Framed sheaves on projective space and Quot schemes

We prove that, given integers m ≥ 3, r ≥ 1 and n ≥ 0, the moduli space of torsion free sheaves on Pm with Chern character (r, 0, . . ., 0,−n ) that are trivial along a hyperplane D ⊂ Pm is isomorphic to the Quot scheme QuotAm (O ⊕r , n ) of 0-dimensional length n quotients of the free sheaf O ⊕r on Am .


INTRODUCTION
This paper builds an identification between two classical moduli spaces in enumerative geometry: the moduli space of framed sheaves on projective space m and Grothendieck's Quot scheme. Unless stated otherwise, we work over the field of complex numbers. If D ⊂ Y is a divisor on a projective variety Y , a D -framed sheaf on Y is a pair (E , φ) where E is a torsion free sheaf on Y and φ is an isomorphism E | D → ⊕r D , where r = rk E . Such pairs (E , φ) are a special case of the more general notion of framed modules introduced by Huybrechts-Lehn [16].
For a fixed coherent sheaf V on Y , the Quot scheme Quot Y (V , P ) parametrises quotients V ։ Q such that Q has Hilbert polynomial P . If P is a constant polynomial, the Quot scheme also exists (as a quasiprojective scheme) for quasi-projective varieties Y such as the affine space m .
The following is the main result of this paper.
It is easy to see that the result does not hold for 2 , as long as r > 1. Indeed, in that case, Fr r,n ( 2 ) is a smooth variety of dimension 2n r containing Quot 2 ( ⊕r , n) as a singular irreducible subvariety of dimension (r + 1)n.
Donaldson [10] constructed a canonical identification between the moduli space of instantons on S 4 = 4 ∪ { ∞ } with SU (r )-framing at ∞ and the moduli space of rank r vector bundles on 2 trivial on a line ℓ ∞ . He defined a partial compactification of the moduli space on the 4-manifold side of the correspondence by allowing connections acquiring singularities. This in turn corresponds to considering torsion free sheaves on the algebro-geometric side, leading to the study of Fr r,n ( 2 ). The 3-dimensional analogue of Donaldson's construction has attracted lots of attention in string theory and hence, after translating in the language of algebraic geometry, in Donaldson-Thomas theory. For instance, in the work of Cirafici-Sinkovics-Szabo [8,Sec. 4.1], the authors construct a correspondence between non-commutative U (r )-instantons on 3 and the 3-dimensional analogue of Donaldson's construction, namely the moduli space Fr r,n ( 3 ). Furthermore, after taking appropriate limits, they relate the construction to the quiver gauge theory of the 'r -framed 3-loop quiver' (Figure 1), which corresponds to Quot 3 ( ⊕r , n) in a precise sense [2]. We briefly review this story in Section 3. Moreover, the very same quiver gauge theory can be engineered from the rank 1 theory in 4 complex dimensions, as shown by Nekrasov and Piazzalunga in [19]. Theorem A formalises this correspondence from an algebraic perspective in the 3-dimensional case, and extends it to higher dimensions.
Framed sheaves and framed modules were mostly studied on surfaces. We do not aim at giving an exhaustive list of references, but we refer the reader to [26,4] for a more complete bibliography. Framed sheaves were also considered on 3-folds by Oprea [20], where a symmetric obstruction theory on their moduli space is constructed -we end Section 2 with a conjecture suggesting that Oprea's obstruction theory might take a very explicit form (Conjecture 2.12). Quot schemes also received a lot of attention lately in enumerative geometry [21,13,25,23], and in the context of motivic invariants [24,17,9].

