Framed sheaves on projective stacks

Given a normal projective irreducible stack X over an algebraically closed field of characteristic zero we consider framed sheaves on $X$, i.e., pairs $(E,phi_E)$, where $E$ is a coherent sheaf on $X$ and $phi_E$ is a morphism from $E$ to a fixed coherent sheaf $F$. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If $X$ is a projective irreducible orbifold of dimension two and $F$ a locally free sheaf on a smooth divisor $Dsubset X$ satisfying certain conditions, we consider $(D,F)$-framed sheaves, i.e., framed sheaves $(E,phi_E)$ with $E$ a torsion-free sheaf which is locally free in a neighbourhood of $D$, and $phi_Eert_D$ an isomorphism. These pairs are μ-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of X. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(D,F)$-framed sheaves on X with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on "stacky" compactifications of them. In particular, in a subsequent paper [20] these results are used to study gauge theories on ALE spaces of type $A_k$.