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Some results on Green’s higher Abel-Jacobi map. (English) Zbl 1053.14006
Let \(X\) be a smooth complex algebraic variety. Then, due to a recent construction by M. Green [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 267–276 (1998; Zbl 0914.14002)], there is a rather explicit way of describing Griffiths’s so-called higher Abel-Jacobi maps \(\psi^p_m: F^m CH^p(X)\to J^p_m(X)\), where \(1\leq m\leq p\leq \dim X\). These maps relate the filtration of the \(p\)th Chow group of \(X\) to the \(p\)th intermediate Jacobian and satisfy the condition \(F^{m+1}CH^p(X)= \ker(\psi^p_m)\).
The aim of the paper under review is to answer some questions raised by M. Green himself, in this context, mainly with a view toward the particular case of zero cycles on an algebraic surface \(S\).
The first result presented here shows that, in general, the higher Abel-Jacobi map \(\psi^2_2\) for a surface \(S\) is not injective. This answers negatively M. Green’s conjecture on the injectivity of \(\psi^2_2\) for surfaces.
The second main result, however, solves another conjecture of M. Green’s. Namely, the author proves that the higher Abel-Jacobi map \(\psi^2_2\) is nontrivial modulo torsion if \(h^{2,0}(S)\neq 0\) for a surface \(S\).

14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
14C25 Algebraic cycles
14K30 Picard schemes, higher Jacobians
14J25 Special surfaces
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
Zbl 0914.14002
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