Fake projective planes. (English) Zbl 1253.14034
The paper under review concerns fake projective planes, that is, smooth compact complex surfaces which are not isomorphic to the complex projective plane, yet have the same Betti numbers
\[
b_0=b_2=b_4 = 1,\;\;\; b_1=b_3=0.
\]
Not only are such surfaces automatically projective and of general type, but they are also known to be quotients of the unit ball in \(\mathbb{C}^2\) by a torsion-free cocompact discrete subgroup of PU\((2,1)\).
The first example of a fake projective plane is due to D. Mumford [Am. J. Math. 101, 233–244 (1979; Zbl 0433.14021)], using \(p\)-adic uniformisation. Later examples were constructed by M.-N. Ishida and F. Kato [Tohoku Math. J., II. Ser. 50, No. 4, 537–555 (1998; Zbl 0962.14031)] and J. Keum [Topology 45, No. 5, 919–927 (2006; Zbl 1099.14031), Sci. China, Math. 54, No. 8, 1665–1678 (2011; Zbl 1238.14027)].
Already Mumford proves that there are only finitely many fake projective planes up to isomorphism. The present paper initiates a classification of fake projective planes, in big part by group-theoretic means. It should be seen in conjunction with the addendum [Invent. Math. 182, No. 1, 213–227 (2010; Zbl 1388.14074)] and the heavily computer-aided work of D. I. Cartwright and T. Steger [C. R., Math., Acad. Sci. Paris 348, No. 1-2, 11–13 (2010; Zbl 1180.14039)]. Altogether, it is proved that there are
– twenty eight classes of fake projective planes,
– fifty fake projective planes up to isometry with respect to the Poincaré metric, and
– one hundred fake projective planes up to biholomorphism (since each isometry class of fake projective planes supports two distinct complex structures as a Riemannian manifold).
The first example of a fake projective plane is due to D. Mumford [Am. J. Math. 101, 233–244 (1979; Zbl 0433.14021)], using \(p\)-adic uniformisation. Later examples were constructed by M.-N. Ishida and F. Kato [Tohoku Math. J., II. Ser. 50, No. 4, 537–555 (1998; Zbl 0962.14031)] and J. Keum [Topology 45, No. 5, 919–927 (2006; Zbl 1099.14031), Sci. China, Math. 54, No. 8, 1665–1678 (2011; Zbl 1238.14027)].
Already Mumford proves that there are only finitely many fake projective planes up to isomorphism. The present paper initiates a classification of fake projective planes, in big part by group-theoretic means. It should be seen in conjunction with the addendum [Invent. Math. 182, No. 1, 213–227 (2010; Zbl 1388.14074)] and the heavily computer-aided work of D. I. Cartwright and T. Steger [C. R., Math., Acad. Sci. Paris 348, No. 1-2, 11–13 (2010; Zbl 1180.14039)]. Altogether, it is proved that there are
– twenty eight classes of fake projective planes,
– fifty fake projective planes up to isometry with respect to the Poincaré metric, and
– one hundred fake projective planes up to biholomorphism (since each isometry class of fake projective planes supports two distinct complex structures as a Riemannian manifold).
Reviewer: Matthias Schütt (Hannover)
MSC:
| 14J29 | Surfaces of general type |
| 11R29 | Class numbers, class groups, discriminants |
| 22E40 | Discrete subgroups of Lie groups |
Keywords:
fake projective planeCitations:
Zbl 0433.14021; Zbl 0962.14031; Zbl 1099.14031; Zbl 1238.14027; Zbl 1180.14039; Zbl 1388.14074References:
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