Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

As an analogy to Gopakumar-Vafa conjecture on CY 3-folds, Klemm-Pandharipande defined GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this paper, we define stable pair type invariants on CY 4-folds and use them to interpret these GV type invariants. Examples are computed for both compact and non-compact CY 4-folds to support our conjectures.

Background. Gromov-Witten invariants are rational numbers counting stable maps from complex curves to algebraic varieties (or symplectic manifolds). They are not necessarily integers because of multiple cover contributions. In [20], Klemm-Pandharipande gave a definition of Gopakumar-Vafa type invariants on Calabi-Yau 4-folds using GW theory and conjectured that they are integers. For dimensional reasons, GW invariants for genus g 2 always vanish on Calabi-Yau 4-folds, so the integrality conjecture only applies in genus 0 and 1. In our previous paper [12], we gave a sheaf-theoretic interpretation of g = 0 GV type invariants using DT 4 invariants [10,4] of one-dimensional stable sheaves, analogous to the work of Katz for 3-folds [18].
In this paper, we propose a sheaf-theoretic approach to both genus 0 and 1 GV type invariants using stable pairs on CY 4-folds. For CY 3-folds, a Pairs/GV conjecture was first developed in work of Pandharipande and Thomas [30,32]. Our paper may be viewed as an analogue of their work in the setting of CY 4-folds. 0.2. GV type invariants on CY 4-folds. Let X be a smooth projective CY 4-fold. As mentioned above, Gromov-Witten invariants vanish for genus g 2 for dimensional reasons, so we only consider the genus 0 and 1 cases.
Then we define stable pair invariants τ (γ i ).
One issue with our current proposal (as in our earlier conjecture [12]) is that we do not have a general mechanism for choosing the orientation in the above conjectures. Currently, in the cases we examine in this paper, we choose orientations on a case-by-case basis to show the correct matching. It would be very interesting to construct canonical choices of orientation for these moduli spaces and study our conjectures using them. (2) Conjecture 0.2 is true for multiple fiber classes β = r[f ] (r 1).
In the above cases, we can directly compute the pair invariants and check the compatibility with the computation of GW invariants in [20].
Product of elliptic curve and Calabi-Yau 3-fold. Let X = Y × E be a product of a Calabi-Yau 3-fold and an elliptic curve E. We check our conjectures when the curve class comes from either Y or E.
Theorem 0.5. (Theorem 2.13, 2.15, Proposition 2.17) Let X = Y × E be given as above. Then (1) Conjecture 0.1 is true for any irreducible curve class β ∈ H 2 (Y ) ⊆ H 2 (X), provided that Y is a complete intersection in a product of projective spaces. (2) Conjecture 0.2 is true for any irreducible curve class β ∈ H 2 (Y ) ⊆ H 2 (X).
The proof of these results is briefly reviewed here. For (1), when β ∈ H 2 (Y ) ⊆ H 2 (X) is an irreducible curve class, we have an isomorphism P n (X, β) ∼ = P n (Y, β) × E.
The corresponding virtual class satisfies (see Proposition 2.11): [P n (X, β)] vir = [P n (Y, β)] vir pair ⊗ [E], for certain choice of orientation in defining the LHS, where the virtual class of P n (Y, β) is defined using the deformation-obstruction theory of pairs (Lemma 2.9) instead of the deformationobstruction theory of complexes in the derived category used by [30].
In this case, we have a forgetful morphism  (3), this is one of few cases where we can compute non-primitive curve classes and form generating series. The point is to identify pair moduli spaces on X with Hilbert schemes of points on Y and use computation of zero dimensional DT invariants of Y .
Hyperkähler 4-folds. When the CY 4-fold X is a hyperkähler 4-fold, GW invariants vanish, and so do the GV type invariants. To verify our conjectures, we are left to prove the vanishing of pair invariants. A cosection map from the (trace-free) obstruction space is constructed and shown to be surjective and compatible with Serre duality (Proposition 2.18). We expect the following vanishing result then follows.
At the moment, a Kiem-Li type theory of cosection localization for D-manifolds is not available in the literature. We believe that when such a theory is established, our claim should follow automatically. Nevertheless, we have the following evidence for the claim.
1. At least when P n (X, β) is smooth, Proposition 2.18 gives the vanishing of virtual class. 2. If there is a complex analytic version of (−2)-shifted symplectic geometry [33] and the corresponding construction of virtual classes [4], one could prove the vanishing result as in GW theory, i.e. taking a generic complex structure in the S 2 -twistor family of the hyperkähler 4-fold which does not support coherent sheaves and then vanishing of virtual classes follows from their deformation invariance. 0.5. Verifications of the conjecture II: local 3-folds and surfaces. For a Fano 3-fold Y , we consider the non-compact CY 4-fold In this case, the stable pair moduli space P n (X, β) is compact (Proposition 3.3), so we can formulate Conjecture 0.1, 0.2 here (even though the target is not projective).
When the curve class β ∈ H 2 (X) is irreducible, we study this as follows. Similar to the case of the product of a CY 3-fold and an elliptic curve, for a certain choice of orientation, the virtual class of P n (X, β) satisfies (Proposition 3.3) pair , under the isomorphism P n (X, β) ∼ = P n (Y, β). And we have a virtual push-forward formula (Proposition 3.2) Similarly for a smooth projective surface S, we consider the non-compact CY 4-fold where L 1 , L 2 are line bundles on S satisfying L 1 ⊗ L 2 ∼ = K S . In particular, when β is irreducible and L i · β < 0 (i = 1, 2), the moduli space P n (X, β) of stable pairs on X is compact and smooth (Lemma 3.6, Proposition 3.7). So pair invariants are well-defined and we can also study our conjectures in this case. In particular, we have be two ample line bundles on S such that L 1 ⊗ L 2 ∼ = K S . Denote β ∈ H 2 (X, Z) ∼ = H 2 (S, Z) to be an irreducible curve class on X = Tot S (L 1 ⊕ L 2 ). Then Conjecture 0.1, 0.2 are true for β.
In fact such a del Pezzo surface must be P 2 or P 1 × P 1 (see the proof of Proposition 3.8), and the corresponding X is given by By using computations due to Kool and Monavari [21], one can check Conjecture 0.1, 0.2 for small degree curve classes on such X (see Section 3.3 for details). 0.6. Verifications of the conjecture III: local curves. Let C be a smooth projective curve. We consider a CY 4-fold X given by can be defined via equivariant residue. Here λ i are the equivariant parameters with respect to the T -action.
On the other hand, there is a two dimensional subtorus T 0 ⊆ (C * ) 3 which preserves the CY 4-form on X. We may define equivariant pair invariants as rational functions in terms of equivariant parameters of T 0 following a localization principle for DT 4 invariants (see Section 4.2, [10], [12,Sect. 4.2]).
When C = P 1 and X = O P 1 (l 1 , l 2 , l 3 ), we explicitly determine P 1,d[C] (X) for d 2 (Proposition 4.5). Note in this case P 0,[P 1 ] (X) = 0 and there are no insertions, so an equivariant analogue of Conjecture 0.1 is given by the following conjecture: Conjecture 0.9. (Conjecture 4.6) Let X = O P 1 (l 1 , l 2 , l 3 ) for l 1 + l 2 + l 3 = −2. Then We can verify the above equivariant conjecture in a large number of examples.
When C is an elliptic curve and L i 's are general degree zero line bundles on C, one can define pair invariants and explicitly compute them.
Theorem 0.11. (Theorem 4.10) Let C be an elliptic curve, L i ∈ Pic 0 (C) (i = 1, 2, 3) general line bundles satisfying Then stable pair invariants P 0,d[C] (X) are well-defined and fit into the generating series d 0 Similarly, if we have n 1,β β ∈ H 2 (X, Z) many such elliptic curves, then they contribute to pair invariants according to the formula: This calculation arises in the heuristic argument for our genus one conjecture (Conjecture 0.2) in the 'ideal' situation as families of rational curves do not contribute to pair invariants P 0,β 's (see Section 1.5 for more details). 0.7. Speculation on the generating series of stable pair invariants. As before, if we allow insertions, we can use the virtual class (0.3) and insertions to define stable pair invariants of P n (X, β) for any n.
For γ ∈ H 4 (X, Z), τ (γ) ∈ H 2 (P n (X, β), Z), so we may define Our computations and geometric arguments indicate that we may have the following formula, which generalizes the formula in Conjecture 0.1, To group these invariants into a generating series, we introduce notation PT(X)(exp(γ)) := n,β P n,β (γ) n! y n q β .
Assuming Conjecture 0.2, then (0.6) is equivalent to the following Gopakumar-Vafa type formula: where n 0,β (γ) and n 1,β are genus 0 and 1 GV type invariants of X (0.1), (0.2) respectively and M (q) = k 1 (1 − q k ) −k is the MacMahon function. As mentioned before, GW invariants on CY 4-folds vanish for g > 1, so they do not form a nice generating series as in the 3-folds case.
Here the advantage of considering stable pair invariants is we can use them to form a generating series which is conjecturally of GV form. A heuristic explanation of the formula will be given in Section 1.5. Some more analysis will be pursued in a future work. 0.8. Notation and convention. In this paper, all varieties and schemes are defined over C. For a morphism π : X → Y of schemes, and for F , G ∈ D b (Coh(X )), we denote by RHom π (F , G) the functor Rπ * RHom X (F , G). We also denote by ext i (F , G) the dimension of Ext i X (F , G). A class β ∈ H 2 (X, Z) is called irreducible (resp. primitive) if it is not the sum of two non-zero effective classes (resp. if it is not a positive integer multiple of an effective class). 0.9. Acknowledgement. We are very grateful to Martijn Kool and Sergej Monavari for helpful discussions on stable pairs on local surfaces and generously sharing their computational results. We would like to thank Dominic Joyce for helpful comments on our preprint and a responsible referee for very careful reading of our paper and helpful suggestions which improves the exposition of the paper. Y. C. is partly supported by the Royal Society Newton International

Definitions and conjectures
Throughout this paper, unless stated otherwise, X is always denoted to be a smooth projective Calabi-Yau 4-fold, i.e. K X ∼ = O X .
1.1. GW/GV conjecture on CY 4-folds. Let M g,n (X, β) be the moduli space of genus g, n-pointed stable maps to X with curve class β. Its virtual dimension is given by For integral classes where ev i : M g,n (X, β) → X is the i-th evaluation map.
For g = 0, the virtual dimension of M 0,n (X, β) is n + 1, and (1.2) is zero unless In analogy with the Gopakumar-Vafa conjecture for CY 3-folds [14], Klemm-Pandharipande [20] defined invariants n 0,β (γ 1 , . . . , γ n ) on CY 4-folds by the identity  For g = 1, the virtual dimension of M 1,0 (X, β) is zero, so no insertions are needed. The genus one GW invariant is also expected to be described in terms of certain integer valued invariants.
Let S 1 , · · · , S k be a basis of the free part of H 4 (X, Z) and be the (4, 4)-component of Künneth decomposition of the diagonal. For β 1 , β 2 ∈ H 2 (X, Z), the meeting number m β1,β2 ∈ Z is introduced in [20] as a virtual number of rational curves of class β 1 meeting rational curves of class β 2 . They are uniquely determined by the following rules: (i) The meeting invariants are symmetric, m β1,β2 = m β2,β1 .
(iv) If β 1 = β 2 = β, we have The invariants n 1,β are uniquely defined by the identity where σ(d) = i|d i. For g 2, GW invariants vanish for dimension reasons, so the GW/GV type integrality conjecture on CY 4-folds only applies for genus 0 and 1. In [20], GW invariants are computed directly in many examples using localization or mirror symmetry to support the conjectures.

1.2.
Review of DT 4 invariants. Let us first introduce the set-up of DT 4 invariants. We fix an ample divisor ω on X and take a cohomology class v ∈ H * (X, Q).
The coarse moduli space M ω (v) of ω-Gieseker semistable sheaves E on X with ch(E) = v exists as a projective scheme. We always assume that M ω (v) is a fine moduli space, i.e. any point [E] ∈ M ω (v) is stable and there is a universal family In [4,10], under certain hypotheses, the authors construct a DT 4 virtual class where χ(−, −) is the Euler pairing. Notice that this class may not necessarily be algebraic.
Roughly speaking, in order to construct such a class, one chooses at every point [E] ∈ M ω (v), a half-dimensional real subspace Ext 2 + (E, E) ⊂ Ext 2 (E, E) of the usual obstruction space Ext 2 (E, E), on which the quadratic form Q defined by Serre duality is real and positive definite. Then one glues local Kuranishi-type models of form , where κ is a Kuranishi map of M ω (v) at E and π + is the projection according to the decomposition Ext 2 (E, E) = Ext 2 In [10], local models are glued in three special cases: (1) when M ω (v) consists of locally free sheaves only; (2) when M ω (v) is smooth; (3) when M ω (v) is a shifted cotangent bundle of a derived smooth scheme. And the corresponding virtual classes are constructed using either gauge theory or algebrogeometric perfect obstruction theory.
The general gluing construction is due to Borisov-Joyce [4] 1 , based on Pantev-Töen-Vaquié-Vezzosi's theory of shifted symplectic geometry [33] and Joyce's theory of derived C ∞ -geometry. The corresponding virtual class is constructed using Joyce's D-manifold theory (a machinery similar to Fukaya-Oh-Ohta-Ono's theory of Kuranishi space structures used in defining Lagrangian Floer theory).
In this paper, all computations and examples will only involve the virtual class constructions in situations (2), (3), mentioned above. We briefly review them as follows: • When M ω (v) is smooth, the obstruction sheaf Ob → M ω (v) is a vector bundle endowed with a quadratic form Q via Serre duality. Then the DT 4 virtual class is given by Here e(Ob, Q) is the half-Euler class of (Ob, Q) (i.e. the Euler class of its real form Ob + ), and PD(−) is its Poincaré dual. Note that the half-Euler class satisfies e(Ob, Q) 2 = (−1) rk(Ob) 2 e(Ob), if rk(Ob) is even, e(Ob, Q) = 0, if rk(Ob) is odd.
• When M ω (v) is a shifted cotangent bundle of a derived smooth scheme, roughly speaking, this means that at any closed point [F ] ∈ M ω (v), we have Kuranishi map of type where κ factors through a maximal isotropic subspace V F of (Ext 2 (F, F ), Q). Then the DT 4 virtual class of M ω (v) is, roughly speaking, the virtual class of the perfect obstruction theory formed by {V F } F ∈Mω(v) . When M ω (v) is furthermore smooth as a scheme, then it is simply the Euler class of the vector bundle On orientations. To construct the above virtual class (1.5) with coefficients in Z (instead of Z 2 ), we need an orientability result for M ω (v), which is stated as follows. Let be the determinant line bundle of M ω (v), equipped with a symmetric pairing Q induced by Serre duality. An orientation of (L, Q) is a reduction of its structure group (from O(1, C)) to SO(1, C) = {1}; in other words, we require a choice of square root of the isomorphism to construct the virtual class (1.5). An orientability result was first obtained for M ω (v) when the CY 4-fold X satisfies Hol(X) = SU (4) and H odd (X, Z) = 0 [11, Theorem 2.2] and it has recently been generalized to arbitrary CY 4-folds by [7]. Notice that, if an orientation exists, the set of orientations forms a torsor for H 0 (M ω (v), Z 2 ).

Stable pair invariants on CY 4-folds.
The notion of stable pairs on a CY 4-fold X can be defined similarly as in the case of threefolds [30]. It consists of data where F is a pure one dimensional sheaf and s is surjective in dimension one. For β ∈ H 2 (X, Z) and n ∈ Z, let P n (X, β) (1.8) be the moduli space of stable pairs (F, s) on X such that [F ] = β, χ(F ) = n. It is a projective scheme parametrizing two-term complexes in the derived category of coherent sheaves on X.
Similar to moduli spaces of stable sheaves, the stable pair moduli space (1.8) admits a deformation-obstruction theory, whose tangent, obstruction and 'higher' obstruction spaces are given by where (−) 0 denotes the trace-free part. Note that Serre duality gives an isomorphism Ext 1 0 ∼ = (Ext 3 0 ) ∨ and a non-degenerate quadratic form on Ext 2 0 . Moreover, we have Lemma 1.3. The stable pair moduli space P n (X, β) can be given the structure of a (−2)-shifted symplectic derived scheme in the sense of Pantev-Töen-Vaquié-Vezzosi [33].
Proof. By [30, Theorem 2.7], P n (X, β) is a disjoint union of connected components of the moduli stack of perfect complexes of coherent sheaves of trivial determinant on X, whose (−2)-shifted symplectic structure is constructed by [ Let I = (O X×Pn(X,β) → F) be the universal pair, the determinant line bundle is endowed with a non-degenerate quadratic form Q defined by Serre duality, where π P : X × P n (X, β) → P n (X, β) is the projection. Similarly as before, the orientability issue for the pair moduli space P n (X, β) is whether the structure group of the quadratic line bundle (L, Q) can be reduced from O(1, C) to SO(1, C) = {1}. By [7], these moduli spaces are always orientable.
Theorem 1.4. Let X be a CY 4-fold and β ∈ H 2 (X, Z) and n ∈ Z be an integer. Then P n (X, β) has a virtual class in the sense of Borisov-Joyce [4], depending on the choice of orientation.
When n = 0, the virtual dimension of the virtual class (1.9) is zero. We define the stable pair invariant as the degree of the virtual class.
When n = 1, the (real) virtual dimension of the virtual class (1.9) is two, so we consider insertions as follows. For integral classes We define the stable pair invariant τ (γ i ).

1.4.
Relations with GW/GV conjecture on CY 4 . We use the stable pair invariants defined in Section 1.3 to give a sheaf-theoretic approach to the GW/GV conjecture in Section 1.1.
1.5. Heuristic approach to conjectures. In this subsection, we give a heuristic argument to explain why we expect Conjecture 1.5, 1.6 (and equality (0.6)) to be true. Even in this heuristic discussion, we ignore questions of orientation. Let X be an 'ideal' CY 4 in the sense that all curves of X deform in families of expected dimensions, and have expected generic properties, i.e.
(1) any rational curve in X is a chain of smooth P 1 's with normal bundle O P 1 (−1, −1, 0), and moves in a compact 1-dimensional smooth family of embedded rational curves, whose general member is smooth with normal bundle O P 1 (−1, −1, 0). (2) any elliptic curve E in X is smooth, super-rigid, i.e. the normal bundle is Furthermore any two elliptic curves are disjoint and disjoint from all families of rational curves on X. (3) there is no curve in X with genus g 2. P 0 (X, β) and genus 1 conjecture. Under our ideal assumptions, a one-dimensional Cohen-Macaulay scheme C supported in one of our families of rational curves has χ(O C ) 1, so for any stable pair I = (O X → F ) ∈ P 0 (X, β), the sheaf F can only be supported on some rigid elliptic curves in X. For a rigid elliptic curve E with [E] = β and 'general' normal bundle (i.e. direct sum of three degree zero general line bundles on E), its contribution to the pair invariant is by a localization calculation (see Theorem 4.10). Similarly, if we have n 1,β β ∈ H 2 (X, Z) many such elliptic curves, then they contribute to pair invariants according to the formula: P 1 (X, β) and genus 0 conjecture. Given a stable pair I = (O X → F ) ∈ P 1 (X, β), F may be supported on a union of rational curves and elliptic curves. Let C := supp(F ), then C = C 1 ⊔ C 2 is a disjoint union of 'rational curve components' and 'elliptic curve components'. Note a Cohen- Thus from the exact sequence when F is supported on elliptic curves, once we include insertions, these stable pairs do not contribute to the invariant So we only consider the case when F ∼ = O C1⊔C2 with C 1 supported on rational curves in a onedimensional family {C t } t∈T . We may further assume the support of C 1 is smooth with normal bundle O P 1 (−1, −1, 0) due to the presence of insertions, at which point it must have multiplicity 1 as well.
Since the families of rational curves are disjoint from the elliptic curves, the moduli space P 1 (X, β) of stable pairs is a disjoint union of product of rational curve families (with curve class β 1 ) and P 0 (X, β 2 ) (where β 1 + β 2 = β). And a direct calculation shows the corresponding virtual class factors as the product of the fundamental class of those rational curve families and [P 0 (X, β 2 )] vir . For γ ∈ H 4 (X), we then have P n (X, β) and generating series. For the moduli space P n,β (X) of stable pairs with n 1, we want to compute For dimension reasons, we may assume for any i = j the rational curves which meet with Z i are disjoint from those with Z j . The insertions cut out the moduli space and pick up stable pairs whose support intersects with all {Z i } n i=1 . We denote the moduli space of such 'incident' stable pairs by where Q 1 (X, β i ; Z i ) is the moduli space of stable pairs supported on rational curves (in class β i ) which meet with Z i .
Indeed let us take a stable pair where F 0 is supported on elliptic curves and each F i for 1 i n is supported on rational curves which meet with Z i . As explained before, a Cohen-Macaulay scheme C supported in the family of rational curves (resp. elliptic curves) satisfies χ(O C ) 1 (resp. χ(O C ) 0), so χ(F 0 ) 0 and χ(F i ) 1 for 1 i n. Hence χ(F 0 ) = 0 and χ(F i ) = 1 for 1 i n. Therefore (1.10) holds.
Moreover each Q 1 (X, β i ; Z i ) consists of finitely many rational curves which meet with Z i , whose number is exactly n 0,βi (γ). By counting the number of points in P 0 (X, β 0 ) and Q 1 (X, β i ; Z i )'s, we obtain n 0,βi (γ).
The above arguments give a heuristic explanation for the formula mentioned in Section 0.7.

Compact examples
In this section, we verify Conjectures 1.5 and 1.6 for certain compact Calabi-Yau 4-folds.
2.1. Sextic 4-folds. Let X be a smooth sextic 4-fold, i.e. a smooth degree six hypersurface of P 5 . By the Lefschetz hyperplane theorem, In order to verify our conjectures, we may use deformation invariance and assume X is general in the (projective) space P H 0 (P 5 , O (6)) of degree six hypersurfaces.
Genus 0. For the genus 0 conjecture, we have: Proof. In such cases, P 0,β (X) = 0 by Proposition 2.2. So we only need to show We consider β = 2[l] as the degree one case follows from the same argument. A Cohen-Macaulay When X is a general sextic, C is either a smooth conic or a pair of distinct intersecting lines (see e.g. [6, Proposition 1.4]). The morphism 2 to the moduli space M 1,β (X) of one dimensional stable sheaves, with [F ] = β and χ(F ) = 1, is an isomorphism. Furthermore, under the isomorphism, we have identifications where C is the support of F and Q is zero dimensional. A Cohen-Macaulay curve C in X with [C] = β has χ(O C ) 1, contradicting with χ(F ) = 0. So P 0 (X, β) = ∅.

Elliptic fibration.
For Y = P 3 , we take general elements Let X be a CY 4-fold with an elliptic fibration given by the equation where [x : y : z] is the homogeneous coordinate of the above projective bundle. A general fiber of π is a smooth elliptic curve, and any singular fiber is either a nodal or cuspidal plane curve. Moreover, π admits a section ι whose image corresond to fiber point [0 : 1 : 0].
Let h be a hyperplane in P 3 , f be a general fiber of π : X → Y and set Genus 0. We consider the stable pair moduli space P 1 (X, [f ]) for the fiber class of π and verify Conjecture 1.5 in this case.
be the fiber class of the elliptic fibration (2.1). Then we have an isomorphism under which the virtual class satisfies where the sign corresponds to the choice of orientation in defining the LHS.
Next, we compare the obstruction theories.
whose cohomology gives an exact sequence where the horizontal and vertical arrows are distinguished triangles. By taking cones, we obtain a distinguished triangle , whose cohomology gives an exact sequence Genus 1. We consider the stable pair moduli space P 0 (X, r[f ]) for multiple fiber classes r[f ] (r 1) of π and confirm Conjecture 1.6 in this case.
Lemma 2.5. For any r ∈ Z 1 , there exists an isomorphism under which the virtual class is given by for certain choice of orientation in defining the LHS, where [Hilb r (P 3 )] vir is the DT 3 virtual class [36].
Proof. The proof is similar to the one in [38,Proposition 6.8]. We show that the natural morphism is an isomorphism. Let (s : O X → F ) ∈ P 0 (X, r[f ]) be a stable pair. By the Harder-Narasimhan and Jordan-Hölder filtrations, we have Here the slope of a zero dimensional sheaf is defined to be infinity. Since F is a pure one dimensional sheaf, so E 1 = F 1 can not be zero dimensional (r 1 1). Therefore ch(E i ) = (0, 0, 0, r i [f ], χ(E i )) for some r i 1. The stability of E i implies that it is scheme theoretically supported on some fiber implies that χ(E i ) = 0 for any i, and hence E ′ n ∼ = O Xp n [15, Proposition 1.2.7]. By diagram chasing, we obtain a morphism I Xp n → F n−1 for the ideal sheaf I Xp n ⊆ O X of X pn , which is surjective in dimension one. Then so is the composition We have the isomorphism In either case, similarly as before, we have E ′ n−1 ∼ = O Xp n−1 . Moreover the morphism (2.7) is a pull-back of a surjection I pn → O pn−1 by π * . By repeating the above argument, we see that each E i is isomorphic to O Xp i , s : O X → F is surjective and given by a pull back of a surjection O P 3 → O Z by π * for some zero dimensional subscheme Z ⊂ P 3 with length n. Using the section ι of π : X → P 3 , we have the morphism ι * : P 0 (X, r[f ]) → Hilb r (P 3 ), which gives an inverse of (2.5). Therefore the morphism (2.5) is an isomorphism.
It remains to compare the virtual classes. We take I Z ∈ Hilb r (P 3 ) and use the spectral sequence Furthermore, Kuranishi maps for deformations of π * I Z on X can be identified with Kuranishi maps for deformations of I Z on P 3 . Similar to [10, Theorem 6.5], we are done. Proof. Combining Lemma 2.5 (where we choose sign to be (−1) r according to the parity of r) and the generating series for zero-dimensional DT invariants [28,24,23], we obtain the formula. Notice from [20, Table 7], we have n 1,[f ] = −20 and n 1,k[f ] = 0 for k = 1 (which can also be checked from GW theory).
In this section, we discuss the irreducible curve class in a quintic fiber for these two examples. Here we only consider genus 1 invariants. Genus 1. Conjecture 1.6 predicts that for an irreducible class β and a suitable choice of orientation, we have P 0,β = n 1,β := GW 1,β + 1 24 GW 0,β (c 2 (X)).
Note that the genus 1 invariants n 1,β for irreducible β are zero for both quintic fibration examples in [20], where their computations of GW 1,β are based on BCOV theory [3]. The pair invariant P 0,β is obviously zero in this case since we have: Lemma 2.7. Let β ∈ H 2 (X, Z) be an irreducible class. The pair moduli space P 0 (X, β) is empty if and only if any curve C ∈ Chow β (X) in the Chow variety is a smooth rational curve.
Proof. ⇐) Given a stable pair (s : O X → F ) ∈ P 0 (X, β), then F is a torsion-free sheaf (in fact a line bundle) over a curve C ∼ = P 1 . Since χ(F ) = 0, so F = O C (−1) contradicting with the surjectivity of s in dimension 1.
⇒) For C ∈ Chow β (X) in an irreducible class β, the restriction map (O X → O C ) gives a stable pair. Since P 0 (X, β) is empty, we have i.e. h 1 (C, O C ) = 0, which implies that C is a smooth rational curve.
With this lemma, we can verify Conjecture 1.6 for irreducible classes in more examples.
Proposition 2.8. Conjecture 1.6 is true for irreducible class β ∈ H 2 (X, Z) when X is either (1) one of the quintic fibrations in [20]; (2) a smooth complete intersection in a projective space; (3) one of the complete intersections in Grassmannian varieties in [13].
2.4. Product of elliptic curve and CY 3-fold. In this subsection, we consider a CY 4-fold of type X = Y × E, where Y is a projective CY 3-fold and E is an elliptic curve.
Genus 0. We study Conjecture 1.5 for an irreducible curve class of X = Y × E. If β = [E], P 1,β = 0, the conjecture is obviously true (in fact for any r 1, one can show Conjecture 1.5 is true for β = r[E]). Below we consider curve classes coming from the CY 3-fold. Lemma 2.9. Let β ∈ H 2 (Y, Z) be an irreducible curve class on a CY 3-fold Y .
Then the pair deformation-obstruction theory of P n (Y, β) is perfect in the sense of [1,25]. Hence we have an algebraic virtual class whose cohomology gives an exact sequence By truncating Ext 2 Y (I Y , F ) = C, the pair deformation theory is perfect.
In particular, when n = 1, the virtual class [P 1 (Y, β)] vir pair has zero degree. We show the following virtual push-forward formula. Proof. Since β is irreducible, there is a morphism Then the universal stable pair is given by where O(1) is the tautological line bundle on P(π M * F) and s is the tautological map.
