On the distribution of the Hodge locus

• 成果类型: Article
• 署名作者: Baldi, Gregorio; Klingler, Bruno; Ullmo, Emmanuel
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Humboldt University of Berlin
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01226-0
• 发表日期: 2024
• 页码: 441-487
• 关键词: theorem; CURVES; density; bounds

摘要

Given a polarizable Z-variation of Hodge structures V over a complex smooth quasi-projective base S, a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of S, called the special subvarieties for V. Our main result in this paper is that, if the level of V is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree d smooth hypersurfaces in P-C(n+1), n >= 3, d >= 5 and (n, d) not equal (4, 5), is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in S-an as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in S of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.