Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperk\"ahler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties.

0. Introduction 0.1. Gopakumar-Vafa invariants. Gromov-Witten invariants of a smooth projective variety X are defined by integration over the virtual class [BF, LT] of the moduli space M g,n (X, β) of genus g degree β ∈ H 2 (X, Z) stable maps: τ k1 (γ 1 ) · · · τ kn (γ n ) GW g,β = [M g,n(X,β) Here ev i : M g,n (X, β) → X is the evaluation map at the i-th marking, ψ i is the i-th cotangent line class, and γ i ∈ H * (X, Q) are cohomology classes. Since M g,n (X, β) is a Deligne-Mumford stack, Gromov-Witten invariants are in general rational numbers, even if all γ i are integral. Moreover the enumerative meaning of Gromov-Witten invariants is often not clear.
For Calabi-Yau 3-folds, Gopakumar and Vafa [GV] found explicit linear transformations which transform the Gromov-Witten invariants to a set of invariants (called Gopakumar-Vafa invariants) which they conjectured to be integers. In an ideal geometry, where all curves are isolated, disjoint and smooth, Gopakumar-Vafa invariants should be the actual count of curves of given genus and degree. The integrality of Gopakumar-Vafa invariants was proven recently in [IP]. A similar transformation of Gromov-Witten invariants into (conjectural) Z-valued invariants has been proposed for Calabi-Yau 4-folds by Klemm and Pandharipande [KP], and for Calabi-Yau 5-folds by Pandharipande and Zinger [PZ]. Universal transformations are expected in every dimension [KP].
Let X be a holomorphic symplectic 4-fold, by which we mean a smooth complex projective 4-fold which is equipped with a non-degenerate holomorphic 2-form σ ∈ H 0 (X, Ω 2 X ). Since the obstruction sheaf has a trivial quotient, the ordinary Gromov-Witten invariants of X vanish for all non-zero curve classes. As a result, also all Klemm-Pandharipande invariants of X vanish. Instead a reduced Gromov-Witten theory is obtained by Kiem-Li's cosection localization [KiL]. It is defined as in (0.1) but by integration over the reduced virtual fundamental class: 1 (0.2) [M g,n (X, β)] vir ∈ A 2−g+n (M g,n (X, β)).
In fact, through a twistor space construction, this definition follows immediately from a similar definition on Calabi-Yau 5-folds given in [PZ] (see §1.3 for more explanations).
In genus 1, the situation is more complicated and does not follow from 5-fold geometry. Since the virtual dimension of (0.2) is 1 + n, we require one marked point and an insertion γ ∈ H 4 (X, Z). Because curves in imprimitive curve classes are very difficult to control in an ideal geometry (see Section 1.4) we will restrict us to a primitive curve class (i.e. where β is not a multiple of a class in H 2 (X, Z)).

(i)
When β is a primitive curve class, When β is a primitive curve class, As in Conjecture 0.4, we verify these equalities in the ideal geometry (see §2.3, §2.4 and also §3 for details). An exception is the last equality involving genus 2 invariants, which we obtain indirectly through stable pair theory [COT22] (see Remark 2.3).
Besides computations in the ideal geometry mentioned above, we study several examples and prove our conjectures in those cases. 0.3. Verification of conjectures I: K3 × K3. Let X = S × T be the product of two K3 surfaces. When the curve class β ∈ H 2 (S × T, Z) is of non-trivial degree over both S and T , then the obstruction sheaf of the moduli space of stable maps has two linearly independent cosections, which implies that the (reduced) Gromov-Witten invariants of X in this class vanish. Therefore we always restrict ourselves to curve classes of form (0.5) β ∈ H 2 (S) ⊆ H 2 (X).
In particular, Conjecture 0.4 holds for X = S × T .
On the Donaldson-Thomas side, a main result of this paper is the explicit computation of all DT 4 invariants of X = S × T for the classes (0.5), see Theorem 5.8 for the formulae. We obtain a perfect match with our prediction: Theorem 0.7 (Corollary 5.9). Conjecture 0.5 holds for X = S ×T and all effective curve classes β ∈ H 2 (S, Z) ⊆ H 2 (X, Z).
Here, since the moduli space M β is connected, there are precisely two choices of orientation. We pick the one specified in Eqn. (5.10) (invariants for the other differ only by an overall sign).
Contrary to the case of Gromov-Witten invariants, the computation of DT 4 invariants on S×T is highly non-trivial. In Theorem 5.7, we first identify the virtual class explicitly. This expresses the DT 4 invariants as tautological integrals on a (smooth) moduli space of one dimensional stable sheaves on the K3 surface S. By Markman's framework of monodromy operators [M08], we then relate such integrals to tautological integrals on the Hilbert schemes of points on S (see §4.3 and §4.4 for details). Finally, we determine these integrals explicitly in §4.1 and §4.2 by a combination of the universality result of Ellingsrud-Göttsche-Lehn [EGL], constraints from Looijenga-Lunts-Verbitsky Lie algebra [LL,Ver13] and known computations of Euler characteristics.
In particular, we found a remarkable closed formula for Fujiki constants of Chern classes of Hilbert schemes S [n] of points on S, which takes the following beautiful form (see also Proposition 4.3 for the formula on generalized Kummer varieties): Theorem 0.8. (Theorem 4.2) Let S be a K3 surface. For any k 0, n k C(c 2n−2k (T S [n] )) q n = (2k)! k!2 k q d dq G 2 (q) k n 1 1 (1 − q n ) 24 .
The right hand side, up to the combinatorical prefactor (2k)!/(k!2 k ), is precisely the generating series of counts of genus k curves on a K3 surface passing through k generic points as considered by Bryan and Leung [BL]. This suggests a relationship to the work of Göttsche on curve counting on surfaces [G98], which will be taken up in a follow-up work. 0.4. Verification of conjectures II: T * P 2 . Let T * P 2 be the total space of the cotangent bundle on P 2 , which is holomorphic symplectic. Let H ∈ H 2 (T * P 2 ) be the pullback of the hyperplane class and use the identification H 2 (T * P 2 , Z) ≡ Z given by taking the degree against H. By Graber-Pandharipande's virtual localization formula [GP], we can compute all genus Gromov-Witten invariants (Proposition 6.1) and determine the Gopakumar-Vafa invariants. if d = 2, 0 otherwise. n 1,1 (H 2 ) = 0, n 2,1 = 0.
In particular, Conjecture 0.4 holds for T * P 2 .
On the sheaf side, we can compute DT 4 invariants for small degree curve classes.
0.5. Verification of conjectures III: K3 [2] . Consider the Hilbert scheme S [2] of two points on a K3 surface S. By a result of Beauville [Bea], S [2] is irreducible hyperkähler, i.e. it is simply connected and the space of its holomorphic 2-forms is spanned by a (unique) symplectic form.
Because the genus 0 Gromov-Witten theory of S [2] is completely known by [O18, O21a, O21c] (see Theorem 7.3 for the primitive case), all genus 0 Gopakumar-Vafa invariants are easily computed. For simplicity, we check the integrality conjecture in the following basic case (ref. §7.7): Theorem 0.11. Conjecture 0.4 holds for all effective curve classes on S [2] in genus 0 and with one marked point.
Higher genus Gromov-Witten invariants are more difficult to compute even for primitive curve classes. Nevertheless there are several conjectures on the structure of these invariants, including (i) a quasi-Jacobi form property, and (ii) a holomorphic anomaly equation (see [O22b,Conj. A & C], see also [O21b] for a progress report). Assuming these conjectures and using several explicit evaluations of Gromov-Witten invariants, we obtain a complete computation of all genus 1 and 2 Gromov-Witten invariants of S [2] in primitive classes, see Theorem 7.4. From this, all Gopakumar-Vafa invariants are computed in Theorems 7.6 and 7.10.
With the help of a computer program, we obtain the following check of integrality: Theorem 0.12. (Corollaries 7.8 and 7.11) Assume Conjectures A and C of [O22b]. , Z), consider the very general deformation (X, β ) of a pair (S [2] , β), where β stays of Hodge type on all fibers. By the deformation theory of hyperkähler varieties, the variety X then has Picard rank 1 and the algebraic classes in H 2 (X, Z) are generated by β . In particular β is irreducible. In this case, it is natural to expect that curves in (X, β ) forms an ideal geometry in the sense of §1.4, §1.5. In other words, after a generic deformation, our Gopakumar-Vafa invariants should give enumerative information about curves in these hyperkähler varieties of K3 [2] -type. In genus 2, this yields the following conjectural formula for the number of isolated (rigid) genus 2 curves on a very general hyperkähler variety of K3 [2] -type of minimal degree. This may be viewed as a 4-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on K3 surfaces: Theorem 0.13. (Theorem 7.10) Assume Conjectures A and C of [O22b]. For any hyperkähler variety X of K3 [2] -type and primitive curve class β ∈ H 2 (X, Z), the genus 2 Gopakumar-Vafa invariant n 2,β is the coefficient determined by β (see Definition 7.1) of the quasi-Jacobi form In genus 1, it is convenient to encode the invariants in the genus 1 Gopakumar-Vafa class where n 1,β (γ) is given in Definition 0.2. In an ideal geometry, n 1,β is the class of the surface swept out by elliptic curves in class β. Theorem 7.6 then yields a conjectural formula for this class. We list the first values of the genus 1 and 2 Gopakumar-Vafa invariants of hyperkähler varieties of K3 [2] -type in Table 1 and Table 2 below. Since the deformation class of a pair (X, β) where β is a primitive curve class, only depends on the square (β, β) (see [O21a]), the Gopakumar-Vafa invariants only depend on (β, β).
It is interesting to compare the enumerative significance of the listed invariants with the known geometry of curves on very general hyperkähler 4-folds of K3 [2] -type with curve class β. In the case (β, β) = −5/2, any curve in class β is a line in a Lagrangian P 2 ⊂ X, see [HT]. In particular, there are no higher genus curves, and indeed we observe the vanishing of the g = 1, 2 Gopakumar-Vafa invariants in this case. Similarly, the case (β, β) = −1/2 corresponds to the exeptional curve class on K3 [2] (the class of the exceptional curve of the Hilbert-Chow morphism K3 [2] → Sym 2 (K3)), and again there are no higher genus curves. The case (β, β) = −2 is similar, see [HT]. The first time we see elliptic curves is in case (β, β) = 0, which corresponds to the fiber class of a Lagrangian fibration X → P 2 . Elliptic curves appear here in fibers over the discriminant. The case (β, β) = 3/2 corresponds to a very general Fano variety of lines on a cubic 4-fold, with β the minimal curve class (of degree 3 against the Plücker polarization). Since there are no cubic genus 2 curves in a projective space (see also Example 1.10), there are no genus 2 curves in this class; again, this matches the vanishing observed in the table. The case (β, β) = 2 are the double covers of EPW sextics [O06]. The first time we should see isolated smooth genus 2 curves is the case (β, β) = 11/2, which are precisely the Debarre-Voisin 4-folds [DV]. Here, the explicit geometry of curves has not been studied yet. It would be very interesting to construct the expected 3465 isolated smooth genus 2 curves explicitly. In fact, to the best of the authors' knowledge, there exists so far no known example of a smooth isolated (rigid) genus 2 curves on a hyperkähler 4-fold, and this may be perhaps the simplest case.
for a hyperkähler 4-fold of K3 [2] -type with primitive curve class β (see §7.2 for the definition of the dual divisor h β ). In an ideal geometry (ref. §1.5), n 1,β is the class of the surface swept out by the elliptic curves in class β. 0.7. Appendix. In the appendix §A, we discuss several cases where we can extend the above GW/GV/DT 4 correspondence to imprimitive curve classes. Notation and convention. All varieties and schemes are defined over C. For a morphism π : X → Y of schemes and objects F, G ∈ D b (Coh(X )) we will use A class β ∈ H 2 (X, Z) is called effective if there exists a non-empty curve C ⊂ X with class [C] = β. An effective class β is called irreducible if it is not the sum of two effective classes, and it is called primitive if it is not a positive integer multiple of an effective class.
A holomorphic symplectic variety is a smooth projective variety together with a non-degenerate holomorphic two form σ ∈ H 0 (X, Ω 2 X ). A holomorphic symplectic variety is irreducible hyperkähler if X is simply connected and H 0 (X, Ω 2 X ) is generated by a symplectic form. A K3 surface is an irreducible hyperkähler variety of dimension 2.

Gopakumar-Vafa invariants
Let X be a holomorphic symplectic 4-fold with symplectic form σ ∈ H 0 (X, Ω 2 X ). In this section we first recall the definition of (reduced) Gromov-Witten invariants, and then give our definition of Gopakumar-Vafa invariants. In Section 1.4, we justify the definition by working in an ideal geometry of curves.
1.1. Gromov-Witten invariants. Let M g,n (X, β) be the moduli space of n-pointed genus g stable maps to X representing the non-zero curve class β ∈ H 2 (X, Z). The moduli space M g,n (X, β) admits a perfect obstruction theory [BF, LT]. By the construction of [MP13,§2.2] the symplectic form σ induces an everywhere surjective cosection of the obstruction sheaf. By Kiem-Li's theory of cosection localization [KiL] it follows that the standard virtual class as defined in [BF, LT] vanishes and instead there exists a reduced virtual fundamental class: [M g,n (X, β)] vir ∈ A 2−g+n (M g,n (X, β)).
In this paper we will always work with the reduced virtual fundamental class which we will hence simply denote by [−] vir .
Given cohomology classes γ i ∈ H * (X) and integers k i 0 the (reduced) Gromov-Witten invariants of X in class β are defined by τ k1 (γ 1 ) · · · τ kn (γ n ) GW g,β = [M g,n(X,β) where ev i : M g,n (X, β) → X is the evaluation map at the i-th marking and ψ i is the i-th cotangent line class. By the properties of the reduced virtual class, the integral (1.1) is invariant under deformations of the pair (X, β) with preserve the Hodge type of the class β. We call the invariant (1.1) a primary Gromov-Witten invariant if all the k i are zero.
1.2. Relations. We record several basic relations among genus 0 Gromov-Witten invariants which will be used later on in the text. For the first reading, this section map be skipped.
Lemma 1.1. Let D be a divisor on X such that d := D · β = 0. Then GW 0,β . Proof. By the divisor equation (e.g. [CK,pp. 305]) GW 0,β . On the other hand, by rewriting ψ 1 in terms of boundary divisors and using the splitting formula for reduced virtual classes as in [MPT,§7.3] one gets Arguing as in Lemma 1.1 we can express both sides in terms of primary Gromov-Witten invariants, which yields the result.
Proof. Let D ∈ H 2 (X) such that d := D · β = 0. Consider the following invariants: Applying topological recursions to the invariants on the left then yields the following relations on the right: Putting all together (using the assistance of a computer) one finds: Lemma 1.4. Assume that all fibers of the universal curve p : C → M 0,0 (X, β) are isomorphic to P 1 . Let π : M 0,1 (X, β) → M 0,0 (X, β) be the forgetful morphism. Then c 1 (ω π ) = ψ 1 .
Recall that we have C ∼ = M 0,1 (X, β). Moreover, since C → M 0,0 (X, β) parametrizes only smooth curves, we have Under this isomorphism, the section s is identified with the diagonal morphism. We have the fiber diagram Hence sinceπ • s = id, we have The second part follows since
The case of genus 1 does not follow from the 5-fold geometry, since the virtual class of the moduli spaces differ by a factor of (−1) g λ g , see [MP13,O21a]. Instead we propose a definition of genus 1 Gopakumar-Vafa invariants based on computations in an ideal geometry of curves in class β. Because curves in imprimitive curve classes are very difficult to control, we restrict hereby to the primitive case (i.e. to those β which are not a multiple in H 2 (X, Z)). Consider the Chern classes of the tangent bundle of X: Definition 1.6. Assume that β ∈ H 2 (X, Z) is primitive. For any γ ∈ H 4 (X, Z), we define the genus 1 Gopakumar-Vafa invariant n 1,β (γ) ∈ Q by τ 0 (γ) GW 1,β = n 1,β (γ) − 1 24 τ 0 (γ)τ 0 (c 2 (T X )) GW 0,β . Next we come to the genus 2 Gopakumar-Vafa invariants. Since the virtual dimension of the moduli space M 2,0 (X, β) is zero, GV invariants are defined without cohomological constraints. In other words, we expect that n 2,β should be given by the enumerative count of genus 2 curves in class β. For the definition we require the following invariant introduced in [NO]: is the class of the diagonal, and • c(T X ) = 1 + c 2 (T X ) + c 4 (T X ) is the total Chern class of T X . The invariant N nodal,β is the expected number of rational nodal curves in class β [NO,Prop. 1.2] Definition 1.7. Assume that β ∈ H 2 (X, Z) is primitive. We define the genus 2 Gopakumar-Vafa invariant n 2,β ∈ Q by ∅ GW 2,β = n 2,β − 1 24 n 1,β (c 2 (X)) + 1 2 · 24 2 τ 0 (c 2 (X))τ 0 (c 2 (X)) GW 0,β + 1 24 N nodal,β .
Remark 1.8. For primitive β ∈ H 2 (X, Z), we obtain the following: It would be interesting to obtain a conceptual understanding for the form of these formulae.
As in the cases of Calabi-Yau 4-folds and 5-folds [KP, PZ], our first main conjecture concerns the integrality of the Gopakumar-Vafa invariants on holomorphic symplectic 4-folds.
1.4. Ideal geometry. We will justify our definition of Gopakumar-Vafa invariants by working in an 'ideal' geometry where we assume curves on X deform in families of expected dimensions and have expected genericity properties. This discussion is inspired by the 'ideal' geometry of curves on Calabi-Yau 4-folds by [KP] and on Calabi-Yau 5-folds by [PZ]. Concretely, since the virtual dimension of M g,0 (X, β) is 2 − g, we expect that: Any genus g curve moves in a smooth compact (2 − g)-dimensional family. In particular, there are no curves of genus g 3.
We discuss now the expected behaviour of the curves in these families. We start with genus zero. Let p : C 0 β → S 0 β be a family of rational curves in class β over a smooth 2-dimensional surface S 0 β , fiberwise embedded in X. Then we can have the following behaviour: (i) All the curves parametrized by S 0 β can be reducible. Reason: Let β = β 1 + β 2 and let C 0 βi → S 0 βi be a 2-dimensional family of rational curves in class β i . Let S 0 β1,β2 be the preimage of the diagonal under the evaluation maps j 1 × j 2 : C 0 β1 × C 0 β2 → X × X. Then S 0 β1,β2 is of expected dimension 3 + 3 − 4 = 2, so by gluing the curves we can obtain a 2-dimensional family of reducible rational curves in class β.
(ii) Given a generic curve C 0 s := p −1 (s) ⊂ X in the family, there exists another curve C 0 s ⊂ X in the family which meets it.
Reason: This follows by the same reasoning as in (i). (iii) For finitely many s ∈ S, we expect the curve C 0 s ⊂ X to be nodal 5 . Reason: The moduli space M 0,2 (X, β) is of expected dimension 4, and hence the preimage of the diagonal under ev 1 × ev 2 is of expected dimension 0. 6 (iv) Even if all fibers of C 0 β → S β are smooth P 1 's, the natural morphism j : C 0 β → X is not necessarily an immersion.
(The differential dj : T C 0 β → j * (T X ) is expected to have a kernel in codimension 2.) Similarly given a family p : C 1 β → S 1 β of elliptic curves in class β over a smooth 1-dimensional curve S 1 β , fiberwise embedded in X, all the curves parametrized by S 1 β can be reducible. The argument is similar to (i) above, by considering the preimage of the diagonal under the evaluation maps ) is a family of rational curves in class β 1 (resp. elliptic curves in class β 2 ).
The genus 2 curves we expect to be smooth and finite. By dimension reasons they should be disjoint from elliptic curves, but can have finite intersection points with the family of rational curves. In the moduli space M 2,0 (X, β) we will hence also see genus 2 curves with rational tails.
In summary, the geometry of curves is more complicated then for both CY 4-folds and CY 5-folds. Especially for imprimitive curve classes β, it becomes increasingly difficult to control.
1.5. Ideal geometry: Primitive case. We make the following additional assumptions: • X is irreducible hyperkähler, • the effective curve class β ∈ H 2 (X, Z) is primitive.
By the global Torelli for (irreducible) hyperkähler varieties [Ver13,Huy] (in fact, the local surjectivity of the period map is sufficient) the pair (X, β) is deformation equivalent (through a deformation with keeps β of Hodge type) to a pair (X , β ) where β ∈ H 2 (X, Z) is irreducible. Hence we may without loss of generality make the following stronger assumption: 7 • the effective curve class β ∈ H 2 (X, Z) is irreducible.
Under these assumptions our ideal geometry of curves simplifies to the following form: (1) The rational curves in X of class β move in a proper 2-dimensional smooth family of embedded irreducible rational curves. Except for a finite number of rational nodal curves, the rational curves are smooth, with normal bundle O P 1 ⊕ O P 1 ⊕ O P 1 (−2). (2) The arithmetic genus 1 curves in X of class β move in a proper 1-dimensional smooth family of embedded irreducible genus 1 curves. Except for a finite number of rational nodal curves, the genus one curves are smooth elliptic curves. For the convenience of computations, we also assume the normal bundle of elliptic curves is L ⊕ L −1 ⊕ O, where L is a generic degree zero line bundle. (3) All genus two curves are smooth and rigid.
(4) There are no curves of genus g 3.
Example 1.10. Let Y ⊂ P 5 be a very general smooth cubic 4-fold and let F (Y ) ⊂ Gr(2, 6) be the Fano variety of lines on Y . By a result of Beauville and Donagi [BD], F (Y ) is an irreducible hyperkähler 4-fold, and together with its Plücker polarization it is the generic member of a locally complete family of polarized hyperkähler varieties deformation equivalent to the second punctual Hilbert scheme of a K3 surface. The algebraic classes in H 2 (F (Y ), Z) are of rank 1. Let β be the generator which pairs positively with the polarization (it is of degree 3 with respect to the Plücker polarization). The geometry of curves in class β has been studied in [OSY, NO, GK]. The Chow variety of curves in class β is given by where S ⊂ F (Y ) is the smooth irreducible surface of lines of second type, and Σ is a smooth curve parametrizing genus 1 curves. There are precisely 3780 rational nodal curves corresponding to the intersection points S ∩ Σ, and all other rational curves are isomorphic to P 1 . Moreover, there are no curves of genus 2 in class β. We see that the curves in F (Y ) of class β satisfy the requirements of the ideal geometry.
1.6. Justification: GV in genus 1. For a given class γ ∈ H 4 (X, Z) let Γ ⊂ X be a generic topological cycle whose class is Poincaré dual to γ. In an ideal geometry, the Gopakumar-Vafa invariant n 1,β (γ) should be the (enumerative) number n(Γ) of arithmetic genus 1 curves in X which are incident to Γ. To derive an expression for it using Gromov-Witten invariants, we start with the genus 1 Gromov-Witten invariant: where β ∈ H 2 (X, Z) is primitive. Assuming the ideal geometry of Section 1.5 we will analyze the contributions from genus 0 and genus 1 curves to it (there are no contributions from genus 2 curves since they never meet the cycle Γ). We show that the contribution from genus 1 curves is precisely n(Γ). This will yield the expression for n 1,β (γ). 7 The statement is false if we do not assume that X is irreducible hyperkähler, for example on X the product of two K3 surfaces, a class β = (β 1 , β 2 ) with both β i non-zero effective, any deformation that keeps β Hodge, will keep both β i Hodge. In particular β stays reducible under deformations.
1.6.1. Contribution from genus one curves. Let p : C 1 β → S 1 β be a 1-dimensional family of elliptic curves of class β as in Section 1.5, and let j : C 1 β → X be the evaluation map. Since Γ (which represents γ ∈ H 4 (X, Z)) is chosen generic, it intersects C 1 β in precisely (C 1 β · γ) many points. Following Section 1.5, we assume that the incident curves are smooth elliptic curves E with normal bundle N E/X = L ⊕ L −1 ⊕ O. We find the contribution of this family to the invariant (1.5) is where ω ∈ H 2 (E, Z) is the class of a point and the trivial factor in the obstruction sheaf does not appear because we used the reduced virtual fundamental class.
1.6.2. Contribution from genus zero curves. Let p : C 0 β → S 0 β be a 2-dimensional family of embedded rational curves of class β in X parametrized by a smooth surface S 0 β . The generic fiber of p is isomorphic to P 1 but over finitely many points we can have a rational nodal curve. The insertion Γ intersects the divisor C 0 β in a curve that we can assume maps to a curve in S 0 β . In particular, it avoids the singular fibers. For simplicity we may hence assume that there are no nodal fibers, and that this is the only family of rational curves in class β. We will compute the contribution of this family to the genus 1 GW invariant (1.5).
Under these assumptions, for any genus 1 degree β stable map f : C → X, the source curve splits canonically as C ∼ = E ∪ P 1 , where E is an elliptic curve glued to P 1 at one point p. The map f is of degree 0 on E, and of degree β on P 1 . Hence By comparing the obstruction theories on the level of virtual classes, we get [M 1,0 and ψ 1 is the usual psi class on the moduli space M 0,1 (X, β). In the last line we have used that the dimension of M 0,k (X, β) is equal to the expected dimension, so Finally, as we will do often, we have suppressed pullback maps along the projection to the factors. Consider the forgetful morphism π : M 1,1 (X, β) → M 1,0 (X, β) which at the same time is the universal curve over the moduli space. In particular, we have a decomposition where E → M 1,1 and P → M 0,1 (X, β) are the universal curves. Since π is flat of relative dimension 1, we have where a : E → M 1,1 (X, β) and b : P → M 1,1 (X, β) are the natural inclusions. We find where for the second summand the γ is pulled back along the evaluation map ev 1 : M 0,1 (X, β) → X (since the map is constant on the elliptic curve).
1.6.3. Conclusion. By the discussion above we have obtained that in the ideal geometry we have Since n(Γ) is the Gopakuma-Vafa invariant n 1,β (γ) in the ideal geometry, this ends the justification for both Definition 1.6 and integrality of genus 1 invariants in Conjecture 1.9.
1.7. Justification: GV in genus 2. In the ideal geometry of Section 1.5, the genus two Gopakumar-Vafa invariant n 2,β should be the (enumerative) number of genus 2 curves in the irreducible curve class β. We hence make the ansatz where the dots stand for the contributions from curves of genus 1. In this section we derive an expression for these lower genus contributions.
1.7.1. Contribution from genus one curves. We consider first the contributions from a 1-dimensional family of elliptic curves C 1 β → S 1 β parametrized by a smooth curve S 1 β , but with the additional assumption that there are no nodal rational curves in the family.
For simplicity of notation we also assume that the family C 1 β parametrizes all curves in class β (so there are no rational or genus 2 curves). We compute the invariant ∅ GW 2,β in this geometry. Under the above assumption we have the isomorphism and with an argument parallel to Section 1.6.2, the virtual class is: where ψ 1 is the cotangent line class on M 1,1 (X, β) and λ 1 ∈ H 2 (M 1,1 ), both pulled back to the product via the projection to the factors. One obtains that: By our assumption there are no family of rational curves in class β, so that we have ψ 1 = τ * (ψ 1 ), where τ : M 1,1 (X, β) → M 1,1 is the forgetful morphism to the moduli space of stable curves, and therefore ψ 2 1 = 0. We conclude that In total hence we see that the family C 1 β → S 1 β contributes − 1 24 n 1,β (c 2 (X)) to the integral (1.8). Assume more generally that there are both rational and elliptic curves in class β, but still no nodal rational curves. Then by the discussion in Section 1.6 and the above computation we have that − 1 24 n 1,β (c 2 (X)) is precisely the contribution from the elliptic curves to (1.8). Hence this contribution remains valid also in the presence of rational curves. 1.7.2. Contribution from genus zero curves. Let p : C 0 β → S 0 β be a family of degree β embedded rational curves in X parametrized by a smooth surface S 0 β . We assume that there are no curves of genus 1 or 2, and that all rational curves parametrized by S 0 β are smooth. Since β is irreducible, this means that all of them are isomorphic to P 1 .
By our assumption, we have an isomorphism of moduli spaces: where the right hand side is the moduli space of genus 2 degree 1 stable maps to the fibers of C 0 β → S 0 β . In particular, we have a diagram: , and q is the structure morphism to the base. By definition the middle square is fibered. The moduli space M 2,0 (C 0 β /S 0 β , 1) carries naturally a virtual fundamental class which we denote by [M ] We also denote the reduced virtual fundamental class of M 2,0 (X, β) by Since f β is fiberwise an embedding we have the subbundle T p ⊂ f * β (T X ). Let N = f * β (T X )/T p be the quotient, which is locally free of rank 3. The key to our discussion is the following comparision of virtual fundamental classes.
Proposition 1.11. We have For the proof we start with the two basic lemmata:

Proof. By the Cohomology and Base Change Theorem we have
where in the second equality we used that N is locally free, and in the forth equality we used flat base change. For the last step we used that S 0 β = M 0,0 (X, β) is smooth with tangent bundle given by p * N (which at each point s ∈ S 0 β has fiber H 0 (C 0 β,s , N C 0 β,s /X )).
Lemma 1.13. We have the exact sequence: Proof. The first statement is just an application of the Leray-Serre spectral sequence for the composition π =p • ρ. For the second statement, we have by flat base change that: By the existence of a global cosection, we have a surjection R 1 p * N → O S 0 β . Since p * N is locally free of rank 2, R 1 p * N is locally free of rank 1, so using the cosection it is isomorphic to O S 0 β .
Proof of Proposition 1.11. The 'standard' virtual tangent bundle 8 of M 2,0 (X, β) relative to the Artin stack of prestable curves M 2 is by definition given by where f = f β •f : C → X is the universal map. The reduced virtual tangent bundle is defined to be the cone: where sr σ is the semi-regularity map associated to the symplectic form σ, see [MP13,MPT].
where the third term is defined as the cone of the first map. By Lemma 1.13 and since the restriction of Similarly, the virtual tangent bundle of the perfect obstruction theory of which induces an isomorphism in degree 0 cohomology. This morphism induces a morphism from the complex (1.11) to the complex (1.9), and combining with Eqn. (1.10), we obtain the distinguished triangle: The claim now follows from the excess intersection formula.
The moduli space M decomposes naturally as the union , and Z 2 acts by interchanging the two factors of M 1,1 and switching the markings on M 0,2 (C 0 β /S 0 β , 1). The class [M ] rel is of dimension 6, but the dimensions of M 1 and M 2 are 7 and 6 respectively. In particular, there exists some class α ∈ A 6 (M 1 ) such that [M ] where ξ i : M i → M are the natural (gluing) morphisms. 9 By Proposition 1.11, we find that: (1.12) These two terms are analyzed as follows: 8 If E • → L M is a perfect obstruction theory, then the associated virtual tangent bundle is T vir The naïve splitting of the virtual class [M 2,0 (X, β)] vir as the sum does not hold. The long detour to the relative virtual class [M ] rel is necessary to decompose the virtual class! Lemma 1.14. We have the vanishing Proof. Let C → M be the universal curve as before, and let C → M 1 be its pull back along ξ 1 : M 1 → M . There exists a natural decomposition C = R ∪ q Z where R is the pullback of the universal curve over M 0,1 (X, β) and Z is the pullback of the universal curve from M 2,1 . The curves R and Z are glued along the marked points v : M 1 → C. In particular, we have the diagram is the projection and E → M 2,1 is the Hodge bundle (pulled back to the product). We obtain that: is the evaluation map to X and we surpressed the pullbacks by the projection to the factors. We conclude that where in the second equality we used the splitting principle and the Mumford relation . Now a straightforward computation (using that M 0,1 (X, β) is of dimension 3 and the Mumford relation, and which may be performed by a computer program) shows that this degree 6 component vanishes.
where R is the universal 2-pointed genus 0 curve, and the E i are the universal genus 1 curves. Let be the image of the marked points under the evaluation map ρ : are the Hodge bundles pulled-back from the first or second copy of M 2 . We argue as in Lemma 1.14, that is first we havẽ . (Here, we need the precompose with the forgetful morphism because the two markings can lie on a bubble in which case the tangent space to a marking maps with zero to the tangent space of the image point; by precomposing with the forgetful map, we contract the bubbles). As in Lemma 1.14 we then obtain that (1.14) .
We determine now a with a test calculation. Let X be the Fano variety of lines on a very general cubic 4-fold, and let β ∈ H 2 (X, Z) be the minimal effective curve class. As we will see in Section 7 we have the evaluations (assuming the conjectural holomorphic anomaly equation): Since there are no genus 2 curves on X in class β (see [NO]) we set n 2,β = 0.
Inserting this into Eqn. (1.16) yields: This conclude the justification of Definition 1.7. While the last step (i.e. §1.7.3) requires two assumption (locality of the contribution of nodal rational curves, and the holomorphic anomaly equation), the remainder of the paper yields plenty of numerical support for this definition.

Donaldson-Thomas invariants
For a holomorphic symplectic 4-fold, we define (reduced) Donaldson-Thomas invariants (DT 4 invariants for short) of one dimensional stable sheaves. We then use them to give a sheaf theoretic approach to Gopakumar-Vafa invariants defined in the previous section. In the last section we justify the definition by computations in the ideal geometry of curves.
Parallel to Gromov-Witten theory, the ordinary virtual class of M β vanishes [KiP, Sav]. For a choice of ample divisor H, one can define a reduced virtual class due to Def. 8.7,Lem. 9.4]: depending on the choice of orientation [CGJ,CL17]. To define descendent invariants, we need insertions: Definition 2.1. For any γ 1 , . . . , γ n ∈ H * (X) and k i ∈ Z 0 the DT 4 invariants are defined by [CMT18,CT20a], we propose the following sheaf theoretic interpretation of all genus Gopakumar-Vafa invariants: Conjecture 2.2. For certain choice of orientation, the following equalities hold. When β is an effective curve class, When β is a primitive curve class, When β is a primitive curve class, By Proposition 1.1, τ 1 (γ) GW 0,β can be deduced by g = 0 primary Gromov-Witten invariants. Therefore these formulae determine all genus Gopakumar-Vafa invariants from primary and descendent DT 4 invariants, which give a sheaf theoretic interpretation for them.
Remark 2.4. Our conjecture implicitly includes the independence of DT 4 invariants on the choice of ample divisor in defining reduced virtual classes (2.1).
2.3. Justification: Primary DT 4 invariants. For Conjecture 2.2 (i), we consider the case γ 1 , γ 2 ∈ H 4 (X, Z) for simplicity. These two 4-cycles (generically) cut out finite number of rational curves and miss high genus curves. As in [CMT18,§1.4], any one dimensional stable sheaf F with [F ] = β is O C for some rational curve C. Their moduli space M β is identified with the moduli space S 0 β of rational curves and [M β for some choice of orientation. After imposing the primary insertion, we have [M β where p : C 0 β → S 0 β is the total space of rational curve family (RCF) of class β and f : C 0 β → X is the evaluation map. Therefore Conjecture 2.2 (i) is confirmed in this ideal setting as both sides of the equation are (virtually) enumerating rational curves of class β incident to cycles dual to γ 1 and γ 2 .
2.4. Justification: Descendent DT 4 invariants. For Conjecture 2.2 (ii), as we put the incident condition with one 4-cycle γ in τ 1 (γ) DT4 β which generically does not intersect genus 2 curves, so we only need to consider the contributions from RCF and ECF (elliptic curve family).
(1) For any RCF of class β, we have an embedding i : C 0 β → S 0 β × X fitting into the diagram: (2.4) By Grothendieck-Riemann-Roch (GRR) formula, we have where ω p is the relative cotangent bundle of p. Combining with Eqn. (2.3), we see RCF in class β contributes to τ 1 (γ) DT4 β by [M β As β is primitive, we may deform it to the irreducible case where RCF consists of smooth rational curves (except at some finite number of fibers of nodal curves which can be ignored by insertion γ ∈ H 4 (X)). By Lemma 1.4, the RHS of Eqn. (2.6) is equal to − 1 2 τ 1 (γ) GW 0,β . This justifies the first term in the RHS of Conjecture 2.2 (ii).
(2) Next we consider the contribution from ECF. Let p : C 1 β → S 1 β be the total space of ECF of class β and j : C 1 β → X be the evaluation map. The insertion γ ∈ H 4 (X) (generically) intersects C 1 β in a finite number of points. We may assume C 1 β = E × S 1 β is the product, p is the projection and j is an embedding in our computations. We further assume E is smooth with normal bundle L ⊕ L −1 ⊕ O for a generic degree zero line bundle L on E.
Lemma 2.5. Let p : C 1 β → S 1 β be a one dimensional family of smooth elliptic curves E on X with normal bundle N E/X = L ⊕ L −1 ⊕ O for a generic L ∈ Pic 0 (E). Then any one dimensional stable sheaf F supported on this family is scheme theoretically supported on a fiber of p.
Proof. By [CMT18,Lem. 2.2], we know F is scheme theoretically supported on Tot E (L ⊕ L −1 ) for a fiber E of p. By [HST,Prop. 4.4], F is scheme theoretically supported on the its zero section, so we are done.
By the above lemma, there exists a morphism whose fiber over {E} is the moduli space M 1,1 (E) of stable bundles on E with rank 1 and χ = 1. Note that M 1,1 (E) ∼ = E. A family version of such isomorphism gives such that the virtual class satisfies [M β for certain choice of orientation.

The embedded rational curve family
As a first illustration of the general case, we work out here all Gromov-Witten, Gopakumar-Vafa and Donaldson-Thomas invariants for a family of smooth irreducible rational curves globally embedding in a holomorphic symplectic 4-fold. We will see that the global embedding assumption forces already almost all of our invariants to vanish.
3.1. Setting. Let X be a holomorphic symplectic 4-fold with symplectic form σ ∈ H 0 (X, Ω 2 X ). Consider a family p : C → S of embedded rational curves in the irreducible curve class β ∈ H 2 (X, Z) parametrized by a smooth surface S.
We make the following assumptions: (i) All fibers of p are non-singular (isomorphic to P 1 ).
(ii) The evaluation map j : C → X is a (global) embedding.
If α vanishes at a point s ∈ S, then for every point x in the fiber C s := p −1 (s) the form σ vanishes on the image of T C,x → T X,j(x) . Since σ j(x) is non-degenerate, it can only vanish on a subspace of at most half the dimension of T X,j(x) , so this is impossible. Hence α does not vanish. We conclude that S is a holomorphic symplectic surface, hence either an abelian or a K3 surface.
Moreover, consider the sequence The form σ = σ| C ∈ H 0 (C, j * Ω 2 X ) is non-degenerate; so the vanishing σ (T p , T C ) = 0 implies that we have an isomorphism σ : Example 3.1. Let S [2] be the Hilbert scheme of two points on a holomorphic symplectic surface S. The Hilbert-Chow map from S [2] to the second symmetric product of S: is a resolution of singularity [F], whose exceptional divisor D fits into the Cartesian diagram where ∆ is the diagonal embedding and p : D → S is a P 1 -bundle. The pair (S [2] , β := j * [D s ]) satisfies the assumptions (i-iii) for the family D → S.

Gromov-Witten invariants.
In the setting (i-iii), we have the following computation of Gromov-Witten invariants. In genus 0, one has the following description: Lemma 3.2. For any γ 1 , . . . , γ n ∈ H * (X), we have Proof. By condition (iii) the evaluation map factors as By restriction to a fiber and using the Aspinwall-Morrison formula (see e.g. [O18,Prop. 7(i)] for our context), we have Consider the fiber diagram where π n and π are the projections to the n-th and the first (n − 1)-factors respectively, and p is the structure morphism. We obtain: where we used that π * π * n (j * γ n ) = p * p * (j * γ n ) and then induction in the last step. The claim follows by putting these two statements together.
In genus 1 and 2, we have: Lemma 3.3. For any γ ∈ H 4 (X, Z) and d 1, we have Proof. Under our assumptions we have an isomorphism of moduli spaces where M 1,1 (C, dF ) is the moduli space of stable maps to the (total space of) C of degree d times the fiber class F , and M 1,1 (C/S, d) is the moduli space of stable maps into fibers of C → S. By comparing the perfect-obstruction theories of the first two moduli spaces one finds that: where the fiber of the bundle V at a point [f : Σ → C, p 1 ] ∈ M 1,1 (C, dF ) is the kernel of the Similarly, the virtual classes of the latter two moduli spaces are related by [M 1,1 Since S is symplectic, we have: e(E ∨ ⊗ p * T S ) = c 2 (T S ) − λ 1 c 1 (T S ) = c 2 (T S ). Hence where in the last step we used that j * (γ) p * c 2 (T S ) = 0 ∈ H * (C) for dimension reasons. The case of genus 2 is similar (using the Mumford relation (1.13)).
By Lemma 3.2 this implies the claim (for all d 1).
We will also require the following evaluation.

Proof. By Lemmata 3.3 and 3.4 and the definition of Gopakumar-Vafa invariants it suffices to
show that N nodal,β vanishes. Since M 0,2 (X, β) = C × S C we have To evaluate the first term we use that the preimage of the diagonal under j × j : C × S C → X × X is equal to C and that the refined intersection has an excess bundle which is an extension of T S and T ∨ p . For the second term we use Eqn. (3.4) and that by Lemma 1.4 we have ψ 1 = −c 1 (T p ) under the isomorphism M 0,1 (X, β) ∼ = C. With this the above becomes: where we used ψ 1 = −c 1 (T p ) under the isomorphism M 0,1 (X, β) ∼ = C by Lemma 1.4.

DT 4 invariants.
Lemma 3.7. In the setting (i-iii), for certain choice of orientation, we have Moreover, all DT 4 invariants vanish in curve class dβ for d > 1.
Proof. The computation is essentially done in §2.4. By [CMT18, Lem. 2.2], any one dimensional stable sheaf in class dβ is scheme theoretically supported on a fiber of p : C → S. Therefore Under the isomorphism (3.6) and the commutative diagram Combining with Eqns. (3.7), (3.8), we are done.
To sum up, combining Lemmata 3.2-3.7, we obtain: Theorem 3.8. Conjecture 1.9 and Conjecture 2.2 hold in the setting specified in §3.1.

Tautological integrals on moduli spaces of sheaves on K3 surfaces
In this section, we compute several tautological integrals on moduli spaces of one dimensional stable sheaves on K3 surfaces. These will be used in Section 5 to compute DT 4 invariants on the product of K3 surfaces, though they are interesting in their own right. 4.1. Fujiki constants. The second cohomology H 2 (M, Z) of an irreducible hyperkähler variety carries a integral non-degenerate quadratic form q : H 2 (M, Z) → Z, called the Beauville-Bogomolov-Fujiki form. By the following result of Fujiki [Fuji] (and its generalization in [GHJ]) it controls the intersection numbers of products of divisors against Hodge cycles which stay Hodge type on all deformations of M : Theorem 4.1. ( [Fuji], [GHJ,Cor. 23.17]) Assume α ∈ H 4j (M, C) is of type (2j, 2j) on all small deformation of M . Then there exists a unique constant C(α) ∈ C depending only on α and called the Fujiki constant of α such that for all β ∈ H 2 (M, C) we have In this section, we consider the Hilbert scheme S [n] of n-points of a K3 surface S, which by the work of Beauville [Bea] is irreducible hyperkähler. We will prove a closed formula for the Fujiki constants of all Chern classes of its tangent bundle.
The first coefficients are listed in Table 3. Remarkablely, the right hand side in Theorem 4.2 is up to the prefactor (2k)!/(k!2 k ) precisely the generating series of counts of genus k curves on a K3 surface passing through k generic points [BL]. This suggests a relationship to the work of Göttsche on curve counting on surfaces [G98]. The proof presented below uses similar ideas as in [G98], but we could not directly deduce it from there. The relationship to curve counting on K3 surfaces will be taken up in a follow-up work.   Table 3. The first non-trivial Fujiki constants of the Chern classes c k := c k (T S [n] ) of Hilbert schemes of points on a K3 surface. The modularity of Theorem 4.2 appears in the diagonals, e.g. the cases k = 0, 1 are the functions: (1 − q n ) −24 = q + 30q 2 + 480q 3 + 5460q 4 + · · · .
Replacing L by L ⊗t for t ∈ Z shows that (1 − q n ) −e(S) exp c 1 (L) 2 t 2 A + c 1 (L)c 1 (S)tB .
Since both sides are power series with coefficients which are polynomials in t, and the equality holds for all t ∈ Z, we find that Eqn. (4.3) also holds for t, a formal variable. We write Φ S,L (t) for the series (4.3). We argue now in two steps.
By taking the t 2k coefficient we obtain that (1 − q n ) −24 .
For completeness we also state the Fujiki constants of Chern classes of the second known infinite family of hyperkähler varieties, the generalized Kummer varieties.
Proof. Using the universality (4.3) and the value of A(q) we computed above, one concludes that for any line bundle L on A, we have: Using q(L n | Kumn−1(A) ) = c 1 (L) 2 we conclude the claim.
Remark 4.4. It is remarkable that all Fujiki constants of c k (T X ) for X ∈ {S [n] , Kum n (A)} are positive integers. By the software package 'bott' of J. Song [Son], the same can be checked numerically for arbitrary products of Chern classes of the tangent bundle (up to n 10). We also refer to [CJ, J] for some general results on positivity of Todd classes of hyperkähler varieties, and to [OSV] for a discussion on positivity of Chern (character) numbers. This suggests the question whether all (non-trivial) Fujiki constants of products of Chern classes on irreducible hyperkähler varieties positive. This question was raised independently and then studied in [BS, Saw].

