Taji, Behrouz
Birational positivity in dimension 4  [ Positivité birational en dimension 4 ]
Annales de l'institut Fourier, Tome 64 (2014) no. 1 , p. 203-216
MR 3330547 | Zbl 06387272
doi : 10.5802/aif.2845
URL stable : http://www.numdam.org/item?id=AIF_2014__64_1_203_0

Classification:  14J35,  14E30
Mots clés: dimension de Kodaira, variétés avec la dimension de Kodaira nulle, théorie du modéle minimal
Dans cet article, nous montrons que pour une variété projective lisse, X, de dimension au plus 4 et de dimension de Kodaira non négative, la dimension de Kodaira des sous-faisceaux cohérents de Ω p est majorée par la dimension de Kodaira de X. Cela implique la finitude du groupe fondamental de X lorsque la dimension de Kodaira de X 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 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.

Bibliographie

[1] Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., 22 (2013) no. 2, p. 201–248 Article  MR 3019449 | Zbl 1267.32017

[2] Campana, Frédéric Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4), 25 (1992) no. 5, p. 539–545 Numdam | MR 1191735 | Zbl 0783.14022

[3] Campana, Frédéric Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom., 4 (1995) no. 3, p. 487–502 MR 1325789 | Zbl 0845.32027

[4] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), 54 (2004) no. 3, p. 499–630 Article  Numdam | MR 2097416 | Zbl 1062.14014

[5] Campana, Frédéric Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu, 10 (2011) no. 4, p. 809–934 Article  MR 2831280 | Zbl 1236.14039

[6] Campana, Frédéric; Peternell, Thomas Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France, 139 (2011) no. 1, p. 41–74 (With an appendix by Matei Toma) Numdam | MR 2815027 | Zbl 1218.14030

[7] Cascini, Paolo Subsheaves of the cotangent bundle, Cent. Eur. J. Math., 4 (2006) no. 2, p. 209–224 (electronic) Article  MR 2221105 | Zbl 1108.14009

[8] Graber, Tom; Harris, Joe; Starr, Jason Families of rationally connected varieties, J. Amer. Math. Soc., 16 (2003) no. 1, p. 57–67 (electronic) Article  MR 1937199 | Zbl 1092.14063

[9] Kollár, János Flips and Abundance for Algebraic Threefolds, Société Mathématique de France, Astérisque, 211 (1992) MR 1225842

[10] Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi Rationally connected varieties, J. Algebraic Geom., 1 (1992) no. 3, p. 429–448 MR 1158625 | Zbl 0780.14026

[11] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, 134 (1998), p. viii+254 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) Article  Zbl 0926.14003

[12] Miyaoka, Yoichi The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, North-Holland, Amsterdam (Adv. Stud. Pure Math.) 10 (1987), p. 449–476 MR 946247 | Zbl 0648.14006

[13] Miyaoka, Yoichi 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.) 46 (1987), p. 245–268 MR 927960 | Zbl 0659.14008

[14] Raynaud, M. Flat modules in algebraic geometry, Compositio Math., 24 (1972), p. 11–31 Numdam | MR 302645 | Zbl 0244.14001

[15] Yau, Shing Tung Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A., 74 (1977) no. 5, p. 1798–1799 Article  MR 451180 | Zbl 0355.32028