Geodesic
The Jordan property for local fundamental groups
Braun, Lukas1 ; Filipazzi, Stefano2 ; Moraga, Joaquín3 ; Svaldi, Roberto2
1 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg im Breisgau, Germany
2 Institute of Mathematics, EPFL, Lausanne, Switzerland
3 Department of Mathematics, Princeton University, Princeton, NJ, United States
Geometry & topology, Tome 26 (2022) no. 1, pp. 283-319.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index 1 covers for n–dimensional klt singularities. We give an application to the study of local class groups of klt singularities.

Keywords: Local fundamental group, klt singularities, Jordan property

Braun, Lukas 1 ; Filipazzi, Stefano 2 ; Moraga, Joaquín 3 ; Svaldi, Roberto 2

1 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg im Breisgau, Germany
2 Institute of Mathematics, EPFL, Lausanne, Switzerland
3 Department of Mathematics, Princeton University, Princeton, NJ, United States 
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Braun, Lukas; Filipazzi, Stefano; Moraga, Joaquín; Svaldi, Roberto. The Jordan property for local fundamental groups. Geometry & topology, Tome 26 (2022) no. 1, pp. 283-319. http://geodesic.mathdoc.fr/item/GT_2022_26_1_a6/

[1] D Abramovich, J Wang, Equivariant resolution of singularities in characteristic 0, Math. Res. Lett. 4 (1997) 427 | DOI

[2] I V Arzhantsev, S A Gaĭfullin, Cox rings, semigroups, and automorphisms of affine varieties, Mat. Sb. 201 (2010) 3 | DOI

[3] P Belmans, A J De Jong, Others, The Stacks project, electronic reference (2005–)

[4] C Birkar, Anti-pluricanonical systems on Fano varieties, Ann. of Math. 190 (2019) 345 | DOI

[5] C Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. 193 (2021) 347 | DOI

[6] C Birkar, P Cascini, C D Hacon, J Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010) 405 | DOI

[7] L Braun, The local fundamental group of a Kawamata log terminal singularity is finite, Invent. Math. 226 (2021) 845 | DOI

[8] F Campana, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011) 809 | DOI

[9] M J Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007) 411 | DOI

[10] D A Cox, J B Little, H K Schenck, Toric varieties, 124, Amer. Math. Soc. (2011) | DOI

[11] A H Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983) 517 | DOI

[12] T De Fernex, J Kollár, C Xu, The dual complex of singularities, from: "Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday" (editors K Oguiso, C Birkar, S Ishii, S Takayama), Adv. Stud. Pure Math. 74, Math. Soc. Japan (2017) 103 | DOI

[13] H Flenner, Divisorenklassengruppen quasihomogener Singularitäten, J. Reine Angew. Math. 328 (1981) 128 | DOI

[14] H Flenner, M Zaidenberg, Locally nilpotent derivations on affine surfaces with a C∗–action, Osaka J. Math. 42 (2005) 931

[15] R M Fossum, The divisor class group of a Krull domain, 74, Springer (1973) | DOI

[16] A Fujiki, R Kobayashi, S Lu, On the fundamental group of certain open normal surfaces, Saitama Math. J. 11 (1993) 15

[17] M Goresky, R Macpherson, Morse theory and intersection homology theory, from: "Analysis and topology on singular spaces, II–III", Astérisque 101–102, Soc. Math. France (1983) 135

[18] M Goresky, R Macpherson, Stratified Morse theory, 14, Springer (1988) | DOI

[19] R M Goresky, Whitney stratified chains and cochains, Trans. Amer. Math. Soc. 267 (1981) 175 | DOI

[20] A Grassi, J Kollár, Reduction to special flips, from: "Flips and abundance for algebraic threefolds" (editor J Kollár), Astérisque 211, Soc. Math. France (1992) 193

[21] D Greb, S Kebekus, T Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J. 165 (2016) 1965 | DOI

[22] R V Gurjar, D Q Zhang, π1 of smooth points of a log del Pezzo surface is finite, I, J. Math. Sci. Univ. Tokyo 1 (1994) 137

[23] R V Gurjar, D Q Zhang, π1 of smooth points of a log del Pezzo surface is finite, II, J. Math. Sci. Univ. Tokyo 2 (1995) 165

[24] J Han, J Liu, V V Shokurov, ACC for minimal log discrepancies of exceptional singularities, preprint (2019)

[25] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[26] C Jordan, Mémoire sur une application de la théorie des substitutions à l’étude des équations différentielles linéaires, Bull. Soc. Math. France 2 (1873/74) 100

[27] M Kapovich, J Kollár, Fundamental groups of links of isolated singularities, J. Amer. Math. Soc. 27 (2014) 929 | DOI

[28] S Keel, J Mckernan, Rational curves on quasi-projective surfaces, 669, Amer. Math. Soc. (1999) | DOI

[29] J Kollár, New examples of terminal and log canonical singularities, preprint (2011)

[30] J Kollár, Singularities of the minimal model program, 200, Cambridge Univ. Press (2013) | DOI

[31] J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) | DOI

[32] A Laface, A Liendo, J Moraga, The fundamental group of a log terminal T–variety, Eur. J. Math. 5 (2019) 937 | DOI

[33] J Milnor, Singular points of complex hypersurfaces, 61, Princeton Univ. Press (1968)

[34] J Moraga, On minimal log discrepancies and Kollár components, preprint (2018)

[35] J Moraga, Fano-type surfaces with large cyclic automorphisms, Forum Math. Sigma 9 (2021) | DOI

[36] D Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961) 5 | DOI

[37] V L Popov, On the Makar–Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, from: "Affine algebraic geometry" (editors D Daigle, R Ganong, M Koras), CRM Proc. Lecture Notes 54, Amer. Math. Soc. (2011) 289 | DOI

[38] Y G Prokhorov, Blow-ups of canonical singularities, from: "Algebra" (editor Y Bahturin), de Gruyter (2000) 301

[39] Y G Prokhorov, V V Shokurov, The first fundamental theorem on complements: from global to local, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001) 99 | DOI

[40] Y G Prokhorov, C Shramov, Jordan property for groups of birational selfmaps, Compos. Math. 150 (2014) 2054 | DOI

[41] Y G Prokhorov, C Shramov, Jordan property for Cremona groups, Amer. J. Math. 138 (2016) 403 | DOI

[42] V V Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 105

[43] D E Speyer, Equivariant normalization?, MathOverflow post (2013)

[44] U Storch, Über die Divisorenklassengruppen normaler komplexanalytischer Algebren, Math. Ann. 183 (1969) 93 | DOI

[45] H Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974) 1 | DOI

[46] Z Tian, C Xu, Finiteness of fundamental groups, Compos. Math. 153 (2017) 257 | DOI

[47] C Xu, Finiteness of algebraic fundamental groups, Compos. Math. 150 (2014) 409 | DOI

[48] C Xu, Z Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021) 149 | DOI