Algebraic geometry/Topology
Conjugate complex homogeneous spaces with non-isomorphic fundamental groups
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1001-1005.

Let X=G/Γ be the quotient of a connected reductive algebraic C-group G by a finite subgroup Γ. We describe the topological fundamental group of the homogeneous space X, which is nonabelian when Γ is nonabelian. Further, we construct an example of a homogeneous space X and an automorphism σ of C such that the topological fundamental groups of X and of the conjugate variety σX are not isomorphic.

Soit X=G/Γ le quotient d'un C-groupe algébrique réductif connexe G par un sous-groupe fini Γ. On décrit le groupe fondamental topologique de l'espace homogène X, qui est non abélien quand Γ est non abélien. Puis on construit un exemple d'espace homogène X et d'automorphisme σ de C tels que les groupes fondamentaux topologiques de X et de la variété conjuguée σX ne sont pas isomorphes.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.010
Keywords: Fundamental group, Conjugate variety, Homogeneous space, Linear algebraic group
Borovoi, Mikhail 1; Cornulier, Yves 2

1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 6997801 Tel Aviv, Israel
2 Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay cedex, France
@article{CRMATH_2015__353_11_1001_0,
     author = {Borovoi, Mikhail and Cornulier, Yves},
     title = {Conjugate complex homogeneous spaces with non-isomorphic fundamental groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1001--1005},
     publisher = {Elsevier},
     volume = {353},
     number = {11},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.010/}
}
TY  - JOUR
AU  - Borovoi, Mikhail
AU  - Cornulier, Yves
TI  - Conjugate complex homogeneous spaces with non-isomorphic fundamental groups
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1001
EP  - 1005
VL  - 353
IS  - 11
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.09.010/
DO  - 10.1016/j.crma.2015.09.010
LA  - en
ID  - CRMATH_2015__353_11_1001_0
ER  - 
%0 Journal Article
%A Borovoi, Mikhail
%A Cornulier, Yves
%T Conjugate complex homogeneous spaces with non-isomorphic fundamental groups
%J Comptes Rendus. Mathématique
%D 2015
%P 1001-1005
%V 353
%N 11
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.09.010/
%R 10.1016/j.crma.2015.09.010
%G en
%F CRMATH_2015__353_11_1001_0
Borovoi, Mikhail; Cornulier, Yves. Conjugate complex homogeneous spaces with non-isomorphic fundamental groups. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1001-1005. doi : 10.1016/j.crma.2015.09.010. http://www.numdam.org/articles/10.1016/j.crma.2015.09.010/

[1] Bauer, I.; Catanese, F.; Grunewald, F. Faithful actions of the absolute Galois group on connected components of moduli spaces, Invent. Math., Volume 199 (2015), pp. 859-888

[2] Baumslag, G. Residually finite groups with the same finite images, Compos. Math., Volume 29 (1974), pp. 249-252

[3] González-Diez, G.; Jaikin-Zapirain, A. The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, Proc. London Math. Soc. (2015) (in press) | DOI

[4] Milne, J.S.; Suh, J. Nonhomeomorphic conjugates of connected Shimura varieties, Amer. J. Math., Volume 132 (2010), pp. 731-750

[5] Rajan, C.S. An example of non-homeomorphic conjugate varieties, Math. Res. Lett., Volume 18 (2011), pp. 937-943

[6] Serre, J.-P. Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris, Volume 258 (1964), pp. 4194-4196

Cited by Sources: