Framed sheaves on projective stacks

Abstract

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, φ E ), where E is a coherent sheaf on X and φ 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 D ⊂ X satisfying certain conditions, we consider (D, F)-framed sheaves, i.e., framed sheaves (E, φ E ) with E a torsion-free sheaf which is locally free in a neighborhood of D, and φ E |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 .

N=2 gauge theories, instanton moduli spaces and geometric representation theory

“…. , w k−1 ) ∈ Z k ≥0 , by using the general theory of framed sheaves on projective stacks developed by [23] one can construct a fine moduli space M u,∆, w (X k ) parameterizing torsionfree sheaves E on X k with a framing isomorphism φ E :…”

Hilbert schemes of points of OP1(−n) as quiver varieties

“…For n ≥ 2, the surface X n is the minimal resolution of the toric singularity of type 1 n (1, 1); it admits the n-th Hirzebruch surface Σ n as a projective compactification. These spaces have been considered in physics in connection with brane counting for string theory compactifications on "local" Calabi-Yau manifolds [1,13,17,34,9], gauge theory on Hirzebruch surfaces and applications to the computation of invariants, such as the Betti numbers of moduli spaces of sheaves on Hirzebruch surfaces [7,24,6] (see also [8,App. D] for some mathematical developments), topological strings and Gromov-Witten invariants (see [37] for a review).…”

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

“…where [ R (s)s ] is the contravariant functor such that for any S-scheme T of finite type, [ R (s)s ](T ) is the set of isomorphism classes of objects of R (s)s (T ). In fact, one can show that two functors M (s)s X /S (F 0 , P, δ) and [ R (s)s ] are isomorphic by following the similar argument in [7,Theorem 4.12]. By [4, Theorem 13.6], we have the morphism [ R (s)s /SL(V )] → M (s)s := R (s)s //SL(V ), which is a good moduli space on the fiber over each geometric point s of S. And we have the morphism R (s)s → M (s)s by the discussion following [7,Proposition 4.11] (see also the proof of [53, Theorem 1.21-(1)]), which factors through R (s)s → [ R (s)s ].…”