Quotients of the complex ball by discrete groups (Q1079243)

We systematically study varieties Q(\(\mu)\), which are compactifications of the space Q of distinct points in (\({\mathbb{P}}^ l)^ r\) given by a sequence of ''weights'' \(\mu\), and which for certain \(\mu\) are also compactification of the quotient of the complex r-ball by discrete subgroups \(\Gamma\) (\(\mu)\) of PU(r,l), as discovered by Deligne and Mostow. We obtain a wealth of topological information about the spaces Q(\(\mu)\) and their desingularizations \(Q^*(\mu)\). In some cases we can completely describe them. Otherwise, we obtain computations of Betti numbers and Hodge numbers. As applications we determine the \(L^ 2\)- cohomology and in many cases the (ordinary) rational cohomology of the groups \(\Gamma\) (\(\mu)\).