Hodge loci

**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 **Hodge**theoretic 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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