On the closure of the Hodge locus of positive period dimension
type: Item
instance of: scholarly article
title: On the closure of the Hodge locus of positive period dimension
zbMATH Open document ID: 1487.14031
author: Bruno Klingler
author: Ania Otwinowska
published in: Inventiones Mathematicae
publication date: 2021-08-27T00:00:00Z
review text: 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.
reviewed by: Salim Tayou
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 7387423
zbMATH Keywords: Hodge theory
zbMATH Keywords: variations of Hodge structures
zbMATH Keywords: o-minimality
MaRDI profile type: Publication
full work available at URL: s00222-021-01042-4
OpenAlex ID: W3142811732
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DOI: 10.1007/S00222-021-01042-4
description: scientific article; zbMATH DE number 7387423