A remark on the Abel–Jacobi morphism for the cubic threefold

Ze Xu

doi:10.1016/j.crma.2012.12.002

Algebraic Geometry

A remark on the Abel–Jacobi morphism for the cubic threefold

Presented by: Claire Voisin

Ze Xu ¹

¹ Academy of Mathematics and Systems Science, Institute of Mathematics, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, 100190, Beijing, China

Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 63-67.

Abstract

Let X be a smooth cubic threefold and $J (X)$ be its intermediate Jacobian. We show that there exists a codimension 2 cycle Z on $J (X) \times X$ with $Z_{t}$ homologically trivial for each $t \in J (X)$ , such that the morphism $ϕ_{Z} : J (X) \to J (X)$ induced by the Abel–Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.

Received: 2012-11-25
Accepted: 2012-12-06
Published online: 2013-01-01

DOI: 10.1016/j.crma.2012.12.002

Author's affiliations:

Ze Xu ¹

¹ Academy of Mathematics and Systems Science, Institute of Mathematics, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, 100190, Beijing, China

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Ze Xu. A remark on the Abel–Jacobi morphism for the cubic threefold. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 63-67. doi : 10.1016/j.crma.2012.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.12.002/

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Cited by Sources:

^☆ This work was performed when the author was visiting Institut de Mathématiques de Jussieu.

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