Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T12:01:08.977Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelianity Conjecture for Special Compact Kähler 3-Folds

Published online by Cambridge University Press:  19 December 2013

Fréderic Campana  and
Benoît Claudon
Fréderic Campana
Affiliation:
Institut Élie Cartan Nancy, Université Henri Poincaré Nancy 1, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France, (frederic.campana@iecn.u-nancy.fr; benoit.claudon@iecn.u-nancy.fr)
Benoît Claudon
Affiliation:
Institut Élie Cartan Nancy, Université Henri Poincaré Nancy 1, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France, (frederic.campana@iecn.u-nancy.fr; benoit.claudon@iecn.u-nancy.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with a negative or numerically trivial canonical bundle and the two-dimensional log minimal model programme, we prove that the fundamental group of special compact Kähler 3-folds is almost abelian. This property was conjectured in all dimensions by Campana in 2004, and also for orbifolds in 2007, where the notion of specialness was introduced. We briefly recall the definition, basic properties and the role of special manifolds in classification theory.

Keywords

Special compact Kähler 3-foldsfundamental grouptwo-dimensional special orbifoldsklt singularitiesPrimary 32J2714J30Secondary 14F3514J1732J1732Q55
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 55 - 78
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Barth, W. P., Hulek, K., Peters, C. A. M. and Yen, A. Van de, Compact complex surfaces, 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 3 (Springer, 2004).CrossRefGoogle Scholar
2.Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom. 18(4) (1983), 755–782.Google Scholar
3.Borzellino, J. E., Orbifolds of maximal diameter, Indiana Univ. Math. J. 42(1) (1993), 37–53.Google Scholar
4.Borzellino, J. E. and Zhu, S.-H., The splitting theorem for orbifolds, Illinois J. Math. 38(4) (1994), 679–691.CrossRefGoogle Scholar
5.Campana, F., Connexité rationnelle des variétés de Fano, Annales Scient. Éc. Norm. Sup. 25(5) (1992), 539–545.CrossRefGoogle Scholar
6.Campana, F., Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Alg. Geom. 4(3) (1995), 487–502.Google Scholar
7.Campana, F., Ensembles de Green–Lazarsfeld et quotients résolubles des groupes de Kähler, J. Alg. Geom. 10(4) (2001), 599–622.Google Scholar
8.Campana, F., Orbifoldes à première classe de Chern nulle, in The Fano Conference, University of Torino, Turin, 2004, pp. 339–352 (Dipartimento di Matematica, Turin, 2004).Google Scholar
9.Campana, F., Orbifolds, special varieties and classification theory, Annales Inst. Fourier 54(3) (2004), 499–630.CrossRefGoogle Scholar
10.Campana, F., Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes, J. Inst. Math. Jussieu 10(4) (2011), 809–934.Google Scholar
11.Campana, F., Quotients résolubles ou nilpotents des groupes de Kähler orbifolde, Manuscr. Math. 135(1) (2010), 117–150.Google Scholar
12.Demailly, J.-P. and Kollár, J., Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Annales Scient. Éc. Norm. Sup. 34(4) (2001), 525–556.Google Scholar
13.Fong, L. and McKernan, J., Log abundance for surfaces, in Flips and abundance for algebraic threefolds (ed. Kollar, J.), Astérisque, Volume 211, pp. 127–137 (Société Mathématique de France, Paris, 1992).Google Scholar
14.Ghigi, A. and Kollár, J., Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82(4) (2007), 877–902.Google Scholar
15.Joyce, D. D., Compact manifolds with special holonomy, Oxford Mathematical Monographs (Oxford University Press, 2000).Google Scholar
16.Kobayashi, R., Uniformization of complex surfaces, in Kähler metric and moduli spaces, Advanced Studies of Pure Mathematics, Volume 18, pp. 313–394 (Academic, 1990).Google Scholar
17.Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113(1) (1993), 177–215.Google Scholar
18.Kollár, J., Shafarevich maps and automorphic forms, in M. B. Por-ter Lectures (Princeton University Press, 1995).Google Scholar
19.Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134 (Cambridge University Press, 1998).Google Scholar
20.Lamotke, K., Regular solids and isolated singularities, Advanced Lectures in Mathematics (Friedrich Vieweg & Sohn, Braunschweig, 1986).Google Scholar
21.Megyesi, G., Generalisation of the Bogomolov–Miyaoka–Yau inequality to singular surfaces, Proc. Lond. Math. Soc. 78(2) (1999), 241–282.Google Scholar
22.Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. IHES 9(1) (1961), 5–22.Google Scholar
23.Nakamura, S., Classification and uniformisation of log-canonical singularities in the presence of branch loci, PhD thesis, Saitama University, Japan (1989) (in Japanese).Google Scholar
24.Namba, M., Branched coverings and algebraic functions, Pitman Research Notes in Mathematics Series, Volume 161 (Longman, New York, 1987).Google Scholar
25.Namikawa, Y. and Steenbrink, J. H. M., Global smoothing of Calabi–Yau threefolds, Invent. Math. 122(2) (1995), 403–419.Google Scholar
26.Siu, Y. T., Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics, DMV Seminar, Volume 8 (Birkhäuser, 1987).CrossRefGoogle Scholar
27.Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Volume 439 (Springer, 1975).Google Scholar
28.Uludaǧ, A. M., Orbifolds and their uniformization, in Arithmetic and geometry around hypergeometric functions, Progress in Mathematics, Volume 260, pp. 373–406 (Birkhäuser, 2007).Google Scholar
29.Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Commun. Pure Appi. Math. 31(3) (1978), 339–411.Google Scholar