Enumeration of the 50 fake projective planes

Cartwright, Donald I.; Steger, Tim

doi:10.1016/j.crma.2009.11.016

Group Theory/Algebraic Geometry

Enumeration of the 50 fake projective planes
[Énumération des 50 faux plans projectifs]

Présenté par : Pierre Deligne

Cartwright, Donald I.¹ ; Steger, Tim ²

¹ School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
² Struttura di Matematica e Fisica, Università di Sassari, Via Vienna 2, 07100 Sassari, Italy

Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 11-13.

Résumé
Abstract

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 $C^{2}$ . 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 $n > 0$ , nous présentons des surfaces algébriques avec $c_{1}^{2} = 3 c_{2} = 9 n$ .

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 $C^{2}$ . 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 $c_{1}^{2} = 3 c_{2} = 9 n$ for any positive integer n.

Reçu le : 2009-08-04
Accepté le : 2009-11-23
Publié le : 2009-12-29

DOI : 10.1016/j.crma.2009.11.016

Affiliations des auteurs :

Cartwright, Donald I. ¹ ; Steger, Tim ²

¹ School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
² Struttura di Matematica e Fisica, Università di Sassari, Via Vienna 2, 07100 Sassari, Italy

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Cartwright, Donald I.; Steger, Tim. Enumeration of the 50 fake projective planes. Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 11-13. doi : 10.1016/j.crma.2009.11.016. http://www.numdam.org/articles/10.1016/j.crma.2009.11.016/

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