  We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic
subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C.
Wall) as the complex projective space of dimension n-1, and show that if n=5,
there are at least four such subgroups, which are in fact torsion-free. This,
in particular, leads to examples of a fake projective space of dimension 4.
Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are
also obtained.
