The DT/PT correspondence for smooth curves
Andrea T. Ricolfi




We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi–Yau threefold. We exploit a local study of the Hilbert–Chow morphism about the cycle of a smooth curve. We determine, via Quot schemes, the global Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3.



Self-forces in arbitrary dimensions
    Abraham I. Harte^1, Peter Taylor^1 and Éanna É. Flanagan^2,3
    December 30, 2023
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§ INTRODUCTION


Let Y be a smooth, projective Calabi–Yau threefold. The Donaldson–Thomas (DT) invariants of Y are enumerative invariants attached to the Hilbert scheme 

    I_m(Y,β)=Z⊂ Y|χ(𝒪_Z)=m, [Z]=β,

viewed as a moduli space of ideal sheaves <cit.>. These numbers are insensitive to small deformations of the complex structure of Y, but they do change when we perturb the stability condition that defines them: the rules that govern these changes are the so called wall-crossing formulas. It is in this spirit that one can interpret the “DT/PT correspondence”, an equality of generating functions

    _β(q)=_0(q)·_β(q)

first conjectured in <cit.> and later proved by Bridgeland <cit.> and Toda <cit.>. 
The left hand side of (<ref>) is the Laurent series encoding 
Donaldson–Thomas invariants of Y in the class β∈ H_2(Y,ℤ), whereas _β encodes the Pandharipande–Thomas (PT) 
invariants of Y, defined through the moduli space of stable pairs <cit.>,

    P_m(Y,β)=(F,s)|χ(F)=m, [ F]=β.

Recall that a pair (F,s), consiting of a one-dimensional coherent sheaf F and a section s∈ H^0(Y,F), is said to be stable when 
F is pure and the cokernel of s is zero-dimensional. Finally, the formula <cit.>

    _0(q)=M(-q)^χ(Y)

determines the zero-dimensional DT theory of Y. Here M(q)=∏_k>0 (1-q^k)^-k is the MacMahon function.



 §.§ Main result

We prove a variant of (<ref>) in this note. 
Let C⊂ Y be a smooth curve of genus g embedded in class β. For integers n≥ 0, we define “local” DT invariants

    _n,C=∫_I_n(Y,C)ν_I χ

where ν_I:I_1-g+n(Y,β)→ is the Behrend function <cit.> on the Hilbert scheme and I_n(Y,C)⊂ I_1-g+n(Y,β) is the closed subset parametrizing subschemes Z⊂ Y containing C. For instance, a generic point of I_n(Y,C) represents a subscheme consisting of C along with n distinct points in Y∖ C.  
Similarly, we consider P_n(Y,C)⊂ P_1-g+n(Y,C), the closed subset parametrizing stable pairs (F,s) such that F=C.
The local invariants on the stable pair side

    _n,C=∫_P_n(Y,C)ν_P χ

have been studied in <cit.>. By smoothness of C, the local moduli space P_n(Y,C) can be identified with the symmetric product ^nC. Let us form the generating functions

    _C(q)=∑_n≥ 0_n,Cq^1-g+n,     _C(q)=∑_n≥ 0_n,Cq^1-g+n.

We say that the DT/PT correspondence holds for C if one has

    _C(q)=_0(q)·_C(q).

We can view (<ref>) as a wall-crossing formula relating the local curve counting invariants attached to C. 

Let n_g,C be the BPS number of C⊂ Y <cit.>. Using the known value of the PT side <cit.>,

    _C(q)=n_g,C· q^1-g(1+q)^2g-2,

we proved that the DT/PT correspondence (<ref>) holds for C smooth and rigid in <cit.>. The goal of this note is to extend the result to all smooth curves. 


Let Y be a smooth, projective Calabi–Yau threefold, C⊂ Y a smooth curve. Then the DT/PT correspondence (<ref>) holds for C.


