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Polyhedral Kahler surfaces are a class of complex surfaces, which are flat everywhere except on a two-dimensional skeleton. They are defined as a generalisation of the "gluing a polygon side by side" construction of flat Riemann surfaces. In this article, we introduce two classes of polyhedral Kahler surfaces with trivial holonomy: products of polyhedral Kahler curves with zero holonomy and ramified coverings of tori, and prove that none of these classes is contained in the other. Existence of other types of polyhedral Kahler surfaces is still open.

Examples and non-examples of polyhedral Kähler surfaces