Birational positivity in dimension 4
[Positivité birational en dimension 4]
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 203-216.

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.

DOI : 10.5802/aif.2845
Classification : 14J35, 14E30
Keywords: Kodaira dimension, varieties of Kodaira dimension zero, minimal model theory
Mot clés : dimension de Kodaira, variétés avec la dimension de Kodaira nulle, théorie du modéle minimal
Taji, Behrouz 1

1 McGill University The Department of Mathematics Burnside Hall, Room 1031 805 Sherbrooke W. Montreal, QC, H3A 0B9 (Canada)
@article{AIF_2014__64_1_203_0,
     author = {Taji, Behrouz},
     title = {Birational positivity in dimension $4$},
     journal = {Annales de l'Institut Fourier},
     pages = {203--216},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     doi = {10.5802/aif.2845},
     mrnumber = {3330547},
     zbl = {1326.14093},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2845/}
}
TY  - JOUR
AU  - Taji, Behrouz
TI  - Birational positivity in dimension $4$
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 203
EP  - 216
VL  - 64
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2845/
DO  - 10.5802/aif.2845
LA  - en
ID  - AIF_2014__64_1_203_0
ER  - 
%0 Journal Article
%A Taji, Behrouz
%T Birational positivity in dimension $4$
%J Annales de l'Institut Fourier
%D 2014
%P 203-216
%V 64
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2845/
%R 10.5802/aif.2845
%G en
%F AIF_2014__64_1_203_0
Taji, Behrouz. Birational positivity in dimension $4$. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 203-216. doi : 10.5802/aif.2845. http://www.numdam.org/articles/10.5802/aif.2845/

[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., Volume 22 (2013) no. 2, pp. 201-248 | DOI | MR | Zbl

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

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

[4] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 3, pp. 499-630 | DOI | Numdam | MR | Zbl

[5] Campana, Frédéric Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu, Volume 10 (2011) no. 4, pp. 809-934 | DOI | MR | Zbl

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

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

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

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

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

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

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

[13] Miyaoka, Yoichi Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.), Volume 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245-268 | MR | Zbl

[14] Raynaud, M. Flat modules in algebraic geometry, Compositio Math., Volume 24 (1972), pp. 11-31 | Numdam | MR | Zbl

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

Cité par Sources :