  We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and
construct such metrics on CP^2 with various orbifold structures. In particular,
in relation to questions of Gromov and Davis-Moussong we construct such metrics
on a compact quotient of the two-dimensional unite complex ball. In the course
of the proof of these results we give criteria for Sasakian 3-manifolds to be
globally CAT(1). We show further that for certain Kummer coverings of CP^2 of
sufficiently high degree their desingularizations are of type K(pi,1).
