On the finite dimensionality of a K3 surface

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 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, i.e 19 or 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) is isomorphic to h(Y), where Y is a desingularization of the quotient surface X= X/< i >. We give several examples of K3 surfaces with a Nikulin involution such that the above isomorphism holds, so giving some evidence to Kimura's conjecture in this case.