 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. 4.1implies that each Wx is con-tained in a unique strict special subvariety Sx of positive period dimension forV. AsSx 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 ofS, hence the result.
Case 2: the period mapShas constant relative dimension d. The proof is the same as in the first case, replacing(Vi)0≥1and “of positive period dimension”
by(Vi)0≥d and “of period dimension at least(d+1)”.
General case: As the period map S is definable in the o-minimal struc-tureRan,exp (see [3]), it follows from the trivialization theorem [25, Theor.
(1.2) p.142] that the locus Sd ⊂ S where the fibers of S are of complex dimension at least d is an Ran,exp-definable subset of S. As Sd 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 Sd is a closed algebraic subvariety of S. Finally we obtain an algebraic filtration S= Sd0 Sd1 · · · Sdk Sdk+1 = ∅.
Suppose that HL(S,V)posis not algebraic. Leti ∈ {0,· · · ,k}be the smallest integer such that(Sdi−Sdi+1)∩HL(S,V)posis not a closed algebraic subvariety ofSdi−Sdi+1. As HL(S−Sdi+1,V⊗|S−S
di+1)pos=HL(S,V)pos∩(S−Sdi+1), to prove that HL(S,V)posis Zariski-dense in Swe can and will assume without loss of generality thati =k (replacingSby S−Sdi+1 if necessary).
Without loss of generality we can assume that HL(S,V)posis contained in Sdi: this is clear ifi =0, asS=Sd0in this case; ifi>0 there are only finitely many maximal special subvarieties of positive period dimension Z1, . . . ,Zm
ofSforVintersectingSdi−1−Sdiand we can without loss of generality replace SbyS−(Z1∪ · · · ∪Zm).
Thus HL(S,V)poscoincide withp((V)0≥di+1∩VQ). Applying Corollary7.5 with d = di +1, it follows that the union Z of irreducible components of HL(S,V)pos
Zar contains a Zariski-open dense setU such that for every point x ∈Uthere exists a weakly special subvarietyWxofSforVof dimension at leastdi +1 passing throughx and contained inZ.
If i > 0 the weakly special subvariety Wx ⊂ Z ⊂ Sdi is strict, and we conclude as above: eachWxis contained in a unique strict special subvarietySx
of positive period dimension forV, thus Z =HL(S,V)pos, which contradicts the assumption that HL(S,V)posis not an algebraic subvariety ofS.
Thusi =0. Hence we are in Case 2 above and we conclude that HL(S,V)pos
is Zariski-dense inSd0 =S. This finishes the proof of Theorem1.5.
Acknowledgements We thank O. Benoist and J. Chen for their remarks on this work.
Funding Open Access funding enabled and organized by Projekt DEAL.
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