FRAMED MODULES AND FRAMED SHEAVES
In this section we briefly review the notion of stability on framed modules introduced by Huybrechts-Lehn [16], and then we show that D -framed sheaves on m (Definition 1.4) are stable with respect to appropriate stability parameters (Lemma 1.7). This confirms representability of their moduli functor.
The Hilbert polynomial of a coherent sheaf E , with respect to H , is defined as P E (k ) = χ(E (k )), where with positive leading coefficient. The framed Hilbert polynomial of a framed module (E , α), depending on the pair (H , δ), is defined as Huybrechts and Lehn defined moduli functors parametrising isomorphism classes of flat families of δ-(semi)stable framed modules with framing datum G and framed Hilbert polynomial P ∈ [k ].
As observed in [16,Lemma 1.7], the study of δ-semistable framed modules reduces to the theory of the Hilbert scheme whenever deg δ ≥ m = dim Y . Thus one focuses attention on the case deg δ < m , and writes Huybrechts and Lehn defined the (H , δ)-slope of a framed module (E , α) with positive rank as the ratio The following is a stronger notion of stability than the one in Definition 1.1.
. A framed module (E , α) of positive rank r = rk E is said to be µ-semistable with respect to δ 1 if ker α is torsion free and for every submodule For framed modules of positive rank, such as those studied in this paper, one has that µ-stability with respect to δ 1 implies δ-stability. Also note that a rank 1 framed module (E , α) with E torsion free is µstable for any choice of (H , δ).
The notion which behaves best in the sense of moduli is δ-stability. We now recall the portion of the main theorem of [16] which we will need in our paper.  (1). Of course D is linearly equivalent to H , so in particular we have D · H m−1 = 1, but we distinguish them because they play different roles.
Indeed, as framing datum we fix the coherent sheaf G = ι * ⊕r D , for a fixed integer r ≥ 1. Note that the framings α ∈ Hom(E ,G ) naturally correspond to morphisms φ α : E | D → ⊕r D via the adjunction ι * ⊣ ι * . Fix an integer n ≥ 0. Consider the Chern character v r,n = (r, 0, . . ., 0, −n) ∈ H * ( m , ). Definition 1.4. Let m ≥ 2 be an integer. A D -framed sheaf of rank r on m is a framed module (E , α) on m with framing datum G = ι * ⊕r D , such that E is torsion free with Chern character ch(E ) = v r,n for some n ≥ 0, and the morphism φ α : E | D → ⊕r D induced by the framing α is an isomorphism.
We will make crucial use of the following result due to Abe and Yoshinaga.
Then there is a natural short exact sequence of sheaves Proof. Since E is torsion free, the natural map E → E ∨∨ to its double dual is injective. Moreover, E ∨∨ is reflexive and α induces a canonical isomorphism E ∨∨ | D ∼ = ⊕r D . By Theorem 1.5 we have that E ∨∨ splits as a direct sum of line bundles, and it is immediate to see that these line bundles are necessarily trivial. So we have a canonical isomorphism E ∨∨ ∼ = ⊕r m , and since E | D ∼ = ⊕r D the quotient Q = ⊕r m /E is supported on finitely many points lying in m \ D .
In the case of projective surfaces it has been proved by Bruzzo and Markushevich that µ (H ,δ) -stability is automatically implied when considering a "good framing" [4, Thm. 3.1]. The strategy of the proof does not extend in full generality to higher dimensional varieties, as observed by Oprea [20]. We shall now provide a new argument for the particular case at hand, but it is still an open question whether it is possible to extend the result to more general settings. Lemma 1.7. Fix integers m ≥ 3, and r ≥ 1. Let (E , α) be a D -framed sheaf of rank r on m , and consider a polynomial δ as in (1.2), such that 0 < δ 1 Clearly ker α → E is torsion free because E is torsion free by definition. Moreover, by means of the diagram We now have to distinguish two cases: . This means ε(α ′ ) = 0. We compute the ordinary H -slope where the inequality is induced by the inclusion E ′ (D ) → E → ⊕r m , again using the semistability of ⊕r m . So we obtain by our assumption δ 1 < r and Equation (1.5).
The proof is complete.
We have defined the functor using the map | D ×B → ⊕r D ×B , but we could have used → (ι × id B ) * ⊕r D ×B instead. Let δ be a rational polynomial as in (1.2). If (E , α) is a D -framed sheaf with ch(E ) = v r,n then, since ε(α) = 1, according to Equation (1.1) we have where P r,n (k ) = χ(E (k )) is the Hilbert polynomial of a coherent sheaf E with Chern character v r,n . Proposition 1.8. Fix integers m ≥ 2, r ≥ 1, and n ≥ 0. Let δ be a polynomial as in (1.2), with 0 < δ 1 < r . Set G = ι * ⊕r D and P = P r,n − δ. Then the moduli functor Fr r,n ( m ) is represented by an open subscheme Proof. The case of 2 is well known [18,4]. Hence, we can restrict to the case m ≥ 3. The locus of framed modules (E , α) ∈ M st δ ( m ;G , P ) such that E is torsion free and the framing induces an isomorphism E | D → ⊕r D is open. But by Lemma 1.7, all D -framed sheaves are δ-stable.

MODULI OF FRAMED SHEAVES AND QUOT SCHEMES
In this section we review the notion of tangent-obstruction theory on a deformation functor [12], and we compare the tangent-obstruction theory on the local Quot functor with that on the D -framed sheaves local moduli functor. This leads to the proof of Theorem A.