In fact, the conclusion of the theorem holds for all Cohen–Macaulay curves, by recent work of Oberdieck <cit.>. While he works with motivic Hall algebras, our method involves a local study of the Hilbert–Chow morphism, and builds upon previous calculations <cit.>, especially the weighted Euler characteristic of the Quot scheme _n(ℐ_C), as we explain in Section <ref>.

The local invariants do not depend on the scheme structure one may put on I_n(Y,C) and P_n(Y,C). However, on the DT side, we will exploit the Hilbert–Chow morphism to endow I_n(Y,C) with a natural scheme structure, and we will prove that it agrees with the Quot scheme studied in <cit.>.
So, in the local theory, we can think of the moduli spaces

    _n(ℐ_C)    and    ^nC

as living on opposite sides of the wall separating DT and PT theory from one another.


Conventions.
All schemes are defined over . The Calabi–Yau condition for us is simply the existence of a trivialization of the canonical line bundle. The Hilbert–Chow morphism _r(X/S)→_r(X/S) is the one constructed by D. Rydh in <cit.>.






§ THE DT/PT CORRESPONDENCE

In this section we outline our strategy to deduce Theorem <ref>.

Let Y be a smooth projective variety, not necessarily Calabi–Yau. We consider the Hilbert–Chow morphism

    _1(Y)→_1(Y)

constructed in <cit.>, sending a 1-dimensional subscheme of Y to its fundamental cycle.  We recall its definition in Section <ref>.
Let I_m(Y,β)⊂_1(Y) be the component parametrizing subschemes Z⊂ Y such that 

    χ(𝒪_Z)=m∈,     [Z]=β∈ H_2(Y,).

Similarly, we let _1(Y,β)⊂_1(Y) be the component parametrizing 1-cycles of degree β. Then (<ref>) restricts to a morphism

    𝗁_m:I_m(Y,β)→_1(Y,β).



Fix an integer n≥ 0. For a Cohen–Macaulay curve C⊂ Y of arithmetic genus g embedded in class β, we let

    I_n(Y,C)⊂ I_1-g+n(Y,β)

denote the scheme-theoretic fibre of 𝗁_1-g+n, over the cycle of C.



We will use that (<ref>) is an isomorphism around normal schemes, at least in characteristic zero <cit.>. Thus, for a smooth curve C⊂ Y, we will identify Chow with Hilb locally around the cycle [C]∈_1(Y) and the ideal sheaf ℐ_C∈_1(Y).
For this reason, we do not need the representability of the global Chow functor, as around the point [C]∈_1(Y,β) we can work with the Hilbert scheme I_1-g(Y,β) instead. 
 

Consider the Quot scheme 

    _n(ℐ_C)

parametrizing quotients of length n of the ideal sheaf ℐ_C⊂𝒪_Y. 
We proved in <cit.> that the association [θ:ℐ_C↠ℰ]↦θ defines a closed immersion

    _n(ℐ_C)↪ I_1-g+n(Y,β).

More precisely, for a scheme S, an S-valued point of the Quot scheme is a flat quotient ℰ=ℐ_C× S/ℐ_Z, and in the short exact sequence

    0→ℰ→𝒪_Z→𝒪_C× S→ 0

over Y× S, the middle term is S-flat, so Z defines an S-point of I_1-g+n(Y,β).
The S-valued points of the image of (<ref>) consist precisely of those flat families Z⊂ Y× S→ S such that Z contains C× S as a closed subscheme. This will be used implicitly in the proof of Theorem <ref>.

The schemes I_n(Y,C) and _n(ℐ_C) have the same -valued points: they both parametrize subschemes Z⊂ Y consisting of C together with “n points”, possibly embedded.
The first step towards Theorem <ref> is the following result, whose proof is postponed to the next section.


Let Y be a smooth projective variety, C⊂ Y a smooth curve of genus g. Then I_n(Y,C)=_n(ℐ_C) as subschemes of I_1-g+n(Y,β).


As an application of Theorem <ref>, in Section <ref> we compute the reduced Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3.

