On the closure of the Hodge locus of positive period dimension
Let \(\mathbb V=\{\mathbb V_{\mathbb Z},\mathcal{V},F^{\bullet},\nabla\}\) be a polarizable variation of Hodge structure over a smooth connected quasi-projective complex variety \(S\). The Hodge locus \(HL(\mathbb V^{\otimes})\) is the set of all points \(s\in S\) such that \(\mathbb V_s\) has more Hodge tensors than \(\mathbb V_{s'}\) for a very general point \(s'\in S\). A classical result of Cattani-Deligne-Kaplan asserts that \(HL(\mathbb V^{\otimes})\) is a countable union of strict algebraic subvarieties of \(S\). The main goal of this paper is to study the Zariski closure of this locus with many striking results. For this matter, the authors introduce the notion of a \emph{special subvariety} of \(S\): it is a closed irreducible algebraic subvariety \(Y\subseteq S\) maximal among those with the same generic Mumford-Tate group. The Hodge locus is a countable union of strict special subvarieties. A special subvariety has \emph{positive period} dimension if its image under the period map of the variation is not a point. The Hodge locus of positive period dimension \(HL(\mathbb V^{\otimes})_{pos}\) is defined then as the union of special subvarieties of positive period dimension. The main result of the paper states that, assuming the simplicity of the generic adjoint Mumford-Tate group, the Zariski closure of \(HL(\mathbb V^{\otimes})_{pos}\) is either Zariski dense or a finite union of strict special subvarieties of \(S\). Several examples are then discussed: Hodge locus in Shimura varieties and the Noether-Lefschetz locus.
scientific article; zbMATH DE number 7387423
scientific article