Comparing tangent-obstruction theories.
We refer the reader to [12, Ch. 6] for a thorough exposition on tangent-obstruction theories on deformation functors.
Fix an algebraically closed field k, and let Art k be the category of local artinian k-algebras with residue field k. A deformation functor is a covariant functor D : Art k → Sets such that D(k) is a singleton. A tangentobstruction theory on a deformation functor D is defined to be a pair (T 1 , T 2 ) of finite dimensional k-vector spaces such that for any small extension I → B ։ A in Art k there is an exact sequence of sets with an additional 0 on the left whenever A = k, and is moreover functorial in small extensions in a precise sense [12, Def. 6.1.21]. The tangent space of the tangent-obstruction theory is T 1 , and is canonical, in the sense that it is determined by the deformation functor as . The obstruction space, T 2 , is not canonical: any larger k-linear space U 2 ⊃ T 2 yields a new tangent-obstruction theory (T 1 ,U 2 ). A deformation functor D is pro-representable if D ∼ = Hom k-alg (R , −) for some local k-algebra R with residue field k. A tangent-obstruction theory on a pro-representable deformation functor always induces a 0 on the left in the sequences (2.1). This can be interpreted by saying that lifts of a given α ∈ D(A), when they exist, form an affine space over T 1 ⊗ k I . It sends a scheme B to the set of isomorphism classes of surjections π * Y V ։ , where π Y : Y × k B → Y is the projection and is a coherent sheaf on Y × k B , flat over B , whose fibres b = | Y ×{ b } have Hilbert polynomial P . Two surjections are 'isomorphic' if they have the same kernel. The Quot functor is represented by a projective k-scheme Q = Quot Y (V , P ). We refer the reader to [12, Ch. 5] for a complete, modern discussion on Quot schemes. Fix a point x 0 ∈ Q(k) corresponding to a quotient V ։ Q with kernel E . One can consider the local Quot functor at x 0 , i.e. the subfunctor Q x 0 ⊂ Q| Art k : Art k → Sets sending a local artinian k-algebra A to the set of families x ∈ Q(Spec A) such that x | m = x 0 , where m ∈ Spec A is the closed point. By representability of Q, the functor Q x 0 is pro-representable, isomorphic to Hom k-alg ( Q,x 0 , −). By [12, Thm. 6.4.9], the pair of k-vector spaces form a tangent-obstruction theory on the deformation functor Q x 0 .
The proof of the following result is included for the sake of completeness (and for lack of a suitable reference).
where the leftmost vertical map is d⊗ k id I and the isomorphism D(A) →D ′ (A) is the induction hypothesis.
We have to show that η B is bijective. To prove injectivity, pick two elements β 1 , β 2 ∈ D(B ) and let We may assume that the images of β 1 and β 2 in D(A) agree, for if they differed, we would have β 1 = β 2 , thus confirming injectivity. By exactness of the bottom exact sequence of sets, we have To prove surjectivity, pick β ′ ∈ D ′ (B ). It maps to 0 ∈ T ′ 2 ⊗ k I , and its image α ′ in D ′ (A) lifts uniquely to an element α ∈ D(A) such that ob(α) goes to 0 ∈ T ′ 2 ⊗ k I . But by the injectivity assumption, we have ob(α) = 0, i.e. α lifts to some β ∈ D(B ).
Note that is B -flat since is B -flat. Such a map is easily seen to be injective on geometric points. If m ≥ 3, we can construct the inverse of η as follows. Given a D -framed sheaf (E , φ), with trivialisation φ : E | D → ⊕r D , we know by the proof of Corollary 1.6 how to construct an isomorphism E ∨∨ → ⊕r m . Thus the inverse of η will send (E , φ) to the isomorphism class of the surjection ⊕r m ։ ⊕r m /E . The proof of the proposition is complete.
We will use an infinitesimal method to prove that η is an isomorphism as long as m ≥ 3.

