Chow motives of genus one fibrations

Abstract: Let

f : X i g h t a r r o w C

be a genus 1 fibration from a smooth projective surface, i.e. its generic fiber

X_{e t a}

is a genus 1 curve. Let

j : J i g h t a r r o w C

be the Jacobian fibration of

f

, e.g. the smooth locus of

j

is the N'eron model of the Jacobian variety of

X_{e t a}

. In this paper, we prove that the Chow motives of

X

and

J

are isomorphic. As an application, we prove the Kimura finiteness for smooth projective surfaces not of general type with geometric genus 0. This can be regarded as a generalization of Bloch-Kas-Lieberman's result to arbitrary characteristic.

Mathematics Subject Classification ID

(K3) surfaces and Enriques surfaces (14J28) Elliptic surfaces, elliptic or Calabi-Yau fibrations (14J27) (Equivariant) Chow groups and rings; motives (14C15)

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