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. A parabolic version of the Kobayshi-Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kahler metrics of nonnegative curvature on CP~(2) with singularities at complex line arrangements.
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机译：在本文中，我们介绍了多面Kahler流形的概念，甚至是具有单一完整性的三维多面流形。我们集中在4维情况下，证明这类流形是光滑的复杂曲面，并对度量的奇点分类。奇异点形成一个除数，除数的补码上的平坦连接的残差为我们提供了一个同调方程组。 T Mochizuki的Kobayshi-Hitchin对应关系的抛物线形式使我们能够描述CP〜（2）上非负曲率的多面Kahler量度，其复杂点处的奇异点。
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