To proceed towards Theorem <ref>, we need to examine the local structure of the Hilbert scheme around subschemes Z⊂ Y whose maximal Cohen–Macaulay subscheme
C⊂ Z is smooth. The result, given below, will be proven in the next section. 


Let Y be a smooth projective variety, C⊂ Y a smooth curve of genus g. 
Then, locally analytically around I_n(Y,C), the Hilbert scheme I_1-g+n(Y,β) is isomorphic to I_n(Y,C)×_1(Y,β).


Roughly speaking, this means that the Hilbert–Chow morphism, locally about the cycle

    [C]∈_1(Y,β),
 
behaves like a fibration with typical fibre I_n(Y,C). 
To obtain this, we first identify  with  locally around C, cf. Remark <ref>. We then need to trivialize the universal curve 𝒞→, which can be done since smooth maps are analytically locally trivial (on the source). However, even if we had 𝒞 = C×, we would not be done: the fibre of Hilbert–Chow (which is the Quot scheme by Theorem <ref>) depends on the embedding of the curve into Y, not just on the abstract curve. So to prove Theorem <ref> we need to trivialize (locally) the embedding of the universal curve into Y×. This is taken care of by a local-analytic version of the tubular neighborhood theorem. After this step, Theorem <ref> follows easily. 

Granting Theorems <ref> and <ref>, we can prove the DT/PT correspondence for smooth curves. So now we assume C is a smooth curve embedded in class β in a smooth, projective Calabi–Yau threefold Y.


By <cit.>, the Hilbert–Chow morphism 

    𝗁_1-g:I_1-g(Y,β)→_1(Y,β)

is (in characteristic zero) an isomorphism over the locus of normal schemes. Under this local identification, the cycle [C] corresponds to the ideal sheaf ℐ_C. We let ν(ℐ_C) be the value of the Behrend function on I_1-g(Y,β) at the point corresponding to ℐ_C.
Since the Behrend function can be computed locally analytically <cit.>, Theorem <ref> implies the identity

    ν_I|_I_n(Y,C)=ν(ℐ_C)·ν_I_n(Y,C),

where ν_I is the Behrend function of I=I_1-g+n(Y,β). After integration, we find

    _n,C=ν(ℐ_C)·χ̃(I_n(Y,C)),

where χ̃(I_n(Y,C)), by Theorem <ref>, agrees with the weighted Euler characteristic of the Quot scheme _n(ℐ_C). But we proved in <cit.> that the relation

    _n,C=ν(ℐ_C)·χ̃(_n(ℐ_C))

is equivalent to the local DT/PT correspondence (<ref>), so the theorem follows. 


As observed in <cit.>, the local DT/PT correspondence says that the local invariants are determined by the topological Euler characteristic of the corresponding moduli space, along with the BPS number of the fixed smooth curve C⊂ Y. The latter can be computed as

    n_g,C=ν(ℐ_C).

For any integer n≥ 0, the formulas are

    _n,C   =n_g,C· (-1)^nχ(I_n(Y,C)),
    _n,C   =n_g,C· (-1)^nχ(P_n(Y,C)).

In particular, the local invariants differ by the Euler characteristic of the corresponding moduli space by the same constant.







§ PROOFS

It remains to prove Theorems <ref> and <ref>.
For Theorem <ref>, we need to review some definitions and results from <cit.>.




 §.§ The fibre of Hilbert–Chow

Rydh has developed a powerful theory of relative cycles and has defined a Hilbert–Chow morphism

    _r(X/S)→_r(X/S)

for every algebraic space X locally of finite type over an arbitrary scheme S. For us X is always a scheme, projective over S. 

We quickly recall the definition of (<ref>). First of all, the Hilbert scheme _r(X/S) parametrizes S-subschemes of X that are proper and of dimension r over S, 
but not necessarily equidimensional, while _r(X/S) parametrizes equidimensional, proper relative cycles of dimension r.
We refer to <cit.> for the definition of relative cycles on X/S. Cycles have a (not necessarily equidimensional) support, which 
is a locally closed subset Z⊂ X. Rydh shows <cit.> that if α is a relative cycle on f:X→ S with support Z, 
then, for every r≥ 0, on the same family there is a unique equidimensional relative cycle α_r with support 

    Z_r=x∈ Z|_xZ_f(x)=r⊂ Z.

Cycles are called equidimensional when their support is equidimensional over the base.
The essential tool for the definition of (<ref>) is the norm family, defined by the following result.


Let X→ S be a locally finitely presented morphism, ℱ a finitely presented 𝒪_X-module which is flat over S. 
Then there is a canonical relative cycle 𝒩_ℱ on X/S, with support equal to ℱ. 
This construction commutes with arbitraty base change. When Z⊂ X is a subscheme which is flat and of finite presentation 
over S, we write 𝒩_Z=𝒩_𝒪_Z.


The Hilbert–Chow functor (<ref>) is defined by Z↦ (𝒩_Z)_r.

Even though we do not recall here the full definition of relative cycle, the main idea is the following. For a locally closed subset Z⊂ X, Rydh defines a projection of X/S adapted to Z to be a commutative diagram

    Urpd    Xdd
    
    Bd   
    
    Trg    S

where U→ X×_ST is étale, B→ T is smooth and p^-1(Z)→ B is finite. A relative cycle α on X/S with support Z⊂ X is the datum, for every projection adapted to Z, of a proper family of zero-cycles on U/B, which Rydh defines as a morphism

    α_U/B/T:B→Γ^⋆(U/B)

to the scheme of divided powers. We refer to <cit.> for the additional compatibility conditions that these data should satisfy.

Let now ℱ be a flat family of coherent sheaves on X/S. If 𝗉=(U,B,T,p,g) denotes a projection of X/S adapted to ℱ⊂ X as in (<ref>), then the zero-cycle defining the norm family 𝒩_ℱ at 𝗉 is

    (𝒩_ℱ)_U/B/T=𝒩_p^∗ℱ/B,

constructed in <cit.>. For us ℱ will always be a structure sheaf, so it will be easy to compare these zero-cycles.

If Z⊂ X is a subscheme that is smooth over S, then the norm family 𝒩_Z is an example of a smooth relative cycle, cf. <cit.>.
The next result states an equivalence, in characteristic zero, between smooth relative cycles and subschemes smooth over the base.



If S is of characteristic zero, then for every smooth relative cycle α on X/S there is a unique subscheme Z⊂ X, smooth over S, such that α=𝒩_Z.


We can now prove Theorem <ref>. We fix Y to be a smooth projective variety, C⊂ Y a smooth curve of genus g in class β, 
and we denote by I_n(Y,C) the fibre over [C] of the Hilbert–Chow morphism

    I_1-g+n(Y,β)→_1(Y,β),

as in Definition <ref>.


We need to show the equality

    I_n(Y,C)=_n(ℐ_C)

as subschemes of I_1-g+n(Y,β). Let S be a scheme over , and set X=Y×_ S. Then a family 

    Z⊂ X→ S

in the Hilbert scheme is an S-valued point of I_n(Y,C) when (𝒩_Z)_1=𝒩_C× S.
The closed immersion (<ref>) from the Quot scheme to the Hilbert scheme factors through I_n(Y,C). Indeed, any S-point ℐ_C× S↠ℐ_C× S/ℐ_Z of the Quot scheme gives a closed immersion C× S↪ Z whose relative ideal is zero-dimensional over S, thus we have (𝒩_Z)_1=(𝒩_C× S)_1=𝒩_C× S, where in the second equality we used that 𝒩_C× S is equidimensional of dimension one over S. So we obtain a closed immersion 

    ι:_n(ℐ_C)↪ I_n(Y,C).

