Fake projective planes
Fake projective planes
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DOI：
10.1007/s00222-007-0034-5
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发表时间：
2005-12
影响因子：
3.1
通讯作者：
Gopal Prasad;Sai-Kee Yeung
中科院分区：
文献类型：
--
作者：
Gopal Prasad;Sai-Kee Yeung
伪射影平面是一个光滑的复曲面，它不是复射影平面，但具有与复射影平面相同的Betti数。这种曲面的第一个例子是由大卫芒福德在1979年使用p-adic均匀化构造的。Ishida和Kato用相关的方法又发现了两个例子。Keum最近给出了一个例子，这个例子可能与之前已知的三个例子不同。确定所有伪射影平面是复代数几何中的一个有趣问题。本文利用伪射影平面基本群的算术性，作者给出的主算术子群的余体积公式，以及一些数论估计，给出了伪射影平面的一个分类.文中给出的构造方法更直接、更自然。它不使用p-adic均匀化。
A fake projective plane is a smooth complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. The first example of such a surface was constructed by David Mumford in 1979 using p-adic uniformization. Two more examples were found by Ishida and Kato by related method. Keum has recently given an example which is possibly different from the three known earlier. It is an interesting problem in complex algebraic geometry to determine all fake projective planes. Using the arithmeticity of the fundamental group of fake projective planes, the formula for the covolume of principal arithmetic subgroups given by the first-named auhor, and some number theoretic estimates, we give a classification of fake projective planes in this paper. Twenty eight distinct classes are found. The construction given in the paper appears to be more direct and more natural. It does not use p-adic uniformization.