Dans cet article, nous montrons que pour une variété projective lisse, , de dimension au plus et de dimension de Kodaira non négative, la dimension de Kodaira des sous-faisceaux cohérents de est majorée par la dimension de Kodaira de . Cela implique la finitude du groupe fondamental de lorsque la dimension de Kodaira de est nulle et sa caractéristique holomorphe d’Euler est non nulle.
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 is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an provided that has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.
@article{AIF_2014__64_1_203_0,
author = {Taji, Behrouz},
title = {Birational positivity in dimension $4$},
journal = {Annales de l'Institut Fourier},
volume = {64},
year = {2014},
pages = {203-216},
doi = {10.5802/aif.2845},
zbl = {06387272},
mrnumber = {3330547},
zbl = {1326.14093},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2014__64_1_203_0}
}
Taji, Behrouz. Birational positivity in dimension $4$. Annales de l'Institut Fourier, Tome 64 (2014) pp. 203-216. doi : 10.5802/aif.2845. http://gdmltest.u-ga.fr/item/AIF_2014__64_1_203_0/
[1] The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., Tome 22 (2013) no. 2, pp. 201-248 | Article | MR 3019449 | Zbl 1267.32017
[2] Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4), Tome 25 (1992) no. 5, pp. 539-545 | Numdam | MR 1191735 | Zbl 0783.14022
[3] Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom., Tome 4 (1995) no. 3, pp. 487-502 | MR 1325789 | Zbl 0845.32027
[4] Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 3, pp. 499-630 http://aif.cedram.org/item?id=AIF_2004__54_3_499_0 | Article | Numdam | MR 2097416 | Zbl 1062.14014
[5] Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu, Tome 10 (2011) no. 4, pp. 809-934 | Article | MR 2831280 | Zbl 1236.14039
[6] Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France, Tome 139 (2011) no. 1, pp. 41-74 (With an appendix by Matei Toma) | Numdam | MR 2815027 | Zbl 1218.14030
[7] Subsheaves of the cotangent bundle, Cent. Eur. J. Math., Tome 4 (2006) no. 2, p. 209-224 (electronic) | Article | MR 2221105 | Zbl 1108.14009
[8] Families of rationally connected varieties, J. Amer. Math. Soc., Tome 16 (2003) no. 1, p. 57-67 (electronic) | Article | MR 1937199 | Zbl 1092.14063
[9] Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Astérisque, Tome 211 (1992) | MR 1225842
[10] Rationally connected varieties, J. Algebraic Geom., Tome 1 (1992) no. 3, pp. 429-448 | MR 1158625 | Zbl 0780.14026
[11] Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 134 (1998), pp. viii+254 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | Article | Zbl 0926.14003
[12] The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, North-Holland, Amsterdam (Adv. Stud. Pure Math.) Tome 10 (1987), pp. 449-476 | MR 946247 | Zbl 0648.14006
[13] Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 46 (1987), pp. 245-268 | MR 927960 | Zbl 0659.14008
[14] Flat modules in algebraic geometry, Compositio Math., Tome 24 (1972), pp. 11-31 | Numdam | MR 302645 | Zbl 0244.14001
[15] Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A., Tome 74 (1977) no. 5, p. 1798-1799 | Article | MR 451180 | Zbl 0355.32028