For every scheme S, we have an injective map of sets

    ι(S):_n(ℐ_C)(S)↪ I_n(Y,C)(S),

and since ι() is a bijection, so far ι is just a bijective closed immersion. We need to show ι(S) is onto, and for the moment we deal with the case where S is a fat point. In other words, assume S is the spectrum of a local artinian -algebra with residue field . Let Z⊂ X→ S be an S-valued point of I_n(Y,C). Consider the finite subscheme F⊂ Y⊂ X given by the support of ℐ_C/ℐ_Z_0, where Z_0 is the closed fibre of Z→ S. Form the open set V=X∖ F⊂ X. Then we have, as relative cycles on V/S, 

    (𝒩_Z)_1|_V=𝒩_C× S|_V=𝒩_(C× S)∩ V.

We claim the left hand side equals the relative cycle 𝒩_Z∩ V.
For sure, these two cycles have the same support, as Z∩ V=Z_1∩ V, and they are determined by the same set of projections; indeed, being equidimensional of dimension one, they are determined by (compatible data of) relative zero-cycles for every projection 𝗉_V/S=(U,B,T,p,g) such that B/T is smooth of relative dimension one. Let us focus on (𝒩_Z)_1 first. Here r=1 is the maximal relative dimension of a point in Z, so the zero-cycle corresponding to a projection 𝗉_X/S as in (<ref>), and adapted to Z_1, is the same as the one defined by the norm family of Z (cf. the proof of <cit.>), namely 𝒩_p^∗𝒪_Z/B. Now we restrict to the open subset i:V→ X. By definition of pullback, the zero-cycle attached to a projection 𝗉_V/S (adapted to Z_1∩ V) is the cycle corresponding to the projection (U,B,T,i∘ p,g) for the full family Z/S, namely

    𝒩_(i∘ p)^∗𝒪_Z/B=𝒩_p^∗𝒪_Z∩ V/B.

The latter is precisely the zero-cycle defining the norm family of Z∩ V/S at the same projection 𝗉_V/S, so the claim is proved, 

    𝒩_Z∩ V=(𝒩_Z)_1|_V.

By the equivalence between smooth cycles and smooth subschemes stated in Theorem <ref>, we conclude that Z∩ V and (C× S)∩ V are the same (smooth) family over S. Moreover, the closure

    (C× S)∩ V⊂ Z

equals C× S, because the open subscheme (C× S)∩ V⊂ C× S is fibrewise dense (intersecting with V is only deleting a finite number of points in the special fibre). We have thus reconstructed a closed immersion C× S↪ Z, giving a well-defined S-valued point of _n(ℐ_C). So ι(S) is onto, and thus a bijection, whenever S is a fat point. This implies ι is étale, by a simple application of the formal criterion for étale maps. The theorem follows because we already know ι is a bijective closed immersion. 







 §.§ Local triviality of Hilbert–Chow


In this section we prove Theorem <ref>. The main tool used in the proof is the following local analytic version of the tubular neighborhood theorem.


Let S be a scheme, j:X→ Y a closed immersion over S. Assume X and Y are both smooth over S, of relative dimension d and n respectively. Then j is locally analytically isomorphic to the standard linear embedding ^d× S→^n× S. 



Let x∈ X and y=j(x)∈ Y. Let ℐ⊂𝒪_Y be the ideal sheaf of X in Y.
The relative smoothness of X, given that of Y, is characterized by the Jacobian criterion <cit.>, asserting that the short exact sequence

    0→ℐ/ℐ^2→ j^∗Ω_Y/S→Ω_X/S→ 0

is split locally around x∈ X. According to loc. cit. this is also equivalent to the following: whenever we choose local sections t_1,…,t_n and g_1,…,g_N of 𝒪_Y,y such that t_1,…, t_n constitute a free generating system for Ω_Y/S,y and g_1,…,g_N generate ℐ_y, after a suitable relabeling we may assume g_d+1,…,g_n generate ℐ about y and 

    t_1,…, t_d, g_d+1,…, g_n

generate Ω_Y/S locally around y. In particular, f_i=t_i∘ j, for i=1,…,d, define a local system of parameters at x. By this choice of local basis for Ω_Y/S around y, we can find open neighborhoods x∈ U⊂ X and y∈ V⊂ Y fitting in a commutative diagram

    U[hook]rj[swap]dét    
    Vdét
    ^d_S[hook]r   ^n_S

where the vertical maps are defined by the local systems of parameters (f_1,…,f_d) and (t_1,…,t_d,g_d+1,…,g_n) respectively, and the lower immersion is defined by sending t_i↦ f_i for i=1,…,d and g_k↦ 0.
Using the analytic topology, the inverse function theorem allows us to translate the étale maps into local analytic isomorphisms, and the statement follows.


