Group Theory/Algebraic Geometry
Enumeration of the 50 fake projective planes
[Énumération des 50 faux plans projectifs]
Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 11-13.

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 C2. 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 c12=3c2=9n.

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 C2. 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 c12=3c2=9n for any positive integer n.

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DOI : 10.1016/j.crma.2009.11.016
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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