Torsion cohomology classes and algebraic cycles on complex projective manifolds

Final version, to appear in Adv. in Mathematics, special issue dedicated to M. Artin. References to related work by C. Schoen added. A mistake mentioned by K. O'Grady corrected.Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincaré dual to an algebraiccycle. We notice that the order ofthese classes are small comparedto the dimension of the manifold. However, building upon a construction of Kollàr, one can provide such examples witharbitrary high prime order, the dimension being fixed. This method alsoprovides examples of torsion algebraiccycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration on Chow groups