Descendent integrals on the Hilbert scheme.
We now turn to integrals over descendents on Hilbert schemes, which are defined for α ∈ H * (S) and d 0 by , where π Hilb , π S are projections from S [n] × S to the factors. We prove the following evaluations: (1 − q n ) −24 .
Proof. This is a special case of [QS], but we can give a direct argument. For any surface S and K-theory class x ∈ K(S) with ch 0 (x) = ch 1 (x) = 0 consider the series By [EGL] and since we know the answer for x = 0, there exists a series A(q) such that Setting x = tO p , we in fact get the equality of (1 − q n ) −24 exp (At) .
Case 1: K3 surfaces. By GRR and taking the t 1 -coefficient, one finds that Case 2: Abelian surfaces. For an abelian surface A, similar as before, we have Here we used that ν * ch 2 (O  (ν × id) −1 (Z) = m −1 13 (Z Kum ), where m 13 is the addition map on the outer factors. Restricting to A × pt × A we find that m 13 | * A×pt×A (np) = n∆ A . Then the claim follows from the definition). We hence obtain that Combining with Eqn. (4.5), we are done.
where G k is given in (4.2).
Proof. Recall that for any hyperkähler variety X, the Looijenga-Lunts-Verbitsky Lie algebra g(X) is isomorphic to so(H 2 (X, Q) ⊕ U Q ), where U = 0 1 1 0 is the hyperbolic lattice [LL,Ver95,Ver96]. The degree 0 part of the Lie algebra splits as g 0 (X) = Qh ⊕ so(H 2 (X, Q)) where h is the degree grading operator. Looijenga and Lunts show that for the natural action of g(X) on cohomology, the subliealgebra so(H 2 (X, Q)) acts by derivations. In other words, if t ⊂ so(H 2 (X)) is a maximal Cartan, we have a decomposition which is multiplicative, i.e. V λ · V µ ⊂ V λ+µ . Here λ runs over all weights of the torus and V λ is the corresponding eigenspace.
For a Hilbert scheme, let δ = c 1 (O

[n]
S ) and recall the natural decomposition We consider the subliealgebra so(H 2 (S)) ⊂ so(H 2 (X, Q)) and for a Cartan t ⊂ so(H 2 (S)) the associated decomposition Since Chern classes are monodromy invariant, they lie in V 0 ; see [LL] for a discussion. If D = 0 there is nothing to prove. Otherwise, we can choose the Cartan t such that D lies in a non-zero eigenspace of the action of so(H 2 (S)) on H 2 (S, Q). Since the map γ → π Hilb * (ch d (O Z )π * S (γ)) is equivariant with respect to the action of so(H 2 (S)) on H * (S [d] ) and H * (S) respectively, we conclude that also G 3 (D) = π Hilb * (ch 3 (O Z )π * S (γ)) is of non-trivial weight with respect to t , i.e. lies in V µ for µ = 0.
By multiplicativity of the decomposition, it follows that the integrand c 2d−2 (S [d] ) · G 3 (D) is of non-zero weight, hence its integral must be zero. This proves (i).
For part (ii) we start with the vanishing from part (i): For any divisor W ∈ H 2 (S, Q) we have In the notation of [NOY,Eqn. (36)] consider the element h F δ = F ∧ δ in so(H 2 (X, Q)) for some F ∈ H 2 (S, Q). Since the integrated degree 0 part of the LLV algebra acts as ring isomorphisms and preserves the Chern classes (see [LL]), we have By [NOY,Prop. 4.4], we have that Taking the derivative d dt | t=0 of Eqn. (4.6), we find that: c(T S [n] )G 2 (p).
Therefore we find as desired (1 − q n ) −24 (5/3G 4 + 20G 2 2 + 2G 2 + 1/24), where we used the following Ramanujan differential equation [BGHZ,pp. 49,Prop. 15] 4.3. Descendent integrals on moduli spaces of 1-dimensional sheaves. Let β ∈ H 2 (S, Z) be an effective curve class and let M S,β be the moduli space of one dimensional stable sheaves F on S with [F ] = β and χ(F ) = 1. By a result of Mukai [M], M S,β is a smooth projective holomorphic symplectic variety of dimension β 2 + 2. Let F be the normalized universal family, i.e. which satisfies det Rπ M * F = O M S,β . For α ∈ H * (S), we define the descendents σ d (α) = π M * (π * S (α) ch d (F)). We have the following evaluations: Proposition 4.7. Let β ∈ H 2 (S, Z) be an effective curve class. For the point class p ∈ H 4 (S) and D ∈ H 2 (S), we have where N 1 (l), N (l) for all l ∈ Z are defined by the generating series 4.4. Transport of integrals to Hilbert schemes. For the proof of Proposition 4.7 we will use the general framework of monodromy operators of Markman [M08] (see also [O22a]) to transport the integrals to the Hilbert schemes. Consider the Mukai lattice, which is the lattice Λ = H * (S, Z) endowed with the Mukai pairing where, if we decompose an element x ∈ Λ according to degree as (r, D, n), we write x ∨ = (r, −D, n). Given a sheaf or a complex of sheaves E on S, its Mukai vector is defined by v(E) := td S · ch(E) ∈ Λ.
Let M (v) be a proper smooth moduli space of stable sheaves on S with Mukai vector v ∈ Λ (where stability is with respect to some fixed polarization). We assume that there exists a universal family F on M (v) × S. If it does not exist, everything below can be made to work by working with the Chern character ch(F) of a quasi-universal family, see [M08] or [O22a]. Let π M , π S be the projections to M (v) and S. One has the Mukai morphism where [−] deg=k stands for extracting the degree k component and where (as we will also do below) have suppressed the pullback maps from the projection to S. The morphism restricts to an lattice isometry (4.10) where on the right we consider the Beauville-Bogomolov-Fujiki form. Define the universal class which is independent of the choice of universal family F. For x ∈ Λ, consider the normalized descendents: where we let δ = π * ch 3 (O Z ) (so that −2δ is the class of the locus of non-reduced subschemes). We define the standard descendents on the Hilbert scheme by For a divisor D ∈ H 2 (S) one finds And for the unit, . By the normalization condition and GRR, we have: Moreover, by a direct computation, one also has: Using the vanishing c 1 (Rπ M * F) = 0 again yields This shows By rewriting the B's in terms of the σ's using the formulae above and then inverting the relation, we obtain for all D ∈ H 2 (S) by a straightforward calculation the following: For later, we also record some pairings with respect to the Beauville-Bogomolov-Fujiki form: Lemma 4.10.
Using the descendents B k (x), one allows to move between any two moduli spaces of stable sheaves on S just by specifying a Mukai lattice isomorphism g : Λ ⊗ Q → Λ ⊗ Q. We give the details in the case of our interest, see [M08, O22a] for the general case.
We want to connect the moduli spaces Define the isomorphism g : for all D ∈ H 2 (S, Z). The isomorphism was constructed so that which shows that it is a lattice isomorphism. Then one has: Theorem 4.11. (Markman [M08,Thm. 1.2], reformulation as in [O22a,Thm. 4]) For any k i 0, α i ∈ H * (S) and any polynomial P ,
We begin with the first evaluation. The strategy is to use Theorem 4.11 and the formulae given in Examples 4.8, 4.9 to move between the standard descendents and Markman's B-classes. We obtain: Observe that: Using the vanishing in Proposition 4.6 (i) we find that as well as ).

