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

Présenté par : 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.

Résumé

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.

Reçu le : 2012-11-25
Accepté le : 2012-12-06
Publié le : 2013-01-01

DOI : 10.1016/j.crma.2012.12.002

Affiliations des auteurs :

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/

Bibliographie
Cité par

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[6] S. Druel Espace des modules des faisceaux de rang 2 semi-stables de classes de Chern $c_{1} = 0$ , $c_{2} = 2$ et $c_{3} = 0$ sur la cubique de $P^{4}$ , Int. Math. Res. Not., Volume 19 (2000), pp. 985-1004

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[9] A. Iliev; D. Markushevich The Abel–Jacobi map for cubic threefold and periods of Fano threefolds of degree 14, Doc. Math., Volume 5 (2000), pp. 23-47

[10] D. Markushevich; A. Tikhomirov The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., Volume 10 (2001), pp. 37-62

[11] A.S. Merkurjev; A.A. Suslin K-cohomology of Severi–Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982) no. 5, pp. 1011-1046 1135–1136 (in Russian)

[12] J.P. Murre Applications of algebraic K-theory to the theory of algebraic cycles, Sitjes 1983 (LNM), Volume vol. 1124 (1985), pp. 216-261

[13] C. Voisin Abel–Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom., Volume 1056 (2012) no. 3011, pp. 1-34

Cité par Sources :

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

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