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Extending work of Klyachko and Perling, we develop a combinatorialdescription of pure equivariant sheaves of any dimension on an arbitrarynonsingular toric variety $X$. Using geometric invariant theory (GIT), thisallows us to construct explicit moduli spaces of pure equivariant sheaves on$X$ corepresenting natural moduli functors (similar to work of Payne in thecase of equivariant vector bundles). The action of the algebraic torus on $X$lifts to the moduli space of all Gieseker stable sheaves on $X$ and we expressits fixed point locus explicitly in terms of moduli spaces of pure equivariantsheaves on $X$. One of the problems arising is to find an equivariant linebundle on the side of the GIT problem, which precisely recovers Giesekerstability. In the case of torsion free equivariant sheaves, we can alwaysconstruct such equivariant line bundles. As a by-product, we get acombinatorial description of the fixed point locus of the moduli space of$\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequelhow these methods can be used to compute generating functions of Eulercharacteristics of moduli spaces of $\mu$-stable torsion free sheaves onnonsingular complete toric surfaces.

Fixed point loci of moduli spaces of sheaves on toric varieties