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.

给定\（{{\ mathbb {V}}} \）可极化\\ {{\ mathbb {Z}}} \）-光滑连接的复拟投影变体S上的霍奇结构，它是\的霍奇轨迹。 （{{\ mathbb {V}}} ^ \ otimes \）是S的闭合点s的集合，其中光纤\（{{\ mathbb {V}}} _ s \）的Hodge张量比一般的张量更多。 。Cattani，德利涅和Kaplan的一个经典的结果指出，霍奇轨迹为\（{{\ mathbb {V}}} ^ \ otimes \）是闭合的不可约代数子簇的可数工会小号，称为特殊子簇小号为\（{{\ mathbb {V}}} \）。假设\（{{\ mathbb {V}}} \）的通用Mumford–Tate组的伴随组很简单，我们证明\（{{\ mathbb {V}}的特殊子变量的并集} \）的期间地图下，其图像是不是一个点或者是一个封闭的代数subvariety小号或是Zariski密集在小号。例如，这意味着以下典型的交集语句：给定一个主要极化的尺寸为g的阿贝尔变数的模空间\（{{{\ mathcal {A}}} _ g \）的Hodge-generic闭合不可约代数子变量S，其并集为S交点的正维不可约分量与严格特殊子簇\（{{\ mathcal {A}}} _克\）或者是一个封闭的代数subvariety小号或是Zariski密集在小号。