Let π P : P 1 (Y, β) × Y → P 1 (Y, β) be the projection, there exists a distinguished triangle By considering a derived extension of the morphism f (2.9), the first two terms in (2.10) are the restriction of cotangent complexes of the corresponding derived schemes to the classical underlying schemes. They are obstruction theories (see [35,Sect. 1.2]), which fit into a commutative diagram where the bottom vertical arrows are truncation functors. Note the above obstruction theories are not perfect. To kill h −2 , as in [16,Sect. 4.4], we consider the top part of trace map By taking cones, we obtain a distinguished triangle ∨ is a vector bundle concentrated in degree −2 and τ −1 (−) has cohomology in degree greater than −2, so we have a commutative diagram To kill h 1 of the left upper term, we consider the inclusion , whose restriction to a closed point I = (O Y → F ) induces an isomorphism C → Hom(F, F ). The cone of the inclusion is τ [0,1] RHom πP (F † , F † ) [1] . Then we have a commutative diagram As (O P1(Y,β) [1]) ∨ is a vector bundle concentrated on degree 1, so we get commutative diagram It is easy to see that φ 1 and φ 2 define perfect obstruction theories. By diagram chasing on cohomology, φ 3 defines a perfect relative obstruction theory. Then we apply Manolache's virtual push-forward formula [26]: where the coefficient c is the degree of the virtual class of the relative obstruction theory φ 3 and can be shown to be 1 by base-change to a closed point. Now we come back to CY 4-fold X = Y × E and show the virtual class [P 1 (Y, β)] vir pair defined using pair deformation-obstruction theory naturally arises in this setting.
Proposition 2.11. Let X = Y × E be a product of a CY 3-fold Y with an elliptic curve E. For an irreducible curve class β ∈ H 2 (Y, Z) ⊆ H 2 (X, Z), we have an isomorphism The virtual class of P n (X, β) satisfies , for certain choice of orientation in defining the LHS. Here [P n (Y, β)] vir pair ∈ A n−1 (P n (Y, β), Z) is the virtual class defined in Lemma 2.9.
Denote i = i t . From the distinguished triangle where the horizontal and vertical arrows are distinguished triangles. By taking cones, we obtain a distinguished triangle Therefore by (2.13), we have the distinguished triangle It follows that we have the distinguished triangle By Serre duality, adjunction and degree shift, (2.16) becomes Combining (2.8), (2.17), we obtain Combining with (2.15) and taking the cohomological long exact sequence, we have We claim the above exact sequence breaks into short exact sequences ) and a dimension counting by Riemann-Roch. The first exact sequence above implies that the map (2.11) induces an isomorphism on tangent spaces. The second exact sequence implies that the obstructions of deforming stable pairs on LHS of (2.11) vanish if and only if those on RHS of (2.11) vanish. Therefore, the map (2.11) induces an isomorphism on formal completions of structure sheaves of both sides at any closed point. So (2.11) must be a scheme theoretical isomorphism.
Next, we show Ext 1 Y (I Y , F ) ⊆ Ext 2 X (I X , I X ) 0 is a maximal isotropic subspace with respect the Serre duality pairing on Ext 2 X (I X , I X ) 0 . For u ∈ Ext 1 Y (I Y , F ), the corresponding element in Ext 2 X (I X , I X ) 0 is given by the composition where the morphism α is the canonical morphism and β is given by (2.12). For another u ′ ∈ Ext 1 Y (I Y , F ), it is enough to show the vanishing of the composition can be written as i * γ. Therefore the composition vanishes, again by Ext 3 Y (I Y , F ) = 0. Moreover, a local Kuranishi map of P n (X, β) at I X can be identified as Similarly as [10, Thm. 6.5], we have the desired equality on virtual classes.
Combining the above result with Proposition 2.10, our genus zero conjecture can be reduced to Katz's conjecture [18]. Proof. To have non-trivial invariants, we only need to consider insertions of form By Proposition 2.10 and 2.11, we have Then Conjecture 1.5 reduces to Katz's conjecture.
Katz's conjecture has been verified for primitive classes in complete intersection CY 3-folds [12, Cor. A.6]. So we obtain Theorem 2.13. Let Y be a complete intersection CY 3-fold in a product of projective spaces, X = Y × E be the product of Y with an elliptic curve E. Then Conjecture 1.5 is true for an irreducible curve class β ∈ H 2 (Y, Z) ⊆ H 2 (X, Z). Furthermore, their degrees fit into the generating series where M (q) = k 1 (1 − q k ) −k is the MacMahon function and we define P 0,0[E] = 1.
We check Conjecture 1.6 for this case.
where σ(d) = i|d i. We have an isomorphism for moduli space M 1,0 (X, r[E]) of genus 1 stable maps to X. Note that M 1,0 (E, r[E]) is smooth of expected dimension and consists of σ(r) r points (modulo automorphisms) (see e.g. [29]). And the genus one invariant for constant map to Y is χ(Y ). So GW 1,r[E] (X) = χ(Y ) · σ(r) r . When the curve class β ∈ H 2 (Y ) ⊆ H 2 (X) comes from Y , we have Proof. We have an isomorphism By divisor equation, one can compute genus zero GW invariants. M 1,0 (X, β) has a similar product structure as (2.19). The obstruction sheaf has a trivial factor T E = O E in E direction. So genus one GW invariants vanish.
Then it is easy to show the following: Proposition 2.17. Let X = Y × E be the product of a CY 3-fold Y with an elliptic curve E. Then Conjecture 1.6 is true for any irreducible class β ∈ H 2 (Y ) ⊆ H 2 (X).
Proof. By Lemma 2.16, we know n 1,β = 0. By Proposition 2.11, the virtual dimension of [P 0 (Y, β)] vir pair is negative, so P 0,β = 0. 2.5. Hyperkähler 4-folds. When the CY 4-fold X is hyperkähler, GW invariants on X vanish as they are deformation-invariant and there are no holomorphic curves for generic complex structures in the S 2 -twistor family. Another way to see the vanishing is via the cosection localization technique developed by Kiem-Li [19].
Roughly speaking, given a perfect obstruction theory [1,25]  To verify Conjectures 1.5 and 1.6 for hyperkähler 4-folds, we only need to show the vanishing of stable pair invariants of P 0 (X, β) and P 1 (X, β).
Cosection and vanishing of DT 4 virtual classes. Fix a stable pair I ∈ P n (X, β), by taking wedge product with square At(I) 2 of the Atiyah class and contraction with the holomorphic symplectic form σ, we get a surjective map φ : Ext 2 (I, I) 0 In fact, we have where Q is non-degenerate on each subspace. Here κ X is the Kodaira-Spencer class which is Serre dual to ch 3 (I).
We claim that the surjectivity of cosection maps leads to the vanishing of virtual classes for stable pair moduli spaces (it also applies to other moduli spaces, e.g. Hilbert schemes of curves/points used in DT/PT correspondence [8,9]). Claim 2.19. Let X be a projective hyperkähler 4-fold and P n (X, β) be the moduli space of stable pairs with n = 0 or β = 0. Then the virtual class satisfies [P n (X, β)] vir = 0.
At the moment, Kiem-Li type theory of cosection localization for D-manifolds is not available in the literature. We believe that when such a theory is established, our claim should follow automatically. Nevertheless, we have the following evidence for the claim.
1. At least when P n (X, β) is smooth, Proposition 2.18 gives the vanishing of virtual class. 2. If there is a complex analytic version of (−2)-shifted symplectic geometry [33] and the corresponding construction of virtual classes [4], one could prove the vanishing result as in GW theory, i.e. taking a generic complex structure in the S 2 -twistor family of the hyperkähler 4-fold which does not support coherent sheaves and then vanishing of virtual classes follows from their deformation invariance.

Non-compact examples
3.1. Irreducible curve classes on local Fano 3-folds. Let Y be a Fano 3-fold. When Y embeds into a CY 4-fold X, the normal bundle of Y ⊆ X is the canonical bundle K Y of Y . By the negativity of K Y , there exists an analytic neighbourhood of Y in X which is isomorphic to an analytic neighbourhood of Y in K Y . Here we simply consider non-compact CY 4-folds of form X = K Y .
Similar to Lemma 2.9, we have Lemma 3.1. Let β ∈ H 2 (Y, Z) be an irreducible curve class on a Fano 3-fold Y . Then the pair deformation-obstruction theory of P n (Y, β) is perfect in the sense of [1,25]. Hence we have an algebraic virtual class whose cohomology gives an exact sequence Then we can apply the construction of [1,25]. When n = 1, similar to Proposition 2.10, we have Proposition 3.2. Let β ∈ H 2 (Y, Z) be an irreducible curve class on a Fano 3-fold Y . Then Proof. The proof is the same as the proof of Proposition 2.11. Just note that as in (2.13), there is a distinguished triangle where the cohomology of RHom X (I X , I X ) 0 [1] is finite dimensional as F has compact support (although X is non-compact) and we may work with a compactification of X.  Genus 1. When any curve C in an irreducible class β ∈ H 2 (Y ) is a smooth rational curve, P 0 (Y, β) = ∅ by Lemma 2.7, so P 0,β (X) = 0 (by Proposition 3.3). In this case, to verify Conjecture 1.6, we are reduced to compute GW invariants and show n 1,β = 0. Proposition 3.5. Let Y = P 3 and X = K Y . Then Conjecture 1.6 is true for any irreducible curve class β ∈ H 2 (X, Z) ∼ = H 2 (Y, Z).
Proof. When Y = P 3 , n 1,β = 0 by [20, be the total space of direct sum of two line bundles L 1 , L 2 on S. Assuming that then X is a non-compact CY 4-fold. For a curve class we can consider the moduli space P n (X, β) of stable pairs on X, which is in general non-compact. In this section, we restrict to the case when the curve class β is irreducible such that L i · β < 0, in which case P n (X, β) is compact and smooth.
Lemma 3.6. Let S be a smooth projective surface and β ∈ H 2 (S, Z) be an irreducible curve class such that K S · β < 0. Then the moduli space P n (S, β) of stable pairs on S is smooth.
Proof. Similar to the proof of Lemma 3.1, for any stable pair I S = (s : whose cohomology gives an exact sequence We claim the map Ext 1 S (F, F ) → H 1 (F ) above is surjective, then Ext 1 S (I S , F ) = 0 follows from the exact sequence (so the smoothness of moduli follows). In fact, we only need to show the surjectivity of where C is the scheme theoretical support of F . However, the above map is simply the multiplication by the section s, which fits into an exact sequence Proposition 3.7. Let S be a smooth projective surface and L 1 , L 2 be two line bundles on S such that L 1 ⊗ L 2 ∼ = K S . Then for any irreducible curve class β ∈ H 2 (X, Z) ∼ = H 2 (S, Z) such that L i · β < 0 (i = 1, 2), we have an isomorphism P n (X, β) ∼ = P n (S, β).

And the virtual class satisfies
for certain choice of orientation in defining the LHS. Here I S = (O S×Pn(S,β) → F) ∈ D b S × P n (S, β) is the universal stable pair and π PS : S × P n (S, β) → P n (S, β) is the projection.
Proof. Under assumption L i · β < 0 and β is irreducible, as in the proof of [12,Prop. 3.1], one can show, for zero section i : S → X, the morphism , where the last isomorphism is deduced similarly as (2.14).
It follows that we have a distinguished triangle where T fits into the distinguished triangle [2]. (3.8) By Serre duality, degree shift and taking dual, (3.8) becomes Combining with (3.4), we obtain a distinguished triangle By taking cohomology of (3.9), we obtain exact sequences By taking cohomology of (3.7), we obtain By the first isomorphism above, we know the map (3.5) induces an isomorphism on tangent spaces. Moreover since P n (S, β) is smooth (Lemma 3.6) and (3.5) is bijective on closed points, so the map (3.5) is an isomorphism.
As in Proposition 3.3, we can show Ext 1 S (F, F ⊗ L 1 ) is a maximal isotropic subspace of Ext 2 X (I X , I X ) 0 with respect to the Serre duality pairing on Ext 2 X (I X , I X ) 0 . Since Ext 0 S (F, F ⊗ L 1 ) = Ext 2 S (F, F ⊗ L 1 ) = 0, Ext 1 S (F, F ⊗ L 1 ) is constant over P n (S, β), so it forms a maximal isotropic subbundle of the obstruction bundle of P n (X, β) whose fiber over I X ∈ P n (X, β) is Ext 2 X (I X , I X ) 0 . Then the virtual class has the desired property [10]. It is easy to check Conjecture 1.5, 1.6 for irreducible curve classes on Tot S (L 1 ⊕ L 2 ) in the following setting.
Proposition 3.8. Let S be a del Pezzo surface and L −1 1 , L −1 2 be two ample line bundles on S such that L 1 ⊗ L 2 ∼ = K S . Denote β ∈ H 2 (X, Z) ∼ = H 2 (S, Z) to be an irreducible curve class on X = Tot S (L 1 ⊕ L 2 ). Then Conjecture 1.5, 1.6 are true for β.
As for the genus 0 conjecture, for any stable pair (s : O S → F ) ∈ P 1 (S, β), F is stable and supported on some C ∼ = P 1 in S. Then F = O C and the morphism φ : P 1 (S, β) to the moduli space M 1,β (S) of 1-dimensional stable sheaves F 's on S with [F ] = β and χ(F ) = 1 is an isomorphism.
As for the moduli space M 0,0 (X, β) of stable maps, we have isomorphisms where the first isomorphism is by the negativity of L i (i = 1, 2) and the second one is defined by mapping f : Next, we compare obstruction theories. By Proposition 3.7, the 'half' obstruction space of P 1 (X, β) at (s : Since S is either P 2 or P 1 × P 1 and β is irreducible, all stable maps are embedding. The obstruction space of M 0,0 (X, β) at f : P 1 → S is H 1 (C, N C/X ) ∼ = H 1 (P 1 , f * T X) with C = f (P 1 ), which fits into the exact sequence A family version of these computations shows the virtual classes satisfy 3.3. Small degree curve classes on local surfaces. We learned from discussions with Kool and Monavari [21] that by using relative Hilbert schemes and techniques developed in Kool-Thomas [22], one can do explicit computations of pair invariants in small degrees for non-compact CY 4-folds We list the results as follows (where pair invariants are defined with respect to certain of choices of orientation).  [20, pp. 22, 24], we know our Conjecture 1.5, 1.6 hold in all above cases.

