On the closure of the Hodge locus of positive period dimension

Abstract

Abstract Given $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> a polarizable variation of $${{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -Hodge structures on a smooth connected complex quasi-projective variety S , the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is a countable union of closed irreducible algebraic subvarieties of S , called the special subvarieties of S for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> 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 $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> 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 $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> is either a closed algebraic subvariety of S or is Zariski-dense in S .

Topics & Concepts