0 M ay 2 00 6 Hodge loci and absolute Hodge classes Claire Voisin
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چکیده
Let π : X → T be a family of smooth projective complex varieties. Assume X , π, T are defined over Q. An immediate consequence of the fact that there are only countably many components of the relative Hilbert scheme for π, and that the relative Hilbert scheme (with fixed Hilbert polynomial) is defined over Q, is the following: if the Hodge conjecture is true, the components of the Hodge locus in T are defined over Q, and their Galois transforms are again components of the Hodge locus. (We recall later on the definition of the components of the Hodge locus.) In [4], it is proven that the components of the Hodge locus (and even the components of the locus of Hodge classes, which is a stronger notion) are algebraic sets, while Hodge theory would give them only a local structure of closed analytic subsets (see [10], 5.3.1). In this paper, we give simple sufficient conditions for components of the Hodge locus to be defined over Q (and their Galois transforms to be also components of the Hodge locus). This criterion of course does not hold in full generality, and it particular does not say anything about the definition field of an isolated point in the Hodge locus. But in practice, it is reasonably easy to check and allows to conclude in some explicit cases, where the Hodge conjecture is not known to hold. We give a few examples of applications in section 3. We will first relate this geometric language to the notion of absolute Hodge classes (as we only deal with the de Rham version, we will not use the terminology of Hodge cycles of [5]), and explain why this notion allows to reduce the Hodge conjecture to the case of varieties defined over Q, thus clarifying a question asked to us by V. Maillot and Ch. Soulé. Let us recall the notion of (de Rham) absolute Hodge class (cf [5]). Let Xan be a complex projective manifold and α ∈ Hdg2k(Xan) be a rational Hodge class. Thus α is rational and
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تاریخ انتشار 2008