Local curves
Let C be a smooth projective curve of genus g(C) = g, and be the total space of a split rank three vector bundle on it. Assuming that then the variety (4.1) is a non-compact CY 4-fold. Below we set l i := deg L i and may assume that l 1 l 2 l 3 without loss of generality.
Let T = (C * ) 3 be the three dimensional complex torus which acts on the fibers of X. Its restriction to the subtorus preserves the CY 4-form on X and also the Serre duality pairing on P n (X, β). In this section, we aim to define equivariant virtual classes of P n (X, β) using a localization formula with respect to the T 0 -action [10,12], and investigate their relations with equivariant GW invariants.
Let • be the point Spec C with trivial T -action, C⊗t i be the one dimensional T -representation with weight 1, and λ i ∈ H * T (•) be its first Chern class. They are generators of equivariant cohomology rings: 4.1. Localization for GW invariants. Let j : C ֒→ X be the zero section of the projection (4.1). We have where [C] is the fundamental class of j(C). For m ∈ Z >0 , we consider the diagram where C is the universal curve and f is the universal stable map.
The T -equivariant GW invariant of X is defined by where N is the T -equivariant normal bundle of j(C) ⊂ X: If g(C) > 0, the vanishing of genus zero GW invariants For example in the d = 1 case, M 0 (P 1 , 1) is one point and In the d = 2 case, a straightforward localization calculation with respect to the (C * ) 2 -action on P 1 gives Here we write l = l for l 0 and l = −l − 1 for l < 0.

4.2.
Localization for stable pairs. Similarly, for m ∈ Z 0 , we want to define (equivariant) stable pair invariant ) is the universal stable pair and π P : X × P n (X, m[C]) → P n (X, m[C]) is the projection. Of course, the above equality is not a definition as the virtual class of the fixed locus as well as the square root needs justification. We will make this precise in specific cases where we compare with GW invariants of X.
Let us first describe stable pairs (s : O X → F ) ∈ P n (X, m[C]) T which are fixed by the fulltorus T : decompose F into T -weight space where the T -weight of F i1,i2,i3 is (i 1 , i 2 , i 3 ). We denote an index set We also have the decomposition The T -equivariance of s induces morphisms in Coh(C) which are surjective in dimension one. It follows that each F −i1,−i2,−i3 is either zero or can be written as for some effective divisor Z i1,i2,i3 ⊂ C. Moreover, the p * O X -module structure on F gives a morphism which commutes with s i1,i2,i3 and s i1+1,i2,i3 . Similar morphisms replacing L 1 by L 2 , L 3 exist and have similar commuting property. Hence, for (i 1 , i 2 , i 3 ) ∈ ∆, we have as divisors in C. So the set ∆ (4.8) is a three dimensional Young diagram, which is finite by the coherence of F .
In general, it is difficult to explicitly determine T 0 -fixed stable pairs. In fact, a T 0 -fixed stable pair is not necessarily T -fixed. Nevertheless, for a T 0 -fixed stable pair (s : O X → F ), O CF := Im s and the corresponding ideal sheaf I CF are actually T -fixed. Proof. Since I CF = H 0 (I), it is T 0 -fixed. For t ∈ T , we have the diagram The above diagram induces the morphism u ∈ Hom(I CF , t * O CF ). It is enough to show u = 0. For a general point c ∈ C, let X c = p −1 (c) = C 3 be the fiber of p at c. Then I CF | Xc is an ideal sheaf of T 0 -fixed zero dimensional subscheme of C 3 . Then it is also T -fixed by [2,Lemma 4.1]. This implies that the morphism restricted to X c is a zero map. Then Im u ⊂ t * O CF is zero on the general fiber of p, hence Im u = 0 by the purity of C F .
Another convenient way to determine T 0 -fixed stable pairs is in the case when P n (X, m[C]) T is smooth and Hom X (I, F ) T0 = Hom X (I, F ) T for any I = (O X → F ) ∈ P n (X, m[C]) T (see e.g. [31,Sect. 3.3] on toric 3-folds). Then one has P n (X, m[C]) T = P n (X, m[C]) T0 . In the examples below, we will explicitly determine the T 0 -fixed locus mainly using Lemma 4.1.

