an:00579762
Zbl 0808.14030
Voisin, Claire
On the Abel-Jacobi map for Calabi-Yau threefolds
FR
Ann. Sci. ??c. Norm. Sup??r. (4) 27, No. 2, 209-226 (1994).
00019019
1994
j
14J30 14D07 14J32
Calabi-Yau threefold; generic deformation; Abel-Jacobi map; variation of mixed Hodge structure
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 \textit{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.
L.Picco Botta (Torino)
Zbl 0772.14015