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Comptes Rendus Mathématique
Volume 348, n° 1-2
pages 11-13 (janvier 2010)
Doi : 10.1016/j.crma.2009.11.016
Received : 4 August 2009 ;  accepted : 23 November 2009
Enumeration of the 50 fake projective planes
Énumération des 50 faux plans projectifs
 

Donald I. Cartwright a , Tim Steger b
a School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia 
b Struttura di Matematica e Fisica, Università di Sassari, Via Vienna 2, 07100 Sassari, Italy 

Abstract

Building upon the classification of Prasad and Yeung [Invent. Math. 168 (2007) 321–370], we have shown that there exist exactly 50 fake projective planes (up to homeomorphism; 100 up to biholomorphism), and exhibited each of them explicitly as a quotient of the unit ball in  . Some of these fake planes admit singular quotients by 3 element groups and three of these quotients are simply connected. Also exhibited are algebraic surfaces with   for any positive integer n .

The full text of this article is available in PDF format.
Résumé

En partant de la classification de Prasad et Yeung [Invent. Math. 168 (2007) 321–370], nous montrons qu’il existe précisément 50 faux plans projectifs (à homéomorphisme près, 100 à biholomorphisme près), et présentons chacun comme un quotient de la boule unité de  . Certains de ces plans admettent des quotients singuliers par des groupes d’automorphismes à 3 éléments, et trois d’entre eux sont simplement connexes. De plus, pour chaque entier  , nous présentons des surfaces algébriques avec  .

The full text of this article is available in PDF format.

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