Summer Lectures in Geometry 

24/07/2012, 15:45 — 16:45 — Room P3.10, Mathematics Building
JongHae Keum, Korean Institute for Advanced Study

Fake projective planes (II)

It is known that a compact complex manifold of dimension $2$ 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 $X$ of general type with $p_g(X)=0$ and $c_1(X)^2=3c_2(X)=9$. By a result of Aubin and Yau, its universal cover is the unit $2$-ball, hence its fundamental group $\pi_1(X)$ is a discrete torsion-free cocompact subgroup of $PU(2,1)$ 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.