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On Grothendieck’s generalized Hodge conjecture for a family of threefolds with trivial canonical bundle. (English) Zbl 0728.14007
Let \(\pi\) : \({\mathcal X}\to U\) be a smooth projective family of threefolds with trivial canonical bundle, \(X_ t=\pi^{-1}(t)\) for \(t\in U\), \(J\to U\) the intermediate Jacobian bundle of \(\pi\), \(J(X_ t)\) its fibre over 0 and \(J_ H(X_ t)\) the maximal Hodge subtorus of \(J(X_ t)\) perpendicular to \(H^{3,0}(X_ t)\). An upper bound for dim \(J_ H(X_ t)\) holding for t general in U and depending on the number of moduli on which the family \(\pi\) depends is given. If the threefolds of the family \(\pi\) are acted on freely by a finite group G then \(J_ H(X_ t)\) is explicitly determined generically over U.
In the rest of the paper these results are used to study a very special case: \(X_ t\) a complete intersection in some \({\mathbb{P}}^ n\) and \(G={\mathbb{Z}}/2{\mathbb{Z}}:\) in this situation it is proved that \(X_ t\) is a complete intersection of four G-invariant quadrics in \({\mathbb{P}}^ 7\) and that generically over U one has \(J_ H(X_ t)=\{the\) skew-symmetric subtorus \(J(X_ t)^-\) of \(J(X_ t)\}\). The monodromy representation of \(\pi_ 1(U,t)\) on \(H^ 3(X_ t,{\mathbb{Q}})\) is studied: as a result it is proved that \(H^ 3(X_ t,{\mathbb{Q}})^+\) and \(H^ 3(X_ t,{\mathbb{Q}})^-\) are the \(\pi_ 1(U,t)\)-irreducible subspaces. Then one finds a family of algebraic 1-cycles on X by looking at the \({\mathbb{P}}^ 4\)’s contained in the rank 6 quadrics through \(X_ t:\) any \(E={\mathbb{P}}_ 4\) as above is contained in one quadric Q, by choosing three other quadrics \(Q_ 1, Q_ 2, Q_ 3\) defining \(X_ t\) together with Q and by taking \(Q_ 1\cap Q_ 2\cap Q_ 3\cap E\) one gets a genus 5 curve C contained in \(X_ t\). The family of all such curves is then used to give a parametrization of \(J(X_ t)^-\) by algebraic cycles: thus proving the generalized Hodge conjecture generically over U. The Abel Jacobi map is then studied more closely and its degree is computed.
Reviewer: F.Bardelli (Pisa)

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
14D07 Variation of Hodge structures (algebro-geometric aspects)
14C25 Algebraic cycles
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