Addendum to Fake Projective Planes

Abstract

A.1. As in [1], let k be a totally real number field and l be a totally complex quadratic extension of k. Let h be a hermitian form on l3, defined in terms of the nontrivial automorphism of l/k, such that h is indefinite exactly at one real place of k. In [1] we proved that if the fundamental group of a fake projective plane is an arithmetic subgroup of PU(h), then the pair (k, l) must be one of the following five: C1, C8, C11, C18, or C21. Around the time [1] was submitted for publication, Tim Steger had shown that (k, l) cannot be C8, and together with Donald Cartwright he had shown that it cannot be C21. Using long and very sophisticated computer-assisted group theoretic computations, Donald Cartwright and Tim Steger have now shown that (k, l) cannot be any of the three remaining pairs C1, C11 and C18. This then proves that the fundamental group of a fake projective plane cannot be an arithmetic subgroup of PU(h). Cartwright and Steger have also shown that there exists a rather unexpected smooth projective complex algebraic surface whose Euler-Poincaré characteristic is 3 but the first Betti-number is nonzero (it is actually 2). This surface is uniformized by the complex 2-ball, and its fundamental group is a cocompact torsion-free arithmetic subgroup of PU(h), h as above, with (k, l) = C11 = (Q( √ 3),Q(ζ12)). In view of the above, we can assert now that there are exactly twenty eight classes of fake projective planes in a classification scheme which is finer than that used in [1], see A.2 and A.5 below. It has been shown recently by Cartwright and Steger using ingenious group theoretic computations that the twenty eight classes of fake projective planes altogether contain fifty distinct fake projective planes up to isometry with respect to the Poincaré metric. Since each such fake projective plane as a Riemannian manifold supports two distinct complex structures [KK, §5], there are exactly one hundred fake projective planes counted up to biholomorphism.

Cite this paper

@inproceedings{Yeung2009AddendumTF, title={Addendum to Fake Projective Planes}, author={S Yeung}, year={2009} }