no. G1
Article de recherche - Géométrie algébrique
Fundamental groups of proper varieties are finitely presented
[Le groupe fondamental d’une variété propre est de présentation finie]
Lara, Marcin1 orcid  ; Srinivas, Vasudevan2 ; Stix, Jakob3
1Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
2TIFR, School of Mathematics, Homi Bhabha Road, Colaba, 400005 Mumbai, India
3Institut für Mathematik, Goethe–Universität Frankfurt, Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany
Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 51-54

It was proven in [1], that the étale fundamental group of a connected smooth projective variety over an algebraically closed field k is topologically finitely presented. In this note, we extend this result to all connected proper schemes over k.

Il a été prouvé dans [1], que le groupe fondamental étale d’une variété projective lisse connexe sur un corps algébriquement clos k est topologiquement de présentation finie. Dans cette note, nous étendons ce résultat à tous les schémas propres connexes sur k.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.518
Classification : 14F35, 14F20

Lara, Marcin  1   ; Srinivas, Vasudevan  2   ; Stix, Jakob  3

1 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
2 TIFR, School of Mathematics, Homi Bhabha Road, Colaba, 400005 Mumbai, India
3 Institut für Mathematik, Goethe–Universität Frankfurt, Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2024__362_G1_51_0,
     author = {Lara, Marcin and Srinivas, Vasudevan and Stix, Jakob},
     title = {Fundamental groups of proper varieties are finitely presented},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {51--54},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G1},
     doi = {10.5802/crmath.518},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.518/}
}
TY  - JOUR
AU  - Lara, Marcin
AU  - Srinivas, Vasudevan
AU  - Stix, Jakob
TI  - Fundamental groups of proper varieties are finitely presented
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 51
EP  - 54
VL  - 362
IS  - G1
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.518/
DO  - 10.5802/crmath.518
LA  - en
ID  - CRMATH_2024__362_G1_51_0
ER  - 
%0 Journal Article
%A Lara, Marcin
%A Srinivas, Vasudevan
%A Stix, Jakob
%T Fundamental groups of proper varieties are finitely presented
%J Comptes Rendus. Mathématique
%D 2024
%P 51-54
%V 362
%N G1
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.518/
%R 10.5802/crmath.518
%G en
%F CRMATH_2024__362_G1_51_0
Lara, Marcin; Srinivas, Vasudevan; Stix, Jakob. Fundamental groups of proper varieties are finitely presented. Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 51-54. doi: 10.5802/crmath.518

[1] Esnault, Hélène; Shusterman, Mark; Srinivas, Vasudevan Finite presentation of the tame fundamental group, Sel. Math., New Ser., Volume 28 (2022) no. 2, 37, 19 pages | DOI | MR | Zbl

[2] Esnault, Hélène; Srinivas, Vasudevan; Stix, Jakob An obstruction to lifting to characteristic 0, Algebr. Geom., Volume 10 (2023) no. 3, pp. 327-347 | DOI | MR | Zbl

[3] Revêtements étales et groupe fondamental (SGA1) (Grothendieck, Alexander, ed.), Documents Mathématiques, 3, Société Mathématique de France, 2003, xviii+327 pages (Séminaire de géométrie algébrique du Bois Marie 1960–61., Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original) | MR

[4] Hironaka, Heisuke Triangulations of algebraic sets, Algebraic geometry (Arcata, Calif., 1974) (Proc. Sympos. Pure Math.), Volume 29, American Mathematical Society (1975), pp. 165-185 | DOI | MR | Zbl

[5] Jarden, Moshe Algebraic extensions of finite corank of Hilbertian fields, Isr. J. Math., Volume 18 (1974), pp. 279-307 | DOI | MR | Zbl

[6] de Jong, Aise Johan Smoothness, semi-stability and alterations, IHES Publ. Math., Volume 83 (1996), pp. 51-93 | MR | Zbl | Numdam

[7] Łojasiewicz, Stanisław Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 18 (1964), pp. 449-474 | MR | Numdam | Zbl

[8] Lubotzky, Alexander Pro-finite presentations, J. Algebra, Volume 242 (2001) no. 2, pp. 672-690 | DOI | MR | Zbl

[9] Shusterman, Mark Balanced presentations for fundamental groups of curves over finite fields, Math. Res. Lett., Volume 29 (2022) no. 4, pp. 1251-1259 | DOI | MR | Zbl

[10] Stix, Jakob A general Seifert-Van Kampen theorem for algebraic fundamental groups, Publ. Res. Inst. Math. Sci., Volume 42 (2006) no. 3, pp. 763-786 | MR | DOI | Zbl

Cité par Sources :