Integral Hodge classes on fourfolds fibered by quadric bundles
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- by Zhiyuan Li and Zhiyu Tian PDF
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Abstract:
We discuss the space of sections and certain bisections on a quadric surfaces bundle $X$ over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As an application, we prove that the integral Hodge conjecture holds for degree $4$ integral Hodge classes (IHC) of fourfolds fibered by quadric bundles over a smooth curve. This gives an alternative proof of a result of Colliot-Thélène and Voisin.References
- M. F. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962), 25–45. MR 145560, DOI 10.1016/0040-9383(62)90094-0
- S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), no. 5, 1235–1253. MR 714776, DOI 10.2307/2374341
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
- Jean-Louis Colliot-Thélène and Manuel Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), no. 1, 141–158 (French). MR 999316, DOI 10.1007/BF01850658
- Jean-Louis Colliot-Thélène and Claire Voisin, Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J. 161 (2012), no. 5, 735–801 (French, with English and French summaries). MR 2904092, DOI 10.1215/00127094-1548389
- A. Conte and J. P. Murre, The Hodge conjecture for fourfolds admitting a covering by rational curves, Math. Ann. 238 (1978), no. 1, 79–88. MR 510310, DOI 10.1007/BF01351457
- Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
- Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040
- Joe Harris and Loring W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84. MR 721453, DOI 10.1016/0040-9383(84)90026-0
- Brendan Hassett and Yuri Tschinkel, Spaces of sections of quadric surface fibrations over curves, Compact moduli spaces and vector bundles, Contemp. Math., vol. 564, Amer. Math. Soc., Providence, RI, 2012, pp. 227–249. MR 2894638, DOI 10.1090/conm/564/11161
- Brendan Hassett, Anthony Várilly-Alvarado, and Patrick Varilly, Transcendental obstructions to weak approximation on general K3 surfaces, Adv. Math. 228 (2011), no. 3, 1377–1404. MR 2824558, DOI 10.1016/j.aim.2011.06.017
- A. N. Tjurin, Five lectures on three-dimensional varieties, Uspehi Mat. Nauk 27 (1972), no. 5, (167), 3–50 (Russian). MR 0412196
- Claire Voisin, Hodge theory and complex algebraic geometry. I, Reprint of the 2002 English edition, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007. Translated from the French by Leila Schneps. MR 2451566
- Claire Voisin, Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), no. 2, 261–296. MR 2342587, DOI 10.1007/s11537-007-0639-x
- Claire Voisin, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom. 22 (2013), no. 1, 141–174. MR 2993050, DOI 10.1090/S1056-3911-2012-00597-9
- Yi Zhu, Homogeneous Fibrations over Curves, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 3067976
Additional Information
- Zhiyuan Li
- Affiliation: Mathematisches Institut, University of Bonn, Endenicher Allee 60, Bonn 53115, Germany
- Email: zhiyli@math.uni-bonn.de
- Zhiyu Tian
- Affiliation: CNRS, Institute Fourier, UMR, 5582, 100 Rue des Mathématiques, BP 74, 38402, Saint-Martin d’Héres, France
- MR Author ID: 975027
- Email: zhiyu.tian@ujf-grenoble.fr
- Received by editor(s): November 8, 2014
- Received by editor(s) in revised form: July 29, 2015, October 12, 2015, and October 18, 2015
- Published electronically: March 17, 2016
- Communicated by: Lev Borisov
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3333-3345
- MSC (2010): Primary 14C25, 14C30
- DOI: https://doi.org/10.1090/proc/12999
- MathSciNet review: 3503702