Infinitesimal method.
Let y 0 = η(x 0 ) ∈ Fr r,n ( m ) be the image of a point x 0 ∈ Quot m ( ⊕r , n) under the bijective morphism η. We obtain an induced natural transformation η 0 : Q x 0 → Fr y 0 between the local moduli functors -Q x 0 was defined in Example 2.1. Both functors are pro-representable and carry a tangent-obstruction theory, cf. (2.2) for the case of the Quot scheme. Our next goal is to show that η 0 is an equivalence when m ≥ 3, using Proposition 2.2. This will be achieved by means of the following two lemmas.
Proof. Consider the short exact sequence of sheaves obtained from the exact sequence of the divisor D . Notice first that For any k > −m we then deduce the following isomorphisms from the long exact sequence in cohomology associated to (2.3): Since both cohomology groups on the right hand side of the isomorphisms vanish for k large enough by Serre's vanishing theorem, we deduce H m−1 ( m , E (−m )) = H m ( m , E (−m )) = 0. Proof. Twisting the exact sequence (1.4) by (−D ) and applying the Hom(E , −) functor we obtain a long cohomology sequence By Serre duality we have and the result follows from Lemma 2.4.
We have thus essentially obtained the proof of the following result. Proposition 2.6. If m ≥ 3, the natural transformation η 0 : Q x 0 → Fr y 0 of local moduli functors induces an isomorphism on tangent spaces and an injection on obstruction spaces. Hence, η 0 is a natural equivalence.
Proof. The first statement follows from Lemma 2.5. The conclusion follows from Proposition 2.2.
We can now finish the proof of our main result. Proof. The morphism η is locally of finite type, since the Quot scheme is of finite type. Next, we check that η is formally étale, using the infinitesimal criterion. Consider a square zero extension S → S of fat points (i.e. spectra of objects A, B of Art ), denote by m the closed point of S and form a commutative diagram where the dotted arrow u is the unique extension of h we have to find in order to establish formal étal- Using pro-representability of Q x 0 and Fr y 0 , the condition that η 0 is a natural equivalence (proved in Proposition 2.6) translates into a commutative diagram Hom y 0 (S , Fr r,n ( m )) Hom y 0 (S , Fr r,n ( m )) where the vertical maps are the isomorphisms η 0,S and η 0,S respectively. Since h ∈ Hom y 0 (S , Fr r,n ( m )) lifts to a map u ∈ Hom x 0 (S, Quot m ( ⊕r , n)) and both u • i and h map to η • h ∈ Hom y 0 (S , Fr r,n ( m )), they must be equal, since the vertical map on the right is also a bijection. Thus u is the unique lift we wanted to find. We conclude that η is étale. Since it is bijective by Proposition 2.3, it is an isomorphism. Remark 2.9. We thank A. Henni for pointing out that it would also possible to give a proof of Theorem A combining the formalism of perfect extended monads [14,15] with the result of Abe-Yoshinaga (Theorem 1.5). The 3-dimensional case is also studied along these lines in [5,Sec. 2.1.2]. Moreover, we note that our proof works over an arbitrary algebraically closed field k of characteristic 0 with no modification. Proof. This follows by combining Theorem 2.7 with [2, Thm. 2.6]. The pair (U r,n,3 , f ) will be given in Remark 3.1.
Remark 2.11. Another Quot scheme on 3 that has been recently proven to be a global critical locus is is the ideal of a line L ⊂ 3 [9]. This was the starting point for the motivic refinement of the local DT/PT (or, ideal sheaves/stable pairs) correspondence around a smooth curve in a 3-fold [23,22]. See [3] for background on symmetric obstruction theories. See also [21,25] for the construction of virtual fundamental classes on several Quot schemes for varieties of dimension at most 3. We propose the following conjecture, essentially a higher rank version of [13, Conj. 9.9]. can be embedded in a smooth quasi-projective variety U r,n,m , called the non-commutative Quot scheme in [2,13], as follows. Consider the m -loop quiver, i.e. the quiver L m with one vertex '0' and m loops. Now consider the quiver L m obtained by adding one additional vertex '∞' along with r edges ∞ → 0 (see Figure 1). This construction is called r -framing -for m = 3 it has some relevance in motivic Donaldson-Thomas theory [6,7] and K-theoretic Donaldson-Thomas theory [13]. It is also performed with care in [15] in the r = 1 case and in [14] for arbitrary r . The space of representations of L m of dimension vector (n, 1) is the affine space Of course, W r,n,m could be defined without reference to quivers, but it is interesting to notice that there exists a quiver stability condition θ on L m such that the open subscheme of Rep (n,1) ( L m ) consisting of θ -stable representations is precisely W r,n,m . The gauge group GL n acts freely on the smooth quasi-affine scheme W r,n,m , by conjugation on the matrices and via the natural action on the vectors. Therefore the quotient U r,n,m = W r,n,m / GL n is a smooth quasi-projective variety, of dimension (m − 1)n 2 + r n. The Quot scheme is realised as the closed subscheme Then the tangent space to Quot S ( ⊕r S , r ) at ξ is given by Hom( ⊕r p , ⊕r p ) = Hom( p , p ) ⊕r 2 ∼ = 2r 2 , using that Hom( p , p ) is 2-dimensional, being the tangent space to the smooth scheme Hilb 1 S = S at p . On the other hand, the Quot scheme Quot 2 ( ⊕r , n) is irreducible of dimension (r + 1)n, as was proven by Ellingsrud and Lehn [11]. Since 2r 2 > (r + 1)r , the point ξ is a singular point.

Remark 3.4.
In the case of 2 , it is well known that Theorem A does not hold. In this case, we do have a closed immersion where B i ∈ End( n ), i ∈ Hom( r , n ) and j ∈ Hom( n , r ). See [18, Thm. 2.1] and the references therein. The inclusion (3.2) is obtained as the locus j = 0.