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 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 : if X has a large Picard number ρ = ρ(X),i.e ρ = 19, 20, then the motive of X is finite dimensional and it is isomorphic to the motive of a Kummer surface. 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. We also prove that, if X and Y are complex K3 surfaces, then the finite dimensionality of h(X) and h(Y ) implies a conjecture of Orlov on the derived categories of X and Y .