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Hodge loci

Hodge loci

Hodge

Hodge loci Claire Voisin Abstract. The goal of this expository article is first of all to show that Hodge theory provides naturally defined subvarieties of any moduli space parameterizing smooth varieties, the “Hodge loci”, although only the Hodge conjecture would guarantee that these subvarieties are defined on a finite extension of the base field. We will show how these subsets can be studied locally in the Euclidean topology and introduce a number of related Hodgetheoretic notions. The article will culminate with two results by Deligne, Cattani-Deligne-Kaplan respectively. The first one says that Hodge classes are absolute Hodge on abelian varieties. This is a statement which we will rephrase in general in terms of Hodge loci and is enough to guarantee that Hodge loci are closed algebraic, defined on a finite extension of the base field. The second tells us that Hodge loci are in general closed algebraic, as predicted by the Hodge conjecture. 1. Introduction These notes are devoted to the study of Hodge loci associated to a family of smooth complex projective varieties π : X → B. The Hodge loci are quite easy to define set theoretically and also, locally on B for the classical topology, as a countable union of analytic subschemes; the local components are indeed endowed with a natural analytic schematic structure, which we will describe and can be studied using Griffiths theory of variations of Hodge structures [16]. On the other hand, the right approach to give a global definition of the Hodge loci is the notion of “locus of Hodge classes” introduced in [8]. This locus of Hodge classes is not a subset of B, but a subset of some Hodge bundle on B, associated to π. Griffiths’ theory has several interesting local consequences on Hodge loci, eg their expected codimension and density properties. However, it only gives a local and transcendental approach to the subject, which makes it hard to understand the global structure of the Hodge loci. As we will explain in section 1.1 below, the Hodge conjecture predicts in fact that, assuming X , B are quasi-projective, Hodge loci are closed algebraic subsets of B, which makes them relevant to the topic of this book. One of our main goals in this paper is to motivate and explain the main Received by the editors . 1991 Mathematics Subject Classification. 14D05, 14D07, 14F25, 14F40, 14C30. Key words and phrases. Variation of Hodge structure, Hodge classes, Monodromy, Mumford- Tate groups, Hodge loci. 1

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