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On the Abel-Jacobi map for Calabi-Yau threefolds. (Sur l’application d’Abel-Jacobi des variétés de Calabi-Yau de dimension trois.) (French) Zbl 0808.14030
The aim of this paper is to prove the following theorem: Let \(X\) be a Calabi-Yau threefold and let \(\{X_ t\}\) be a generic deformation of \(X\), then the Abel-Jacobi map \(\Phi_{X_ t}\) is nonzero modulo torsion.
The methods and the results of C. Voisin [Int. J. Math. 3, No. 5, 699-715 (1992; Zbl 0772.14015)] are used and extended here. Let \(\Sigma\) be a smooth surface of a suitable degree in \(X\) and let \((X_ t, \Sigma_ t)\) be a deformation of the pair \((X, \Sigma)\). The proof is based on a description of the vanishing of \(\Phi_{X_ t}\) for a general \(t\) in terms of the variation of the mixed Hodge structure of \(X - \Sigma\), and leads to the study of a system of quadrics whose dimension is bounded. Then the result follows by contradiction.

MSC:
14J30 \(3\)-folds
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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