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Pseudoeffective and nef classes on abelian varieties

Part of: Abelian varieties and schemes Cycles and subschemes

Published online by Cambridge University Press:  01 June 2011

Olivier Debarre ,
Lawrence Ein ,
Robert Lazarsfeld  and
Claire Voisin
Olivier Debarre
Affiliation:
École Normale Supérieure, Département de Mathématiques et Applications, UMR CNRS 8553, 45 rue d’Ulm, 75230 Paris cedex 05, France (email: odebarre@dma.ens.fr)
Lawrence Ein
Affiliation:
Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA (email: ein@math.uic.edu)
Robert Lazarsfeld
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (email: rlaz@umich.edu)
Claire Voisin
Affiliation:
Institut de Mathématiques de Jussieu, Case 247, 4 Place Jussieu, 75005 Paris, France (email: voisin@math.jussieu.fr)
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Abstract

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We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian surface with itself. In both cases, we find for instance the existence of nef classes that are not pseudoeffective, answering in the negative a question raised by Grothendieck in correspondence with Mumford. We also discuss several problems and questions for further investigation.

Keywords

pseudoeffective and nef cyclesabelian variety

MSC classification

Secondary: 14C25: Algebraic cycles 14K99: None of the above, but in this section

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1793 - 1818
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Bar02]Barvinok, A., A course in convexity, Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, RI, 2002).CrossRefGoogle Scholar
[BCHM10]Birkar, C., Cascini, P., Hacon, C. D. and Mc Kernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.Google Scholar
[BL04]Birkenhake, C. and Lange, H., Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, vol. 302, second edition (Springer, Berlin, 2004).Google Scholar
[Cha08]Chan, K., A characterization of double covers of curves in terms of the ample cone of second symmetric product, J. Pure Appl. Algebra 212 (2008), 2623–2632.CrossRefGoogle Scholar
[Cut00]Cutkosky, S. D., Irrational asymptotic behavior of Castelnuovo–Mumford regularity, J. Reine Angew. Math. 522 (2000), 93–103.Google Scholar
[Dem]Demailly, J.-P., Complex analytic and differential geometry, book draft available at http://www-fourier.ujf-grenoble.fr/∼demailly/manuscripts/agbook.pdf.Google Scholar
[Ful09]Fulger, M., The cones of effective cycles on projective bundles over curves, arXiv:0910.3703v1 [math.AG], Math. Z., to appear.Google Scholar
[FL83]Fulton, W. and Lazarsfeld, R., Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), 35–60.CrossRefGoogle Scholar
[Gro64]Grothendieck, A., Letter to David Mumford, 31 August 1964, in David Mumford, Selected papers, Volume II (Springer, New York, 2010).Google Scholar
[Kle66]Kleiman, S., Towards a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344.CrossRefGoogle Scholar
[Kou93]Kouvidakis, A., Divisors on symmetric products of curves, Trans. Amer. Math. Soc. 337 (1993), 117–128.CrossRefGoogle Scholar
[Law75]Lawson, H. B., The stable homology of a flat torus, Math. Scand. 36 (1975), 49–73.Google Scholar
[Laz04]Lazarsfeld, R., Positivity in algebraic geometry I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vols. 48 and 49 (Springer, Berlin, 2004).Google Scholar
[Lie68]Lieberman, D., Numerical and homological equivalence of algebraic cycles on Hodge manifolds, Amer. J. Math. 90 (1968), 366–374.CrossRefGoogle Scholar
[Mum70]Mumford, D., Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5 (Oxford University Press, London, 1970), reprinted by Hindustan Book Agency, New Delhi, 2008.Google Scholar
[Mus11]Mustopa, Y., Residuation of linear series and the effective cone of C d, Amer. J. Math. 133 (2011), 393–416.CrossRefGoogle Scholar
[Pac03]Pacienza, G., On the nef cone of symmetric products of a generic curve, Amer. J. Math. 125 (2003), 1117–1135.CrossRefGoogle Scholar
[Pet09]Peternell, T., Submanifolds with ample normal bundles and a conjecture of Hartshorne, in Interactions of classical and numerical algebraic geometry, Contemporary Mathematics, vol. 496 (American Mathematical Society, Providence, RI, 2009), 317–330.Google Scholar
[Rib83]Ribet, K., Hodge classes on certain types of abelian varieties, Amer. J. Math. 105 (1983), 523–538.CrossRefGoogle Scholar
[Tan82]Tankeev, S. G., Cycles on simple abelian varieties of prime dimension, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 155–170, 192 (in Russian).Google Scholar
[Voi10]Voisin, C., Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal. 19 (2010), 1494–1513.Google Scholar