Cocompact Arithmetic Subgroups of Pu

We will say that a smooth projective complex algebraic variety V (of dimension n − 1) is a fake Pn−1 if V is the quotient of the unit ball Bn−1 by a torsion-free cocompact discrete subgroup of PU(n − 1, 1), and all the Betti numbers of V are equal to those of Pn−1 C . If the fundamental goup of a fake P n−1 is an arithmetic subgroup of PU(n − 1, 1), then we will say that it is an arithmetic fake Pn−1. It follows from the results of this paper that arithmetic fake Pn−1 can exist only if n = 3 or 5. We have shown in [PY] that there are twelve distinct classes of fake P. We will show here that there exists an arithmetic fake P (in fact, there are four distinct classes of these varieties).