On the finite dimensionality of a K3 surface

TL;DR

This paper investigates the finite dimensionality of Chow motives for complex K3 surfaces, providing new results for surfaces with high Picard number, certain group actions, and involutions, supporting Kimura's conjecture.

Contribution

It proves finite dimensionality for K3 surfaces with high Picard number, specific group actions, and explores implications of involutions, advancing understanding of Kimura's conjecture.

Findings

Finite dimensionality holds for K3 surfaces with Picard number 19 or 20.

Finite dimensionality is established when a non-symplectic group acts trivially.

For K3 surfaces with a Nikulin involution, the motive is isomorphic to that of a desingularized quotient.

Abstract

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.