On the finite dimensionality of a K3 surface

Abstract

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by 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 ρ = ρ(X), i.e. ρ = 19,20, 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 $${h(X) \simeq h(Y)}$$ , where Y is a desingularization of the quotient surface $${X/\langle i \rangle }$$ . We give several examples of K3 surfaces with a Nikulin involution such that the isomorphism $${h(X) \simeq h(Y)}$$ holds, so giving some evidence to Kimura’s conjecture in this case.