# Summer Lectures in Geometry

### Fake projective planes

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 ${p}_{g}\left(X\right)=0$ and ${c}_{1}\left(X{\right)}^{2}=3{c}_{2}\left(X\right)=9$. By a result of Aubin and Yau, its universal cover is the unit $2$-ball, hence its fundamental group ${\pi }_{1}\left(X\right)$ is a discrete torsion-free cocompact subgroup of $\mathrm{PU}\left(2,1\right)$ 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.
Session I

For detailed overviews of each course see http://camgsd.ist.utl.pt/encontros/slg/.