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23/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building

JongHae Keum, *Korean Institute for Advanced Study*

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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 ${p}_{g}(X)=0$ and ${c}_{1}(X{)}^{2}=3{c}_{2}(X)=9$. 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 $\mathrm{PU}(2,1)$ 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.

Session I