On the closure of the Hodge locus of positive period dimension

• 成果类型: Article
• 署名作者: Klingler, B.; Otwinowska, A.
• 署名单位: Humboldt University of Berlin
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01042-4
• 发表日期: 2021
• 页码: 857-883
• 关键词: density; subvarieties; CURVES

摘要

Given V a polarizable variation of Z-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for V-circle times is the set of closed points s of S where the fiber 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 V-circle times is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for V. Under the assumption that the adjoint group of the generic Mumford-Tate group of V is simple we prove that the union of the special subvarieties for 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 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 A(g) is either a closed algebraic subvariety of S or is Zariski-dense in S.