Fake projective planes

By Gopal Prasad, Sai-Kee Yeung. 2005

189 citations recorded by the source corpus.

A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler Poincare characteristic among smooth surfaces of general type. The first example was constructed by Mumford. Later on two more examples were found by Ishida and Kato. A fourth possible one was recently proposed by Keum. In a recent joint work with Gopal Prasad, we showed that there are seventeen non-empty classes of fake projective planes and there can at most be four more specific classes. Higher dimensional analogues and examples were also obtained. The main purpose of the talk is to explain the joint work with Prasad and other related results such as arithmeticity of the lattices involved obtained earlier by Klingler and Yeung independently.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3a97d1411388eb0be46cf9828a0fe13922807325

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. JongHae Keum
2. John R. Parker
3. John Christian Ottem
4. Martin Deraux
5. Julien Paupert

In Computational Theory and Mathematics

1. JongHae Keum
2. John R. Parker
3. John Christian Ottem
4. Martin Deraux
5. Julien Paupert

In Geometry and Topology

1. JongHae Keum
2. John R. Parker
3. John Christian Ottem
4. Martin Deraux
5. Julien Paupert

Citations in the corpus

• New non-arithmetic complex hyperbolic lattices (2014, methodology): “For instance, the determination of explicit group presentations for small covolume arithmetic lattices in PU(2, 1) allowed Cartwright and Steger to complete the classification of fake projective planes (see [PY07] and [CS10]).”
• Quotients of fake projective planes (2008, methodology): “I thank Gopal Prasad, Sai-Kee Yeung, Donald Cartwright for many helpful conversations, and especially Tim Steger for informing me of Lemma 4.3.”
• Enumeration of the 50 fake projective planes = Énumération des 50 faux plans projectifs (2010, unknown): “, [11, Proposition 8.8] shows that there are at most 5 possible pairs ( k ,(cid”
• Ample subvarieties and q-ample divisors (2011, background): “that the fundamental group is isomorphic to a torsion free lattice in PU(2, 1), so in particular it is infinite (see [22]).”