23/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building
JongHae Keum, Korean Institute for Advanced Study
Fake projective planesIt is known that a compact complex manifold of dimension with the same Betti numbers as the complex projective plane is projective. Such a manifold is called "a fake projective plane" if it is not isomorphic to the complex projective plane. So a fake projective plane is exactly a smooth surface of general type with and . By a result of Aubin and Yau, its universal cover is the unit -ball, hence its fundamental group is a discrete torsion-free cocompact subgroup of satisfying certain conditions. The classification of such subgroups has been done by G. Parasad and S.-K. Yeung, with the computer based computation by D. Cartwright and J. Steger. This has settled the arithmetic side of the classification problem. I will go over this, and then report recent progress on the other side of the problem-geometric construction.