Algebraic Geometry
A remark on the Abel–Jacobi morphism for the cubic threefold
Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 63-67.

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)×X with Zt homologically trivial for each tJ(X), such that the morphism ϕZ:J(X)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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.12.002
Xu, Ze 1

1 Academy of Mathematics and Systems Science, Institute of Mathematics, Chinese Academy of Sciences, No. 55 East Zhongguancun Road, 100190, Beijing, China
@article{CRMATH_2013__351_1-2_63_0,
     author = {Xu, Ze},
     title = {A remark on the {Abel{\textendash}Jacobi} morphism for the cubic threefold},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--67},
     publisher = {Elsevier},
     volume = {351},
     number = {1-2},
     year = {2013},
     doi = {10.1016/j.crma.2012.12.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.12.002/}
}
TY  - JOUR
AU  - Xu, Ze
TI  - A remark on the Abel–Jacobi morphism for the cubic threefold
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 63
EP  - 67
VL  - 351
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.12.002/
DO  - 10.1016/j.crma.2012.12.002
LA  - en
ID  - CRMATH_2013__351_1-2_63_0
ER  - 
%0 Journal Article
%A Xu, Ze
%T A remark on the Abel–Jacobi morphism for the cubic threefold
%J Comptes Rendus. Mathématique
%D 2013
%P 63-67
%V 351
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.12.002/
%R 10.1016/j.crma.2012.12.002
%G en
%F CRMATH_2013__351_1-2_63_0
Xu, Ze. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.12.002/

[1] Beauville, A. Vector bundles on the cubic threefold (Bertram, A. et al., eds.), Symposium in Honor of C.H. Clemens, Contemporary Mathematics, vol. 312, 2002, pp. 71-86

[2] Bloch, S.; Ogus, A. Gerstenʼs conjecture and the homology of schemes, Ann. Sci. Ec. Norm. Supér., IV. Sér., Volume 7 (1974), pp. 181-201

[3] Bloch, S.; Srinivas, V. Remarks on correspondences and algebraic cycles, Amer. J. Math., Volume 105 (1983), pp. 1235-1253

[4] Castravet, A.-M. Rational families of vector bundles on curves, Internat. J. Math., Volume 15 (2004) no. 1, pp. 13-45

[5] Clemens, C.; Griffiths, P. The intermediate Jacobian of the cubic threefold, Ann. of Math., Volume 95 (1972) no. 2, pp. 281-356

[6] Druel, S. Espace des modules des faisceaux de rang 2 semi-stables de classes de Chern c1=0, c2=2 et c3=0 sur la cubique de P4, Int. Math. Res. Not., Volume 19 (2000), pp. 985-1004

[7] Fulton, W. Intersection Theory, Ergebnisse der Mathematik (3), vol. 2, Springer, Berlin, 1984

[8] Huybrechts, D.; Lehn, M. The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, vol. 31, Max-Planck-Institut für Mathematik, Bonn, 1997

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

[10] Markushevich, D.; Tikhomirov, A. 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] Merkurjev, A.S.; Suslin, A.A. 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] Murre, J.P. Applications of algebraic K-theory to the theory of algebraic cycles, Sitjes 1983 (LNM), Volume vol. 1124 (1985), pp. 216-261

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

Cited by Sources:

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