On the transcendental part of the motive of a surface

In this paper we introduce a birational invariant of a smooth projective surface S over a field k, the transcendental part t_2(S) of the motive h(S), and prove a formula for the endomorphism ring of t_2(S) which is a higher dimensional analogue of a classical result of Weil concerning divisorial correspondences. We also prove that t_2(S) =0 iff the Albanese kernel T(S_{k(S)} of the surface S over the field k(S) vanishes. Over the complex field C this is equivalent to Bloch’s conjecture ; p_g(S)=0 implies T(S)=0.