Hodge theory on some invariant threefolds of even degree. (English) Zbl 0879.14015
The Grothendieck-Hodge conjecture for threefolds suggests that it should be possible to construct a family of curves whose Abel-Jacobi image generates the subtorus of the intermediate Jacobian spanned by the maximal rational sub-Hodge structure inside \(H^3\) which is abelian, that is, having Hodge numbers \(h^{3,0} = h^{0,3} =0\). In this paper, the author considers the universal family of the hypersurfaces of degree \(2p\) and dimension three invariant under a certain action of the group of \(p\)-th roots of unity, and proves the Grothendieck-Hodge conjecture for the general point of some special subfamilies.
Reviewer: Min Ho Lee (Cedar Falls)
MSC:
| 14J30 | \(3\)-folds |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14K30 | Picard schemes, higher Jacobians |
| 14L30 | Group actions on varieties or schemes (quotients) |
Keywords:
invariant threefolds; intermediate Jacobian; Hodge structures; Abel-Jacobi map; action of the group of roots of unity; Grothendieck-Hodge conjectureReferences:
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