Complex hyperbolic orbifolds and hybrid lattices

A class of complex hyperbolic lattices inPU(2, 1) called the Deligne-Mostow lattices has been reinterpreted by Hirzebruch (see [1, 10] and [24]) in terms of line arrangements. They use branched covers over a suitable blow up of the complete quadrilateral arrangement of lines in P2 to construct the complex hyperbolic surfaces over the orbifolds associated to the lattices. In [18] and [19], fundamental domains for these lattices have been built by Pasquinelli. Here we show how the fundamental domains can be interpreted in terms of line arrangements as above. This parallel is then applied in the following context. Wells in [25] shows that two of the Deligne-Mostow lattices inPU(2, 1) can be seen as hybrids of lattices inPU(1, 1). Here we show that he implicitly uses the line arrangement and we complete his analysis to all possible pairs of lines. In this way, we show that three more Deligne-Mostow lattices can be given as hybrids.