Note that Lemma <ref> does not hold globally. For a closed immersion X⊂ Y of smooth complex projective varieties, it is not true in general that one can find a global tubular neighborhood. The obstruction lies in ^1(N_X/Y,T_X).

Before the proof of Theorem <ref>, we introduce the following notation.
If Z⊂ Y is a 1-dimensional subscheme corresponding to a point in the fibre I_n(Y,C) 
of Hilbert–Chow, we can attach to Z its “finite part”, the finite subset F_Z⊂ Z which is the support of the maximal zero-dimensional subsheaf of 𝒪_Z, namely the quotient ℐ_C/ℐ_Z. 


By <cit.> the Hilbert–Chow map is a local isomorphism around normal schemes, so we may identify an open neighborhood of the cycle of C in the Chow scheme with an open neighborhood U of [C] in the Hilbert scheme I_1-g(Y,β). We then consider the Hilbert–Chow map 

    𝗁=𝗁_1-g+n:I_1-g+n(Y,β)→_1(Y,β)

and we fix a point in the fibre [Z_0]∈ I_n(Y,C). It is easy to reduce to the case where the finite part F_0=F_Z_0⊂ Z_0 is confined on C, that is, Z_0 has only embedded points. 
We need to show that the Hilbert scheme is locally analytically isomorphic to U× I_n(Y,C) about [Z_0]. By Lemma <ref>, the universal embedding 𝒞⊂ Y× U, locally around the finite set of points F_0⊂ C⊂𝒞, is locally analytically isomorphic to the embedding of the zero section C× U⊂ C× U×^2 of the trivial rank 2 bundle. 
In particular we can find, in C× U×^2 and in Y× U, analytic open neighborhoods V and V' of F_0, fitting in a commutative diagram

    (C× U)∩ Vd[hook]r    V[hook]ropend    C× U×^2
    𝒞∩ V'[hook]r    V'[hook]ropen    Y× U

where the vertical maps are analytic isomorphisms. Now consider the open subset 

    A={(Z,u)∈ I_n(Y,C)× U | F_Z⊂ V_u}⊂ I_n(Y,C)× U.

Letting φ denote the isomorphism V → V', given a pair (Z,u)∈ A we can look at Z'=𝒞_u∪φ(F_Z), which is a new subscheme of Y, mapping to u under Hilbert–Chow.
The association (Z,u)↦ Z' defines an isomorphism between A and the open subset B⊂𝗁^-1(U) parametrizing subschemes Z'⊂ Y such that F_Z' is contained in V'_u, where u is the image of [Z'] under Hilbert–Chow. Note that [Z_0]∈ B corresponds to (Z_0,C)∈ A under this isomorphism. The theorem is proved.









§ THE DT THEORY OF AN ABEL–JACOBI CURVE

In this section we fix a non-hyperelliptic curve C of genus 3, embedded in its Jacobian 

    Y=( C,Θ)

via an Abel–Jacobi map. We let β=[C]∈ H_2(Y,) be the corresponding curve class. For n≥ 0, we let 

    ℋ^n_C⊂ I_n-2(Y,β)

be the component of the Hilbert scheme parametrizing subschemes Z⊂ Y whose fundamental cycle is algebraically equivalent to [C].


