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Fake projective plane

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Dec 9, 2021

In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.

For Freedman’s example of a non-smoothable manifold with the same homotopy type as the complex projective plane, see 4-manifold.

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Severi asked if there was a complex surface homeomorphic to the projective plane but not biholomorphic to it. Yau (1977) showed that there was no such surface, so the closest approximation to the projective plane one can have would be a surface with the same Betti numbers (b0,b1,b2,b3,b4) = (1,0,1,0,1) as the projective plane. The first example was found by Mumford (1979) using p-adic uniformization introduced independently by Kurihara and Mustafin. Mumford also observed that Yau’s result together with Weil’s theorem on the rigidity of discrete cocompact subgroups of PU(1,2) implies that there are only a finite number of fake projective planes. Ishida & Kato (1998) found two more examples, using similar methods, and Keum (2006) found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface. Prasad & Yeung (2007), Prasad & Yeung (2010) found a systematic way of classifying all fake projective planes, by showing that there are twenty-eight classes, each of which contains at least an example of fake projective plane up to isometry, and that there can at most be five more classes which were later shown not to exist. The problem of listing all fake projective planes is reduced to listing all subgroups of appropriate index of an explicitly given lattice associated to each class. By extending these calculations Cartwright & Steger (2010) showed that the twenty-eight classes exhaust all possibilities for fake projective planes and that there are altogether 50 examples determined up to isometry, or 100 fake projective planes up to biholomorphism.

A surface of general type with the same Betti numbers as a minimal surface not of general type must have the Betti numbers of either a projective plane P2 or a quadric P1×P1. Shavel (1978) constructed some “fake quadrics”: surfaces of general type with the same Betti numbers as quadrics. Beauville surfaces give further examples.

Higher-dimensional analogues of fake projective surfaces are called fake projective spaces.

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