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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)

##### 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
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