Let -1:Y→ Y be the automorphism y↦ -y, and let -C denote the image of C. As C is non-hyperelliptic, the cycle of C is not algebraically equivalent to the cycle of -C <cit.>. 
The Hilbert scheme  I_n-2(Y,β) consists of two connected components, which are interchanged by -1. Moreover, the Abel–Jacobi embedding C⊂ Y has 
unobstructed deformations, and there is an isomorphism Y → ℋ^0_C given by translations <cit.>.




As remarked in <cit.>, the morphism

    ℋ^1_C→ℋ^0_C× Y

sending T_x(C)∪ y↦ (T_x(C),y), where T_x denotes translation by x, is the Albanese map.
It can be easily checked that ℋ^1_C is isomorphic to the blow-up

    _𝒰(ℋ^0_C× Y),

where 𝒰 is the universal family. In particular, ℋ^1_C is smooth of dimension 6.









The quotient of the Hilbert scheme by the translation action of Y gives a Deligne–Mumford stack I_m(Y,β)/Y. 
In fact, since the Y-action is free, this is an algebraic space. The reduced Donaldson–Thomas invariants

    ^Y_m,β=∫_I_m(Y,β)/Yν χ∈

were introduced in <cit.> for arbitrary abelian threefolds. We consider their generating function

    _β(p)=∑_m∈^Y_m,βp^m.

We state the following result as a corollary of Theorem <ref>.


Let C⊂ Y be non-hyperelliptic, embedded in class β. Then

    _β(p)=2p^-2(1+p)^4.




As the Hilbert–Chow morphism is an isomorphism around normal schemes, we have an isomorphism

    I_-2(Y,β) → _1(Y,β).

On the other hand, the Hilbert scheme is the disjoint union of two copies of ℋ^0_C, where ℋ^0_C≅ Y because C is not hyperelliptic. Focusing on the component parametrizing translates of C, the Hilbert–Chow morphism ℋ^n_C→ℋ^0_C induces an isomorphism 

    Y×_n(ℐ_C) → ℋ^n_C

by Theorem <ref>. This shows that the quotient space ℋ^n_C/Y is isomorphic to the Quot 
scheme _n(ℐ_C). Keeping into account the second component 
of I_n-2(Y,β), still isomorphic to ℋ^n_C, we find 

    ^Y_n-2,β=2·χ̃(_n(ℐ_C)),

where χ̃ denotes the Behrend weighted Euler characteristic. 
Then 

    _β(p)=∑_n≥ 0^Y_n-2,βp^n-2=2p^-2∑_n≥ 0χ̃(_n(ℐ_C))p^n=2p^-2(1+p)^4,

where the last equality follows from <cit.>.



If one considers homology classes of type (1,1,d) for all d≥ 0, on an arbitrary abelian threefold Y, one has the formula

    ∑_d≥ 0∑_m∈ℤ^Y_m,(1,1,d)(-p)^mq^d=-K(p,q)^2,

where K is the Jacobi theta function

    K(p,q)=(p^1/2-p^-1/2)∏_m≥ 1(1-pq^m)(1-p^-1q^m)/(1-q^m)^2.

Relation (<ref>) was conjectured in <cit.> and proved in <cit.>.
Corollary <ref> confirms the coefficient of q via Quot schemes, when Y is the Jacobian of a general curve. Indeed, in this case the Abel–Jacobi class is of type (1,1,1).


The local DT theory of a general Abel–Jacobi curve C of genus 3 is determined as follows. 
Using again the isomorphism Y≅ℋ^0_C, we can compute the BPS number

    n_3,C=ν(ℐ_C)=-1,

thus the DT/PT correspondence at C (Theorem <ref>) yields

    _C(q)=_C(q)=-q^-2(1+q)^4.

In other words, the global theory is related to the local one by

    _β(q)=-2·_C(q).



Acknowledgements.
I would like to thank my advisors Martin G. Gulbrandsen and Lars H. Halle, and Letterio Gatto, for many valuable discussions. Thanks to David Rydh for sharing many useful comments and a key example that was useful to understand the norm family. Finally, I wish to thank Richard Thomas for patiently sharing his insights and commenting on a previous draft of this work.



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