A splitting theorem for good complexifications
[Un théorème de décomposition pour les bonnes complexifications]
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1965-1985.

Le but de cet article est de montrer qu’il existe des restrictions aux groupes fondamentaux que peuvent avoir les variétés admettant une bonne complexification, en démontrant le théorème suivant de décomposition, de type Cheeger–Gromoll : Toute variété fermée M admettant une bonne complexification a un recouvrement fini M 1 , possédant un structure de fibré de base (S 1 ) k et de fibre N ayant une bonne complexification et un premier nombre de Betti virtuel nul. On donne plusieurs applications de ce théorème aux variétés de dimension au plus 5.

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold M admitting a good complexification has a finite-sheeted regular covering M 1 such that M 1 admits a fiber bundle structure with base (S 1 ) k and fiber N that admits a good complexification and also has zero virtual first Betti number. We give several applications to manifolds of dimension at most 5.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3054
Classification : 14P25,  57M05,  14F35,  20F65,  57M50,  57M07,  20F67
Mots clés : bonne complexification, théorème de décomposition, variété affine, groupe fondamental, théorème de géométrisation, nombre de Betti virtuel.
@article{AIF_2016__66_5_1965_0,
     author = {Biswas, Indranil and Mj, Mahan and Parameswaran, A. J.},
     title = {A splitting theorem for good complexifications},
     journal = {Annales de l'Institut Fourier},
     pages = {1965--1985},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     doi = {10.5802/aif.3054},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3054/}
}
TY  - JOUR
AU  - Biswas, Indranil
AU  - Mj, Mahan
AU  - Parameswaran, A. J.
TI  - A splitting theorem for good complexifications
JO  - Annales de l'Institut Fourier
PY  - 2016
DA  - 2016///
SP  - 1965
EP  - 1985
VL  - 66
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3054/
UR  - https://doi.org/10.5802/aif.3054
DO  - 10.5802/aif.3054
LA  - en
ID  - AIF_2016__66_5_1965_0
ER  - 
Biswas, Indranil; Mj, Mahan; Parameswaran, A. J. A splitting theorem for good complexifications. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1965-1985. doi : 10.5802/aif.3054. https://aif.centre-mersenne.org/articles/10.5802/aif.3054/

[1] Agol, Ian The virtual Haken conjecture, Doc. Math., Tome 18 (2013), pp. 1045-1087 (With an appendix by Agol, Daniel Groves, and Jason Manning)

[2] Amorós, J.; Burger, M.; Corlette, K.; Kotschick, D.; Toledo, D. Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, Tome 44, American Mathematical Society, Providence, RI, 1996, xii+140 pages | Article

[3] Arapura, Donu; Nori, Madhav Solvable fundamental groups of algebraic varieties and Kähler manifolds, Compositio Math., Tome 116 (1999) no. 2, pp. 173-188 | Article

[4] Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry 3-manifold groups (preprint, http://arxiv.org/abs/1205.0202v2)

[5] Biswas, Indranil; Mj, Mahan Quasiprojective three-manifold groups and complexification of three-manifolds, Int. Math. Res. Not., Tome 2015 (2015) no. 20, pp. 10041-10068 | Article

[6] Farb, B. Relatively hyperbolic groups, Geom. Funct. Anal., Tome 8 (1998) no. 5, pp. 810-840 | Article

[7] Fujino, Osamu On quasi-Albanese maps (preliminary version, https://www.math.kyoto-u.ac.jp/~fujino/quasi-albanese.pdf.)

[8] Gromov, Michael Curvature, diameter and Betti numbers, Comment. Math. Helv., Tome 56 (1981) no. 2, pp. 179-195 | Article

[9] Gromov, Michael Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.) Tome 8, Springer, New York, 1987, pp. 75-263 | Article

[10] Gromov, Michael Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.) Tome 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295

[11] Gromov, Michael Number of Questions (2014) (preprint)

[12] Haglund, Frédéric; Wise, Daniel T. Coxeter groups are virtually special, Adv. Math., Tome 224 (2010) no. 5, pp. 1890-1903 | Article

[13] Hatcher, Allen E. A proof of the Smale conjecture, Diff (S 3 )O(4), Ann. of Math. (2), Tome 117 (1983) no. 3, pp. 553-607 | Article

[14] Iitaka, Shigeru Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 23 (1976) no. 3, pp. 525-544

[15] Kulkarni, R. S. On complexifications of differentiable manifolds, Invent. Math., Tome 44 (1978) no. 1, pp. 46-64 | Article

[16] Lamotke, Klaus The topology of complex projective varieties after S. Lefschetz, Topology, Tome 20 (1981) no. 1, pp. 15-51 | Article

[17] McLean, Mark The growth rate of symplectic homology and affine varieties, Geom. Funct. Anal., Tome 22 (2012) no. 2, pp. 369-442 | Article

[18] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi The second main theorem for holomorphic curves into semi-abelian varieties, Acta Math., Tome 188 (2002) no. 1, pp. 129-161 | Article

[19] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, Tome 171, Springer, New York, 2006, xvi+401 pages

[20] Sageev, Michah Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3), Tome 71 (1995) no. 3, pp. 585-617 | Article

[21] Serre, Jean-Pierre Morphismes universels et variété d’Albanese, Séminaire Claude Chevalley (1958-1959), Secrétariat mathématique, 1960 no. 10, pp. 1-22

[22] Smale, Stephen Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., Tome 10 (1959), pp. 621-626

[23] Spieß, Michael; Szamuely, Tamás On the Albanese map for smooth quasi-projective varieties, Math. Ann., Tome 325 (2003) no. 1, pp. 1-17 | Article

[24] Suzuki, Masakazu Sur les opérations holomorphes du groupe additif complexe sur l’espace de deux variables complexes, Ann. Sci. École Norm. Sup. (4), Tome 10 (1977) no. 4, pp. 517-546

[25] Tibăr, Mihai Polynomials and vanishing cycles, Cambridge Tracts in Mathematics, Tome 170, Cambridge University Press, Cambridge, 2007, xii+253 pages | Article

[26] Totaro, Burt Complexifications of nonnegatively curved manifolds, J. Eur. Math. Soc. (JEMS), Tome 5 (2003) no. 1, pp. 69-94 | Article

[27] Wise, Daniel T The structure of groups with a quasi-convex hierarchy (2012) (preprint)

Cité par Sources :