Abstract
Given {{mathbb {V}}} a polarizable variation of {{mathbb {Z}}}-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for {{mathbb {V}}}^otimes is the set of closed points s of S where the fiber {{mathbb {V}}}_s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for {{mathbb {V}}}^otimes is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for {{mathbb {V}}}. Under the assumption that the adjoint group of the generic Mumford–Tate group of {{mathbb {V}}} is simple we prove that the union of the special subvarieties for {{mathbb {V}}} 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 {{mathcal {A}}}_g 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 {{mathcal {A}}}_g is either a closed algebraic subvariety of S or is Zariski-dense in S.
Highlights
1.1 Motivation Hodge lociLet (VZ, V, F, ∇) be a polarizable variation of Z-Hodge structure (ZVHS)of arbitrary weight on a smooth connected complex quasi-projective variety S. VZ is a finite rank locally free ZSan-local system on the complex manifold San associated to S; and (V, F, ∇) is the unique algebraic regular filtered flat connection filtration FonanSdwthheosheoalnoamlyotripfihciactifloanticsoVn⊗neZcStainoOnSan endowed ∇an defined with its Hodge by V, see [23,(4.13)])
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(Vs,Q) has maximal dimension when s ranges through Y an
In this paper we investigate the geometry of the Zariski-closure of the Hodge locus HL(S, V⊗)
Summary
Of arbitrary weight on a smooth connected complex quasi-projective variety S. Hodge structure Vs admits more Hodge classes than the very general fiber Vs (for us a Hodge class of a pure Z-Hodge structure H = (HZ, F) is a class in HZ whose image in HC lies in F0 HC, or equivalently a morphism of Hodge structures Z(0) → H ) It is empty if V contains no non-trivial weight zero factor. In [4] Cattani, Deligne and Kaplan proved the following unconditional celebrated result (see [5], we 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. 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(Vs,Q) has maximal dimension when s ranges through Y an. 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
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