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Approximately Rationally or Elliptically Connected Varieties

Published online by Cambridge University Press:  17 December 2013

Claire Voisin
Claire Voisin*
Affiliation:
Institut de Mathématiques de Jussieu, TGA Case 247, 4 Place Jussieu, 75005 Paris, France ([email protected])
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Abstract

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We discuss a possible approach to the study of the vanishing of the Kobayashi pseudo-metric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighbourhood of X in projective space for the Euclidean topology.

Keywords

rational curvesrational connectednessKobayashi hyperbolicityPrimary 32Q4514M22
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 281 - 297
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Boucksom, S., Demailly, J.-P., Paun, M. and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension,J. Alg. Geom. 22 (2013), 201–248.CrossRefGoogle Scholar
2.Brody, R., Compact manifolds and hyperbolicity, Trans. Am. Math. Soc. 235 (1978), 213–219.Google Scholar
3.Campana, F., Connexitéabélienne des variétés kählériennes compactes, Bull. Soc. Math. France 126(4) (1998), 483–506.Google Scholar
4.Campana, F., Orbifolds, special varieties and classification theory, Annales Inst. Fourier 54(3) (2004), 499–630.Google Scholar
5.Chen, X. and Lewis, J., Density of rational curves on K3 surfaces, Math. Ann. 365(1) (2013), 331–354.CrossRefGoogle Scholar
6.Clemens, H., Curves on higher-dimensional complex projective manifolds, in Proceedings of the International Congress of Mathematicians, 1986, Volume 1, pp. 634–640 (American Mathematical Society, Providence, RI, 1986).Google Scholar
7.Clemens, H., Curves in generic hypersurfaces, Annales Scient. Éc. Norm. Sup. 19 (1986), 629–636.Google Scholar
8.Clemens, H., Kollár, J. and Mori, S., Higher-Dimensional Complex Geometry: A Summer Seminar at the University of Utah, Salt Lake City, 1987, Astérisque, Volume 166 (Société Mathématique de France, Paris, 1988).Google Scholar
9.Demailly, J.-P., Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, in Algebraic Geometry – Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, Volume 62, Part 2, pp. 285–360 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
10.Diverio, S., Merker, J. and Rousseau, E., Effective algebraic degeneracy, Invent. Math. 180(1) (2010), 161–223.Google Scholar
11.Duval, J., Singularités des courants d'Ahlfors, Annales Scient. Éc. Norm. Sup. 39(3) (2006), 527–533.Google Scholar
12.Duval, J., Sur le lemme de Brody, Invent. Math. 173 (2008), 305–314.Google Scholar
13.Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Am. Math. Soc. 16(1) (2003), 57–67.Google Scholar
14.Kawamata, Y., On Bloch's conjecture, Invent. Math. 57 (1980), 97–100.Google Scholar
15.Kobayashi, S., Intrinsic distances, measures and geometric function theory, Bull. Am. Math. Soc. 82 (1976), 357–416.CrossRefGoogle Scholar
16.Kollár, J., Miyaoka, Y. and Mori, S., Rationally connected varieties. J. Alg. Geom. 1(3) (1992), 429–448.Google Scholar
17.Lang, S., Hyperbolic and diophantine analysis, Bull. Am. Math. Soc. 14(2) (1986), 159–205.CrossRefGoogle Scholar
18.Pacienza, G., Rational curves on general projective hypersurfaces, J. Alg. Geom. 12(2) (2003), 245–267.CrossRefGoogle Scholar
19.Păun, M., Courants d'Ahlfors et localisation des courbes entières, in Séminaire Bourbaki, Volume 2007/2008, Exposés 982–996, Astérisque, Volume 326, pp. 281–297 (Société Mathématique de France, Paris, 2009).Google Scholar
20.Soulé, C. and Voisin, C., Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198(1) (2005), 107–127.Google Scholar
21.Voisin, C., On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom. 44(1) (1996), 200–213 (erratum: J. Diff. Geom. 49(3) (1998), 601–611).Google Scholar
22.Voisin, C., On some problems of Kobayashi and Lang: algebraic approaches, in Current developments in mathematics, 2003, pp. 53–125 (International Press, Somerville, MA, 2003).Google Scholar
23.Voisin, C., Hodge theory and complex algebraic geometry, II, Cambridge Studies in Advanced Mathematics, Volume 77 (Cambridge University Press, 2003).Google Scholar
24.Voisin, C., Intrinsic pseudo-volume forms and K-correspondences, in The Fano Conference, University of Torino, Turin, 2004, pp. 761–792 (Dipartimento di Matematica, Turin, 2004).Google Scholar
25.Voisin, C., A geometric application of Nori's connectivity theorem, Annali Scuola Norm. Sup. Pisa 3(3) (2004), 637–656.Google Scholar
26.Voisin, C., On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157(2) (2004), 329–343.Google Scholar