P 1,m[C]
(X) and genus zero conjecture. Let C = P 1 be a smooth rational curve and X = O P 1 (l 1 , l 2 , l 3 ) with l 1 + l 2 + l 3 = −2. This serves as the local model for a neighbourhood of a rational curve in a CY 4-fold.
For some special choice of (l 1 , l 2 , l 3 ), we can determine P 1,m[C] (X) for all m.  By (4.7), we have Then the calculation is straightforward.
By comparing the above computations with the corresponding GW invariants, we obtain the following equivariant analogue of Conjecture 1.5 (note from the above proof, we know P 0,m[C] (X) = 0 (m 1) since P 0 (X, m[C]) = ∅).
Comparing with Proposition 4.2, we are done.
Degree one class. When m = 1, it is easy to show the canonical section gives the only T 0 -fixed stable pair in P 1 (X, [P 1 ]). Similar to Proposition 4.2, we have which coincides with corresponding GW invariant (4.5). Here we have chosen the plus sign in defining P 1,[P 1 ] (X).
Degree two class. When m = 2, let (s : O X → F ) ∈ P 1 (X, 2[P 1 ]) be a T 0 -fixed stable pair. Then F is thickened into one of the L i -direction, i.e.
i , where s 0 and s i are injective, and surjective in dimension one, φ defines the p * O X -module structure (which is also injective, and surjective in dimension one by the diagram). Denote , the above diagram is equivalent to a commutative diagram where s 0 , s i and φ are injective. These impose conditions where the last equality is because χ(F ) = 1. It is not hard to show the following Lemma 4.4. We have the following isomorphism where Pic (a,b) (P 1 ) denotes the moduli space of triples (L, L ′ , ι), (L, L ′ ) ∈ Pic a (P 1 ) × Pic b (P 1 ), ι : L ֒→ L ′ , and ι is an inclusion of sheaves.
To determine the virtual class [P 1 (X, 2[P 1 ]) T0 ] vir and the square root in (4.7), we take a T 0fixed stable pair I = (s : O X → F ) and view it as an element in the T 0 -equivariant K-theory of X. Then where both sides of the equality can be written using grading into T 0 -weight space.
Similar to [12,Sect. 4.4], we set and j is the inclusion of zero section of X. The T 0 -fixed and movable part satisfies where χ(F, F ) 1/2,fix and χ(F, F ) 1/2,mov was computed in [12,Sect. 4.4]. In particular, So dim C (−χ(I, I) 1/2,fix 0 ) = d i is the dimension of P 1 (X, 2[P 1 ]) T0 . Thus the virtual class of the associated T 0 -fixed locus P 1 (X, 2[P 1 ]) T0 may be defined to be its usual fundamental class.
In the above definition, the second integration can be easily shown to be 1 and the first one has been explicitly determined before [12, Corollary 4.9]. So we obtain Proposition 4.5. Let X = O P 1 (l 1 , l 2 , l 3 ) with l 1 + l 2 + l 3 = −2 and l 1 l 2 l 3 . Then We pose the following equivariant version of Conjecture 1.5 (note in this case P 0,[P 1 ] (X) = 0 as χ(O P 1 ) > 0). It is consistent with our previous conjecture on one dimensional stable sheaves [12,Conj. 4.10].    Furthermore, if p * Ext 1 X (I, I) and p * Ext 2 X (I, I) are locally free, then H 1 (X, Ext 1 X (I, I)) ∼ = H 0 (X, Ext 2 X (I, I)) ∨ .
And H 0 (X, Ext 2 X (I, I)), H 1 (X, Ext 1 X (I, I)) are maximal isotropic subspaces of Ext 2 X (I, I) 0 with respect to Serre duality pairing.
Proof. We have the local to global spectral sequence for the projection p : X → C (4.1).
We describe the torus fixed locus P 0 (X, m[C]) T0 as follows. Lemma 4.9. Let C be an elliptic curve and L i ∈ Pic 0 (C). Then (O X → F ) ∈ P 0 (X, m[C]) is T 0 -fixed if and only if it is of the form for some three dimensional Young diagram ∆ ⊂ Z 3 0 . In particular, we have in this case.
Proof. The stable pair (4.13) is obviously T -fixed, hence T 0 -fixed. Conversely, for T 0 -fixed stable pair (s : O X → F ) with χ(F ) = 0, we denote O Z = Im s and then I Z is T -fixed by Lemma 4.1.
Since c 1 (L i ) = 0 and F/O Z is zero dimensional, then We determine stable pair invariants for X = Tot C (L 1 ⊕ L 2 ⊕ L 3 ) when line bundles L i ∈ Pic 0 (C) over the elliptic curve C are general. We denote I to be the ideal sheaf of W . Let U ⊂ C be an open subset on which L i are trivial. Then p −1 (U ) ∼ = U × C 3 and I| p −1 (U) is isomorphic to π * I Z for the T -fixed zero-dimensional subscheme Z ⊂ C 3 corresonding to ∆. Therefore we have an isomorphism of T -equivariant sheaves on U p * Ext k X (I, I)| U ∼ = Ext k C 3 (I Z , I Z ) ⊗ C O U . (4.14) Let be the decomposition into T -weight spaces. By (4.14), we have The relation (4.2) and Lemma 4.8 imply that Ext 2 X (I, I) mov For a general choice of (L 1 , L 2 ), we have H 0 (L a 1 ⊗ L b 2 ) = H 1 (L a 1 ⊗ L b 2 ) = 0 for any (a, b) = (0, 0), so the movable part also vanishes. Thus P 0,m[C] (X) = ♯ P 0 (X, m[C]) T0 , which is the number of three dimensional partitions of m.

Appendices
5.1. Stable pairs and one dimensional sheaves for irreducible curve classes. When β ∈ H 2 (X, Z) be an irreducible curve class on a smooth projective CY 4-fold X, we have a morphism φ n : P n (X, β) → M n,β (X) to the moduli scheme of 1-dimensional stable sheaves with Chern character (0, 0, 0, β, n) (e.g. [32, pp. 270]), whose fiber over [F ] is P (H 0 (X, F )). Note that M n,β (X) is in general a stack instead of scheme when β is arbitrary. The virtual dimension of M n,β (X) satisfies vir.dim R (M n,β (X)) = 2, by [4,12]. One could use the virtual class to define invariants.
For integral classes γ i ∈ H mi (X, Z), 1 i l, let τ : H m (X) → H m−2 (M n,β (X)), τ (γ) = π P * (π * X γ ∪ ch 3 (F)), where π X , π M are projections from X × M n,β (X) to corresponding factors, F → X × M n,β (X) is the universal sheaf, and ch 3 (F) is the Poincaré dual to the fundamental cycle of F. We propose the following conjecture. In all compact examples studied in this paper, one can check Conjecture 5.1 holds in these cases. In particular, when X = Y × E is the product of a CY 3-fold Y with an elliptic curve E and the irreducible class β ∈ H 2 (Y, Z) ⊆ H 2 (X, Z) sits inside Y , then Conjecture 5.1 reduces to a special case of the multiple cover formula ([17, Conjecture 6.20], [38,Conjecture 6.3]): N n,β = k 1,k|(n,β) 1 k 2 N 1,β/k , for any β in a CY 3-fold Y , where N n,β ∈ Q is the generalized DT invariant [17] which counts one dimensional semistable sheaves E on Y with [E] = β, χ(E) = n. The above formula is proved when β is primitive in [37,Lemma 2.12] (see also [12,Appendix A.]).
It is an interesting question to define 'generalized DT 4 type invariant' counting semistable sheaves on CY 4-folds and search for similar multiple cover formula on CY 4-folds.

An orientability result for moduli spaces of stable pairs on CY 4-folds.
Let X be a smooth projective CY 4-fold and c ∈ H even (X). For a moduli stack M c of coherent sheaves on X with Chern character c, we define  [7], where the proof uses involved tools like semi-topological K-theory. To be self-contained, we include here a simpler proof of an orientability result for CY 4-folds with technical assumptions Hol(X) = SU (4) and H odd (X, Z) = 0. Proof. By the work of Joyce-Song [17,Thm. 5.3], the moduli stack M c is 1-isomorphic to a finite type moduli stack of holomorphic vector bundles on X via Seidel-Thomas twists, under which the universal family can be identified (so is the determinant line bundle and Serre duality pairing). Thus we may assume M c to be a moduli stack of (rank n) holomorphic bundles without loss of generality.
Fix a base point x 0 ∈ X, a framing φ of a vector bundle E is an isomorphism There is a natural GL(n, C)-action on φ changing the framing. Let M framed c denote the moduli stack of framed holomorphic bundles with Chern character c, on which GL(n, C) acts by changing framings. Note that M framed Then orientability for moduli spaces of stable pairs follows from the orientability of moduli stacks of one dimensional sheaves. Theorem 5.3. Let X be a CY 4-fold with Hol(X) = SU (4) and H odd (X, Z) = 0. Then for any β ∈ H 2 (X, Z) and n ∈ Z, the quadratic line bundle (L, Q) over P n (X, β) is orientable.