In this paper we show that for a complex K3 surface X with a large Picard number ρ, the finite dimensionality of the transcendental part of the motive t2(X) implies the isomorphism of t2(X) with t2(Y ), where Y is a Kummer surface.Therefore the motive of X belongs to the subcategory generated by motives of abelian varieties. On the other hand, if X and Y are complex distinct K3 surfaces, which are general members of smooth projective families {Xt} and {Ys} over the disk (hence ρ(X) = ρ(Y ) = 1), then Murre’s Conjecture implies that Hom (t2(X), t2(Y ) = 0 in the category of Chow Motives.
On the motive of a K3 surface
PEDRINI, CLAUDIO
2010-01-01
Abstract
In this paper we show that for a complex K3 surface X with a large Picard number ρ, the finite dimensionality of the transcendental part of the motive t2(X) implies the isomorphism of t2(X) with t2(Y ), where Y is a Kummer surface.Therefore the motive of X belongs to the subcategory generated by motives of abelian varieties. On the other hand, if X and Y are complex distinct K3 surfaces, which are general members of smooth projective families {Xt} and {Ys} over the disk (hence ρ(X) = ρ(Y ) = 1), then Murre’s Conjecture implies that Hom (t2(X), t2(Y ) = 0 in the category of Chow Motives.
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