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Let X be a complex projective surface with arbitrary singularities. We construct a generalized Abel-Jacobi map A 0 (X) → J 2 (X) and show that it is an isomorphism on torsion subgroups. Here A 0 (X) is the appropriate Chow group of smooth 0-cycles of degree 0 on X, and J 2 (X) is the intermediate Jacobian associated with the mixed Hodge structure on H 3 (X). Our result generalizes a theorem of Roitman for smooth surfaces: if X is smooth then the torsion in the usual Chow group A 0 (X) is isomorphic to the torsion in the usual Albanese variety J 2 (X) ∼ = Alb(X) by the classical Abel-Jacobi map.

For a smooth projective surface X the finite dimensionality of the Chow motive h(X ), as conjectured by Kimura, has several geometric consequences. For a complex surface of general type with p g = 0 it is equivalent to Bloch's conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we prove some results on Kimura's conjecture for complex K3 surfaces. If X has a large Picard number ρ = ρ(X ), i.e. ρ = 19, 20, then the motive of X is finite dimensional. If X has a non-symplectic group acting trivially on algebraic cycles then the motive of X is finite dimensional. If X has a symplectic involution i, i.e. a Nikulin involution, then the finite dimensionality of h(X ) implies h(X ) h(Y ), where Y is a desingularization of the quotient surface X/ i . We give several examples of K3 surfaces with a Nikulin involution such that the isomorphism h(X ) h(Y ) holds, so giving some evidence to Kimura's conjecture in this case.

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