Birational positivity in dimension 4

Behrouz Taji[1]

  • [1] McGill University The Department of Mathematics Burnside Hall, Room 1031 805 Sherbrooke W. Montreal, QC, H3A 0B9 (Canada)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 1, page 203-216
  • ISSN: 0373-0956

Abstract

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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of Ω p is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an X provided that X has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

How to cite

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Taji, Behrouz. "Birational positivity in dimension $4$." Annales de l’institut Fourier 64.1 (2014): 203-216. <http://eudml.org/doc/275526>.

@article{Taji2014,
abstract = {In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $\Omega ^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.},
affiliation = {McGill University The Department of Mathematics Burnside Hall, Room 1031 805 Sherbrooke W. Montreal, QC, H3A 0B9 (Canada); Université Henri Poincaré Nancy 1 Institut Élie Cartan de Nancy, UMR 7502 B.P. 70239 54506 Vandoeuvre-lès-Nancy Cedex (France)},
author = {Taji, Behrouz},
journal = {Annales de l’institut Fourier},
keywords = {Kodaira dimension; varieties of Kodaira dimension zero; minimal model theory},
language = {eng},
number = {1},
pages = {203-216},
publisher = {Association des Annales de l’institut Fourier},
title = {Birational positivity in dimension $4$},
url = {http://eudml.org/doc/275526},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Taji, Behrouz
TI - Birational positivity in dimension $4$
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 1
SP - 203
EP - 216
AB - In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $\Omega ^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.
LA - eng
KW - Kodaira dimension; varieties of Kodaira dimension zero; minimal model theory
UR - http://eudml.org/doc/275526
ER -

References

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  1. Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, Thomas Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), 201-248 Zbl1267.32017MR3019449
  2. Frédéric Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), 539-545 Zbl0783.14022MR1191735
  3. Frédéric Campana, Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom. 4 (1995), 487-502 Zbl0845.32027MR1325789
  4. Frédéric Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499-630 Zbl1062.14014MR2097416
  5. Frédéric Campana, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011), 809-934 Zbl1236.14039MR2831280
  6. Frédéric Campana, Thomas Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France 139 (2011), 41-74 Zbl1218.14030MR2815027
  7. Paolo Cascini, Subsheaves of the cotangent bundle, Cent. Eur. J. Math. 4 (2006), 209-224 (electronic) Zbl1108.14009MR2221105
  8. Tom Graber, Joe Harris, Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57-67 (electronic) Zbl1092.14063MR1937199
  9. János Kollár, Flips and Abundance for Algebraic Threefolds, 211 (1992), Société Mathématique de France Zbl0782.00075MR1225842
  10. János Kollár, Yoichi Miyaoka, Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), 429-448 Zbl0780.14026MR1158625
  11. János Kollár, Shigefumi Mori, Birational geometry of algebraic varieties, 134 (1998), Cambridge University Press, Cambridge Zbl0926.14003
  12. Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985 10 (1987), 449-476, North-Holland, Amsterdam Zbl0648.14006MR946247
  13. Yoichi Miyaoka, Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) 46 (1987), 245-268, Amer. Math. Soc., Providence, RI Zbl0659.14008MR927960
  14. M. Raynaud, Flat modules in algebraic geometry, Compositio Math. 24 (1972), 11-31 Zbl0244.14001MR302645
  15. Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799 Zbl0355.32028MR451180

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