Abstract
The aim of this short note is to show how to construct a rational Remmert reduction (the \({\widetilde \Gamma}\) -reduction) for the universal cover of smooth orbifolds (X/Δ). Doing this, we are led to introduce some singular Kähler metrics on (X/Δ) adapted to the \({\mathbb {Q}}\) -divisor Δ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barlet, D.: Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975). Lecture Notes in Math., vol.~482, pp. 1–158. Springer, Berlin (1975)
Borzellino, J.: Riemannian geometry of orbifolds. Ph.D. thesis, University of California, Los Angeles (1992). Available at http://www.calpoly.edu/~jborzell/Publications/publications.html
Campana F.: Coréduction algébrique d’un espace analytique faiblement kählérien compact. Invent. Math. 63(2), 187–223 (1981)
Campana F.: Remarques sur le revêtement universel des variétés kählériennes compactes. Bull. Soc. Math. France 122(2), 255–284 (1994)
Campana, F.: Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes (preprint) arXiv:0705.0737 (2007)
Griffiths, P.A.: Entire Holomorphic Mappings in One and Several Complex Variables. Princeton University Press, Princeton (1976). (the fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N.J., October and November 1974, Annals of Mathematics Studies, No. 85)
Kato, M.: On uniformizations of orbifolds, Homotopy theory and related topics (Kyoto 1984). Adv. Stud. Pure Math., vol. 9, pp. 149–172. North-Holland, Amsterdam (1987)
Kollár J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113(1), 177–215 (1993)
Lieberman, D.I.: Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977). Lecture Notes in Math., vol. 670, pp. 140–186. Springer, Berlin (1978)
Namba M.: Branched coverings and algebraic functions. Pitman Research Notes in Mathematics Series, vol. 161. Longman Scientific & Technical, Harlow (1987)
Satake I.: On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. USA 42, 359–363 (1956)
Shafarevich, I.R.: Basic Algebraic Geometry. Springer, New York (1974). (translated from the Russian by K. A. Hirsch, Die Grundlehren der mathematischen Wissenschaften, Band 213)
Stolzenberg G.: Volumes, limits, and extensions of analytic varieties. Lecture Notes in Mathematics, No. 19. Springer, Berlin (1966)