Toric sheaves and polyhedra

Over a smooth projective toric variety we study toric sheaves, that is, reflexive sheaves equivariant with respect to the acting torus, from a polyhedral point of view. One application is the explicit construction of the torus invariant universal extension of two nef line bundles via polyhedral inclusion/exclusion sequences. Second, we link the cohomology of toric sheaves to the cohomology of certain constructible sheaves explicitly built out of the associated polyhedra. For the latter we define a concrete double complex and a spectral sequence which computes the cohomology of toric sheaves from the reduced cohomology of polyhedral subsets living in the realification of the character lattice of the toric variety.