Abstract
We prove that, given integers mge 3, rge 1 and nge 0, the moduli space of torsion free sheaves on {mathbb {P}}^m with Chern character (r,0,ldots ,0,-n) that are trivial along a hyperplane D subset {mathbb {P}}^m is isomorphic to the Quot scheme mathrm{Quot}_{{mathbb {A}}^m}({mathscr {O}}^{oplus r},n) of 0-dimensional length n quotients of the free sheaf {mathscr {O}}^{oplus r} on {mathbb {A}}^m. The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.
Highlights
This paper builds an identification between two classical moduli spaces in algebraic geometry: the moduli space of framed sheaves on projective space Pm and Grothendieck’s Quot scheme
D ⊂ Y is a divisor on a projective variety Y, a D-framed sheaf on Y is a pair (E, φ) where is a torsion free sheaf on and φ is an isomorphism
Donaldson [10] constructed a canonical identification between the moduli space of instantons on S4 = R4∪{∞} with SU (r )-framing at ∞ and the moduli space of rank r holomorphic vector bundles on P2 trivial on a line ∞
Summary
This paper builds an identification between two classical moduli spaces in algebraic geometry: the moduli space of framed sheaves on projective space Pm and Grothendieck’s Quot scheme. Donaldson [10] constructed a canonical identification between the moduli space of instantons on S4 = R4∪{∞} with SU (r )-framing at ∞ and the moduli space of rank r holomorphic vector bundles on P2 trivial on a line ∞ He defined a partial compactification of the moduli space on the 4-manifold side of the correspondence by allowing connections acquiring singularities. In the work of Cirafici–Sinkovics–Szabo [8, Sec. 4.1], the authors construct a correspondence between non-commutative U (r )-instantons on A3 and the 3-dimensional analogue of Donaldson’s construction, namely the moduli space Frr,n(P3) They relate the construction to the quiver gauge theory of the ‘r -framed 3-loop quiver’ (Fig. 1), which corresponds to QuotA3 (O⊕r , n) in a precise sense [2]. Quot schemes received a lot of attention lately in enumerative geometry [13,21,23,25], and in the context of motivic invariants [9,17,24]
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