On Grothendieck’s generalized Hodge conjecture for a family of threefolds with trivial canonical bundle.

*(English)*Zbl 0728.14007Let \(\pi\) : \({\mathcal X}\to U\) 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.

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)

##### MSC:

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 |