Examples and non-examples of polyhedral K\"ahler surfaces

TL;DR

This paper introduces two classes of polyhedral K"ahler surfaces with trivial holonomy, explores their properties, and shows that these classes are distinct, highlighting open questions about other possible types.

Contribution

It defines and distinguishes two new classes of polyhedral K"ahler surfaces with trivial holonomy, expanding understanding of their structure and relationships.

Findings

Two classes are mutually exclusive.

Existence of other polyhedral K"ahler surfaces remains open.

Provides foundational results on trivial holonomy cases.

Abstract

Polyhedral K\"ahler 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 K\"ahler surfaces with trivial holonomy: products of polyhedral K\"ahler 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 K\"ahler surfaces is still open.