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

Examples and non-examples of polyhedral Kähler surfaces