For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by S.I Kimura, has several geometric consequences. For a complex surface of general type with pg = 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 give some evidence to Kimura’s conjecture for a K3 surface :...