Framed sheaves on projective stacks

Given a normal projective irreducible stack $\mathcal{X}$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathcal{X}$, i.e., pairs $(\mathcal{E},{\varphi }_{\mathcal{E}})$, where $\mathcal{E}$ is a coherent sheaf on $\mathcal{X}$ and ${\varphi }_{\mathcal{E}}$ is a morphism from $\mathcal{E}$ to a fixed coherent sheaf $\mathcal{F}$. 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 $\mathcal{X}$ is a projective irreducible orbifold of dimension two and $\mathcal{F}$ a locally free sheaf on a smooth divisor $\mathscr D\subset \mathscr X$satisfyingcertainconditions,weconsider$(\mathscr{D}, $\mathcal{F}$)$-framedsheaves,i\mathrm{.}e\mathrm{.},framedsheaves$(\mathcal E,\phi_{\mathcal E})$with$\mathcal E$ a torsion-free sheaf which is locally free in a neighborhood of $\mathcal{D}$, and ${{\varphi }_{\mathcal{E}}}_{\mathrm{\mid }\mathcal{D}}$ an isomorphism. These pairs are $\mu$-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of $\mathscr X$. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(\mathcal{D},\mathcal{F})$-framed sheaves on $\mathcal{X}$ 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 we will use these results to study gauge theories on ALE spaces of type $A_k$.