Finite quotients of three-dimensional complex tori
[Quotients finis des tores complexes de dimension trois]
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 881-914.

Nous fournissons une caractérisation des quotients des tores complexes de dimension trois par l’action libre en codimension un d’un groupe fini, par une condition d’annulation de la première et deuxième classe de Chern orbifolde. Nous traitons aussi le cas des actions libres en codimension deux, utilisant la deuxième classe de Chern « birationelle », comme nous l’appelons, au lieu de la classe de Chern orbifolde.

Toutes les deux notions des classes de Chern sont introduites ici dans le cadre des espaces complexes compacts avec des singularités klt. Dans cette généralité, le sujet n’a pas été traité dans la littérature jusqu’à maintenant. Nous discutons aussi le rapport de notre définition aux classes de Chern classiques de Schwartz–MacPherson.

We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of free action in codimension two, using instead the “birational” second Chern class, as we call it.

Both notions of Chern classes are introduced here in the setting of compact complex spaces with klt singularities. In such generality, this topic has not been treated in the literature up to now. We also discuss the relation of our definitions to the classical Schwartz–MacPherson Chern classes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3326
Classification : 32J27, 32S20, 53C55, 14E30
Keywords: Complex tori, torus quotients, vanishing Chern classes, second orbifold Chern class, Minimal Model Program, klt singularities
Mot clés : Tores complexes, quotients des tores, annulation des classes de Chern, deuxième classe de Chern orbifolde, Programme des Modèles Minimaux, singularités klt
Graf, Patrick 1 ; Kirschner, Tim 2

1 Universität Bayreuth Lehrstuhl für Mathematik I 95440 Bayreuth (Germany)
2 Universität Duisburg–Essen Fakultät für Mathematik 45117 Essen (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AIF_2020__70_2_881_0,
     author = {Graf, Patrick and Kirschner, Tim},
     title = {Finite quotients of three-dimensional complex tori},
     journal = {Annales de l'Institut Fourier},
     pages = {881--914},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     doi = {10.5802/aif.3326},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3326/}
}
TY  - JOUR
AU  - Graf, Patrick
AU  - Kirschner, Tim
TI  - Finite quotients of three-dimensional complex tori
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 881
EP  - 914
VL  - 70
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3326/
DO  - 10.5802/aif.3326
LA  - en
ID  - AIF_2020__70_2_881_0
ER  - 
%0 Journal Article
%A Graf, Patrick
%A Kirschner, Tim
%T Finite quotients of three-dimensional complex tori
%J Annales de l'Institut Fourier
%D 2020
%P 881-914
%V 70
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3326/
%R 10.5802/aif.3326
%G en
%F AIF_2020__70_2_881_0
Graf, Patrick; Kirschner, Tim. Finite quotients of three-dimensional complex tori. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 881-914. doi : 10.5802/aif.3326. https://aif.centre-mersenne.org/articles/10.5802/aif.3326/

[1] Bhatt, Bhargav; Carvajal-Rojas, Javier; Graf, Patrick; Schwede, Karl; Tucker, Kevin Étale fundamental groups of strongly F-regular schemes, Int. Math. Res. Not. (2019), pp. 4325-4339 | DOI

[2] Bingener, Jürgen On deformations of Kähler spaces. I, Math. Z., Volume 182 (1983) no. 4, pp. 505-535 | DOI | Zbl

[3] Campana, Frédéric Orbifoldes à première classe de Chern nulle, The Fano Conference, Univ. Torino, Turin, 2004, pp. 339-351 | Zbl

[4] Campana, Frédéric; Höring, Andreas; Peternell, Thomas Abundance for Kähler threefolds, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 4, pp. 971-1025 | DOI | Zbl

[5] Druel, Stéphane The Zariski–Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc., Volume 46 (2014) no. 4, pp. 827-835 | DOI | MR | Zbl

[6] Graf, Patrick Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann., Volume 371 (2018), pp. 487-516 | DOI | MR | Zbl

[7] Graf, Patrick; Kovács, Sándor J. An optimal extension theorem for 1-forms and the Lipman-Zariski Conjecture, Doc. Math., Volume 19 (2014), pp. 815-830 | MR | Zbl

[8] Grauert, Hans; Remmert, Reinhold Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer, 1984, xviii+249 pages | MR | Zbl

[9] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 114 (2011), pp. 1-83 | MR | Zbl

[10] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz The Miyaoka–Yau inequality and uniformisation of canonical models (2016) (http://arxiv.org/abs/1511.08822v2, to appear in Ann. Sci. Éc. Norm. Supér.) | Zbl

[11] Greb, Daniel; Stefan, Kebekus; Peternell, Thomas Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties, Duke Math. J., Volume 165 (2016) no. 10, pp. 1965-2004 | DOI | Zbl

[12] Höring, Andreas; Peternell, Thomas Minimal models for Kähler threefolds, Invent. Math., Volume 203 (2016) no. 1, pp. 217-264 | DOI | Zbl

[13] Iversen, Birger Cohomology of sheaves, Universitext, Springer, 1986 | Zbl

[14] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987 (Kanô Memorial Lectures, 5) | MR | Zbl

[15] Kollár, János Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, 2007 | MR | Zbl

[16] Kollár, János; Mori, Shigefumi Classification of Three-Dimensional Flips, J. Am. Math. Soc., Volume 5 (1992) no. 3, pp. 533-703 | DOI | MR | Zbl

[17] Kollár, János; Mori, Shigefumi Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 | MR | Zbl

[18] Lazarsfeld, Robert Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004 | MR | Zbl

[19] MacPherson, Robert D. Chern classes for singular algebraic varieties, Ann. Math., Volume 100 (1974), pp. 423-432 | DOI | MR | Zbl

[20] Mumford, David Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II (Progress in Mathematics), Volume 36, Birkhäuser, 1983, pp. 271-328 | DOI | MR | Zbl

[21] Nakayama, Noboru Zariski-decomposition and Abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004 | MR | Zbl

[22] Platonov, Vladimir Petrovich A certain problem for finitely generated groups, Dokl. Akad. Nauk BSSR, Volume 12 (1968), pp. 492-494 | MR

[23] Satake, Ichiro On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA, Volume 42 (1956), pp. 359-363 | DOI | MR | Zbl

[24] Schwartz, Marie-Hélène Classes caractéristiques définies par une stratification d’une variété analytique complexe, C. R. Math. Acad. Sci. Paris, Volume 260 (1965), pp. 3535-3537 | Zbl

[25] Shepherd-Barron, N. I.; Wilson, P. M. H. Singular threefolds with numerically trivial first and second Chern classes, J. Alg. Geom., Volume 3 (1994), pp. 265-281 | MR | Zbl

[26] Shin-Yi Lu, Steven; Taji, Behrouz A Characterization of Finite Quotients of Abelian Varieties, Int. Math. Res. Not. (2018), pp. 292-319 | Zbl

[27] Vâjâitu, Viorel Kählerianity of q-Stein spaces, Arch. Math., Volume 66 (1996) no. 3, pp. 250-257 | DOI | Zbl

[28] Varouchas, Jean Kähler spaces and proper open morphisms, Math. Ann., Volume 283 (1989) no. 1, pp. 13-52 | DOI | MR | Zbl

[29] Yau, Shing-Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | Zbl

Cité par Sources :