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We prove that a smooth projective variety |$X$| of dimension |$n$| with strictly nef third, fourth, or |$(n-1)$| -th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for |$X$| under the assumption that |$\rho (X)\ne 1$|⁠. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]