Enumeration of the 50 fake projective planes

Donald I. Cartwright; Tim Steger

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

Donald I. Cartwright ¹ ; Tim Steger 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, Volume 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 :

Donald I. Cartwright ¹ ; Tim Steger 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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     title = {Enumeration of the 50 fake projective planes},
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     pages = {11--13},
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     language = {en},
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Donald I. Cartwright; Tim Steger. Enumeration of the 50 fake projective planes. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 11-13. doi : 10.1016/j.crma.2009.11.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.016/

Bibliographie
Cité par

[1] M.A. Armstrong The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc., Volume 64 (1968), pp. 299-301

[2] D.I. Cartwright; T. Steger Application of the Bruhat–Tits tree of ${SU}_{3} (h)$ to some ${\tilde{A}}_{2}$ groups, J. Aust. Math. Soc., Volume 64 (1998), pp. 329-344

[3] F. Hirzebruch, Automorphe Formen und der Satz von Riemann–Roch, in: 1958 Symposium Internacional de Topologia Algebraica, UNESCO, pp. 129–144

[4] M.-N. Ishida; F. Kato The strong rigidity theorem for non-Archimedean uniformization, Tohoku Math. J., Volume 50 (1998), pp. 537-555

[5] J. Keum A fake projective plane with an order 7 automorphism, Topology, Volume 45 (2006), pp. 919-927

[6] J. Keum Quotients of fake projective planes, Geom. Topol., Volume 12 (2008), pp. 2497-2515

[7] V.S. Kharlamov; V.M. Kulikov On real structures on rigid surfaces, Izv. Math., Volume 66 (2002), pp. 133-150

[8] B. Klingler Sur la rigidité de certains groupes fondamentaux, l'arithméticité des réseaux hyperboliques complexes, et les « faux plans projectifs », Invent. Math., Volume 153 (2003), pp. 105-143

[9] D. Mumford An algebraic surface with K ample, $K^{2} = 9$ , $p_{g} = q = 0$ , Amer. J. Math., Volume 101 (1979), pp. 233-244

[10] G. Prasad Volumes of S-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math., Volume 69 (1989), pp. 91-117

[11] G. Prasad; S.-K. Yeung Fake projective planes, Invent. Math., Volume 168 (2007), pp. 321-370

[12] G. Prasad, S.-K. Yeung, Fake projective planes, Addendum, in press

[13] R. Rémy, Covolume des groupes S-arithmétiques et faux plans projectifs [d'après Mumford, Prasad, Klingler, Yeung, Prasad–Yeung], Séminaire Bourbaki, 60ème année, 2007–2008, no. 984

[14] S.-T. Yau Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA, Volume 74 (1977), pp. 1798-1799

[15] S.-K. Yeung Integrality and arithmeticity of co-compact lattices corresponding to certain complex two-ball quotients of Picard number one, Asian J. Math., Volume 8 (2004), pp. 107-130

[16] S.-K. Yeung, Classification of fake projective planes, in: Handbook of Geometric Analysis, vol. 2, in press

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