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.
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.
Accepted:
Published online:
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
@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, Volume 70 (2020) no. 2, pp. 881-914. doi : 10.5802/aif.3326. https://aif.centre-mersenne.org/articles/10.5802/aif.3326/
[1] Étale fundamental groups of strongly -regular schemes, Int. Math. Res. Not. (2019), pp. 4325-4339 | DOI
[2] On deformations of Kähler spaces. I, Math. Z., Volume 182 (1983) no. 4, pp. 505-535 | DOI | Zbl
[3] Orbifoldes à première classe de Chern nulle, The Fano Conference, Univ. Torino, Turin, 2004, pp. 339-351 | Zbl
[4] Abundance for Kähler threefolds, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 4, pp. 971-1025 | DOI | Zbl
[5] The Zariski–Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc., Volume 46 (2014) no. 4, pp. 827-835 | DOI | MR | Zbl
[6] Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann., Volume 371 (2018), pp. 487-516 | DOI | MR | Zbl
[7] An optimal extension theorem for -forms and the Lipman-Zariski Conjecture, Doc. Math., Volume 19 (2014), pp. 815-830 | MR | Zbl
[8] Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer, 1984, xviii+249 pages | MR | Zbl
[9] Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 114 (2011), pp. 1-83 | MR | Zbl
[10] 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] 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] Minimal models for Kähler threefolds, Invent. Math., Volume 203 (2016) no. 1, pp. 217-264 | DOI | Zbl
[13] Cohomology of sheaves, Universitext, Springer, 1986 | Zbl
[14] 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] Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, 2007 | MR | Zbl
[16] Classification of Three-Dimensional Flips, J. Am. Math. Soc., Volume 5 (1992) no. 3, pp. 533-703 | DOI | MR | Zbl
[17] Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 | MR | Zbl
[18] Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004 | MR | Zbl
[19] Chern classes for singular algebraic varieties, Ann. Math., Volume 100 (1974), pp. 423-432 | DOI | MR | Zbl
[20] 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] Zariski-decomposition and Abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004 | MR | Zbl
[22] A certain problem for finitely generated groups, Dokl. Akad. Nauk BSSR, Volume 12 (1968), pp. 492-494 | MR
[23] On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA, Volume 42 (1956), pp. 359-363 | DOI | MR | Zbl
[24] 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] Singular threefolds with numerically trivial first and second Chern classes, J. Alg. Geom., Volume 3 (1994), pp. 265-281 | MR | Zbl
[26] A Characterization of Finite Quotients of Abelian Varieties, Int. Math. Res. Not. (2018), pp. 292-319 | Zbl
[27] Kählerianity of -Stein spaces, Arch. Math., Volume 66 (1996) no. 3, pp. 250-257 | DOI | Zbl
[28] Kähler spaces and proper open morphisms, Math. Ann., Volume 283 (1989) no. 1, pp. 13-52 | DOI | MR | Zbl
[29] 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
Cited by Sources: