  In this article we introduce the notion of Polyhedral Kahler manifolds, even
dimensional polyhedral manifolds with unitary holonomy. We concentrate on the
4-dimensional case, prove that such manifolds are smooth complex surfaces, and
classify the singularities of the metric. The singularities form a divisor and
the residues of the flat connection on the complement of the divisor give us a
system of cohomological equations. Parabolic version of Kobayshi-Hitchin
correspondence of T. Mochizuki permits us to characterize polyhedral Kahler
metrics of non-negative curvature on CP^2 with singularities at complex line
arrangements.
