Polyhedral complexes and topology of complex-projective varieties

I will explain how to use polyhedral complexes of constant
curvature and  hyperbolic orbifolds
in order to construct some interesting complex-projective varieties.
Among other things,I will show how to construct projective varieties
with normal crossing singularities which have prescribed
finitely-presented fundamental groups. This is partly joint work with
Janos Kollar.