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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 $$S$$.

##### MSC:
 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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