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Griffiths’ infinitesimal invariant and the Abel-Jacobi map. (English) Zbl 0692.14003
Let X be a hypersurface in \({\mathbb{P}}^ n\). Let us name Noether-Lefschetz locus the locus of smooth subvarieties of X of a fixed codimension. - If \(m\geq 3\) and one considers codimension one subvarieties of X the Lefschetz theorems show immediately that the \(N-L\quad locus\) is empty in all degrees. In higher codimension the situation is more interesting. There is the following conjecture due to P. A. Griffiths and J. Harris: On a general 3-fold X of degree \(\geq 6\), the Abel-Jacobi map \(\alpha\) from algebraic 1-cycles on X homologically equivalent to zero to the intermediate Jacobian \(J^ 2(X)\) is zero.
The author draws a three steps program for proving the above conjecture and does the first two, getting the following partial result: If X is a 3-fold as above then the image of \(\alpha\) is contained in the set of torsion points of the intermediate Jacobian.
The proof is mainly based on an improvement of an infinitesimal invariant of normal functions introduced by P. A. Griffiths [Compos. Math. 50, 267-324 (1983; Zbl 0576.14009)] and a vanishing theorem for Koszul cohomology due to the author [J. Differ. Geom. 27, 155-159 (1988; Zbl 0674.14005)].
Reviewer: A.Del Centina

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K30 Picard schemes, higher Jacobians
14J99 Surfaces and higher-dimensional varieties
14C15 (Equivariant) Chow groups and rings; motives
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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