Variations of Hodge Structure and Algebraic Cycles

Claire Voisin
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Claire Voisin: Université d’Orsay

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 706-715 from Springer

Abstract: Abstract Let X be a complex projective variety. Then each cohomology group of X admits a Hodge structure, that is a decomposition of H k (X, ℂ) = H k (X, ℤ) ⊗ ℂ into a direct sum $$\mathop \oplus \limits_{p + q = k} {H^{p,q}}(X)$$ ⊕ p + q = k H p , q ( X ) , where $${H^{p,q}}(X) \simeq {H^q}(\Omega _X^p) \subset {H^k}(X,\mathbb{C})$$ H p , q ( X ) ≃ H q ( Ω X p ) ⊂ H k ( X , ℂ ) is the set of classes that can be presented by a closed k-form everywhere of type (p, q). We will be concerned in this paper with the relations between Hodge structures and Chow groups CH. (X), where CHℓ(X) is the group of ℓ-cycles (= arbitrary integral combinations of ℓ-dimensional subvarieties) modulo rational equivalence [5].

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_64

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