Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T15:31:06.407Z Has data issue: false hasContentIssue false

ON THE INTEGRAL HODGE AND TATE CONJECTURES OVER A NUMBER FIELD

Published online by Cambridge University Press:  12 September 2013

BURT TOTARO*
Affiliation:
DPMMS, Wilberforce Road, Cambridge CB3 0WB, England UCLA Department of Mathematics, Box 951555, Los Angeles, CA 90095-1555, [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Hassett and Tschinkel gave counterexamples to the integral Hodge conjecture among 3-folds over a number field. We work out their method in detail, showing that essentially all known counterexamples to the integral Hodge conjecture over the complex numbers can be made to work over a number field.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

References

Artin, M., ‘Supersingular K3 surfaces’, Ann. Sci. Éc. Norm. Supér. 7 (1974), 543567.Google Scholar
Atiyah, M. and Hirzebruch, F., ‘Analytic cycles on complex manifolds’, Topology 1 (1962), 2545.CrossRefGoogle Scholar
Birkenhake, C. and Lange, H., Complex Abelian Varieties (Springer, 2004).CrossRefGoogle Scholar
Charles, F., The Tate conjecture for K3 surfaces over finite fields. http://arxiv.org/abs/1206.4002.Google Scholar
Colliot-Thélène, J.-L. and Szamuely, T., ‘Autour de la conjecture de Tate à coefficients ${\mathbf{Z} }_{l} $ pour les variétés sur les corps finis’, in The Geometry of Algebraic Cycles (AMS/Clay Institute Proceedings, 2010), 8398.Google Scholar
Colliot-Thélène, J.-L. and Voisin, C., ‘Cohomologie non ramifiée et conjecture de Hodge entière’, Duke Math. J. 161 (2012), 735801.CrossRefGoogle Scholar
Debarre, O., Hulek, K. and Spandaw, J., ‘Very ample linear systems on abelian varieties’, Math. Ann. 300 (1994), 181202.Google Scholar
Deligne, P., ‘La conjecture de Weil. I’, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273307.CrossRefGoogle Scholar
Deligne, P., ‘La conjecture de Weil. II’, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.Google Scholar
de Jong, A. J. and Katz, N., ‘Monodromy and the Tate conjecture: Picard numbers and Mordell–Weil ranks in families’, Israel J. Math. 120 (2000), 4779.Google Scholar
Fulton, W., Intersection Theory (Springer, 1984).CrossRefGoogle Scholar
Grabowski, C., On the integral Hodge conjecture for 3-folds. PhD Thesis, Duke University (2004).Google Scholar
Griffiths, P. and Harris, J., Principles of Algebraic Geometry (Wiley, 1978).Google Scholar
Griffiths, P. and Harris, J., ‘On the Noether–Lefschetz theorem and some remarks on codimension two cycles’, Math. Ann. 271 (1985), 3151.Google Scholar
Katz, N. and Mazur, B., Arithmetic Moduli of Elliptic Curves (Princeton, 1985).CrossRefGoogle Scholar
Kollár, J., ‘Trento examples’, in Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Mathematics, vol. 1515 (Springer, 1992), 134135.Google Scholar
Kollár, J., Rational Curves on Algebraic Varieties (Springer, 1996).CrossRefGoogle Scholar
Kollár, J., ‘Singularities of pairs’, Proc. Symp. Pure Math. 62, Part 1 (Amer. Math. Soc., 1997), 221287.Google Scholar
Maulik, D., Supersingular K3 surfaces for large primes. http://arxiv.org/abs/1203.2889.Google Scholar
Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic. http://arxiv.org/abs/1301.6326.Google Scholar
Milne, J., Étale Cohomology (Princeton, 1980).Google Scholar
Nori, M., ‘Algebraic cycles and Hodge-theoretic connectivity’, Invent. Math. 111 (1993), 349373.CrossRefGoogle Scholar
Nygaard, N. and Ogus, A., ‘Tate’s conjecture for K3 surfaces of finite height’, Ann. of Math. (2) 122 (1985), 461507.CrossRefGoogle Scholar
Rathmann, J., ‘The uniform position principle for curves in characteristic $p$ ’, Math. Ann. 276 (1987), 565579.Google Scholar
Schoen, C., ‘An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite fields’, Math. Ann. 311 (1998), 493500.Google Scholar
Serre, J.-P., ‘Zeta and $L$ functions’, in Arithmetical Algebraic Geometry (Purdue, 1963) (Harper & Row, 1965), 8292.Google Scholar
Shioda, T., ‘Algebraic cycles on a certain hypersurface’, in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, 1016 (Springer, 1983), 271294.Google Scholar
Soulé, C. and Voisin, C., ‘Torsion cohomology classes and algebraic cycles on complex projective manifolds’, Adv. Math. 198 (2005), 107127.Google Scholar
Tate, J., ‘Endomorphisms of abelian varieties over finite fields’, Invent. Math. 2 (1966), 134144.Google Scholar
Totaro, B., ‘Torsion algebraic cycles and complex cobordism’, J. Amer. Math. Soc. 10 (1997), 467493.Google Scholar
Totaro, B., ‘Non-injectivity of the map from the Witt group of a variety to the Witt group of its function field’, J. Inst. Math. Jussieu 2 (2003), 483493.CrossRefGoogle Scholar
Voisin, C., ‘On integral Hodge classes on uniruled and Calabi–Yau threefolds’, in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, 45 (2006), 4373.Google Scholar