Let us write
which is well-defined since the above shows that the right hand side only depends on d. Taking generating series and using the evaluations of descendents on S [d] (in particular, the expression (4.8) and the differential equation (4.7)), we conclude where we denote M (q) = n 1 (1 − q n ) −24 . This proves the first evaluation (after shifting the generating series by q).
For the second case, one argues similarly, and obtains In the third case, one obtains

Product of K3 surfaces
In this section, we consider the product of two K3 surfaces S and T : If the curve class β ∈ H 2 (S × T, Z) is of non-trivial degree over both S and T , then one can construct two linearly independent cosections, which imply that the reduced invariants of X in this class vanish. 10 Because of that we always take β in the image of the natural inclusion where ι : S × {t} → X is the inclusion of a fiber. In §5.1, we first discuss the computations of GW/GV invariants. Then we completely determine all DT 4 invariants. By comparing them, we prove Conjecture 2.2 for X = S × T .

Gromov-Witten invariants.
For β ∈ H 2 (S, Z) ⊆ H 2 (X, Z), by the product formula in Gromov-Witten theory [B99], the reduced virtual classes satisfy The Gromov-Witten theory of K3 surfaces in low genus is well-known.
10 Of course, one may work with 2-reduced invariants but the moduli spaces becomes more difficult to handle.
We leave the study of the 2-reduced theory to a future work.
In particular, Conjecture 1.9 holds for X = S × T .
The nodal invariant is computed as follows: If β h is a primitive curve class of square β 2 h = 2h − 2, we conclude: where we used ∆(q) = q n 1 (1 − q n ) 24 and the identity (ref. Eqn. (4.7)): Using the definition of n 2,β , we conclude that: This is exactly the desired result.
We will also need the following later (in the appendix): Lemma 5.2. For any effective curve class β ∈ H 2 (S, Z) ⊆ H 2 (X, Z), we have Proof. Consider a divisor D = pr * 1 (α) ∈ H 2 (X) with d := α · β = 0. By Lemma 1.1 and Eqn. (5.1) we have where M S,β is the moduli space of one dimensional stable sheaves F on S with [F ] = β and χ(F ) = 1. By a result of Mukai [M], M S,β is a smooth projective holomorphic symplectic variety of dimension β 2 + 2. In order to determine the DT 4 virtual class of M β , we first recall: Sw,Ex. 16.52,pp. 410], [EG,Lem. 5]) Let E be a SO(2n, C)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M . Denote E + to be its positive real form 11 . The half Euler class of (E, Q) is Definition 5.4. ( [EG], [KiP,Def. 8.7]) Let E be a SO(2n, C)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M . An isotropic cosection of (E, Q) is a map is zero. If φ is furthermore surjective, we define the (reduced) half Euler class: as the half Euler class of the isotropic reduction. HereQ denotes the induced non-degenerate symmetric bilinear form on We show reduced half Euler classes are independent of the choice of surjective isotropic cosection.
Lemma 5.5. Let E be a SO(2n, C)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M and φ : E → O M be a surjective isotropic cosection. Then we can write the positive real form E + of E as red (E, Q) = ± e(E + ). Moreover, it is independent of the choice of surjective cosection.

In particular, when
In the diagram: has an induced non-degenerate symmetric bilinear form, so Let E + := V + /R, by the metric Q| V+ on V + , we may write By using the metric Q| E+ on E + , we may write Combining with Eqn. (5.9), we have By definition, the reduced half Euler class is the Euler class of E + .
Given two surjective cosections , then the bundle E + they determine are the same, so are the reduced half Euler classes.
, it is easy to see the corresponding E + has a trivial subbundle R, so both reduced half Euler classes vanish. ( , whose half Euler classes are the same. Therefore we know the reduced half Euler class is independent of the choice of surjective isotropic cosection. The last statement when Recall a Sp(2r, C)-bundle (or symplectic vector bundle) is a complex vector bundle of rank 2r with a non-degenerate anti-symmetric bilinear form. One class of quadratic vector bundles is given by tensor product of two symplectic vector bundles V 1 , V 2 . Their half Euler classes can be computed using Chern classes of V 1 , V 2 . For our purpose, we restrict to the following case.
Finally, we can determine the (reduced) virtual class of M β .
Theorem 5.7. For certain choice of orientation, we have (5.10) [M β Proof. Under the isomorphism (5.8): (5.11) where F S is a universal sheaf of M S,β and ∆ T denotes the diagonal in T × T .
The above theorem immediately implies the following: Corollary 5.9. Conjecture 2.2 holds for the product X = S × T and β ∈ H 2 (S, Z) ⊆ H 2 (X, Z).
Proof. This follows by inspection using Theorem 5.8 on DT 4 invariants and Proposition 5.1 and Lemma 5.2 for the GV/GW invariants respectively.
Another remarkable consequence of Theorem 5.8 is that all DT 4 invariants of S × T depend upon the curve class β only via the square β 2 and not the divisibility. More precisely, given pairs (S, β) and (S , β ) of a K3 surface and an effective curve class such that β 2 = β 2 , let ϕ : H 2 (S, R) → H 2 (S , R) be any real isometry such that ϕ(β) = β . Extend ϕ to the full cohomology by setting ϕ(1) = 1 and ϕ(p S ) = p S where p S ∈ H 4 (S, Z) is the point class.
This raises the question whether a similar independence of the divisibility holds for Donaldson-Thomas invariants of holomorphic symplectic 4-folds more generally. 5.5. Proof of Theorem 5.8. We split the proof in two parts.
For (i), we similarly have Proof of Theorem 5.8 Parts (ii-iv) and (vi-viii). We first express the DT 4 descendent invariants as integrals on M S,β . Let We obtain By base change to a point, we have π M S,β * (ch 2 (F norm S )) = 1.
Combining with Theorem 5.7, we obtain that Part (ii) now follows from Proposition 4.7(i).
Similarly, for (iii) we have Hence where the last equality is by base change to a point. Using Theorem 5.7, we obtain Thus with Proposition 4.7, we obtain that For part (iv), one similarly establishes: For (vi), we compute using Lemma 4.10 that Since M S,β and S [d] are deformation equivalent they share the same Fujiki constants: where β 2 = 2d − 2. This implies the claim. Finally for (vii) and (viii), we similarly find: = 48(θ 1 · β)N 1 (β 2 /2).

Cotangent bundle of P 2
We consider the geometry X = T * P 2 . There is a natural identification of curve classes: where ⊂ P 2 is a line.
6.1. GW and GV invariants. Let H ∈ H 2 (T * P 2 ) be the pullback of hyperplane class. We identify H 2 (T * P 2 , Z) ≡ Z by its degree against H.
Proof. This follows by a direct calculation using Graber-Pandharipande virtual localization formula [GP]. We refer to [PZ,§3] for a computation with parallel features.
In particular, Conjecture 1.9 holds for T * P 2 .
Lemma 6.3. Let ι : P 2 → T * P 2 be the zero section. Then the pushforward map is an isomorphism.
Proof. The map ι * is obviously injective. We show that ι * is also surjective. As T * P 2 admits a birational contraction T * P 2 → Y which contracts the zero section P 2 → T * P 2 to 0 ∈ Y and Y is affine, any one dimensional sheaf on T * P 2 is set theoretically supported on the zero section. It is enough to show that any one dimensional stable sheaf F on T * P 2 is scheme theoretically supported on the zero section.
Recall the following fact as stated in [CMT18, Lem. 2.2]: let g : Z → T be a morphism of C-schemes, and take a closed point t ∈ Z. Let Z t ⊂ Z be the scheme theoretic fiber of g at t. Suppose that F ∈ Coh(Z) is set theoretically supported on Z t and satisfies End(F ) = C. Then F is scheme theoretically supported on Z t . It should be well-known (and easy) that the scheme theoretic fiber of T * P 2 → Y at 0 ∈ Y is the reduced zero section, then surjectivity of ι * follows from the above fact. As we cannot find its reference, we give another argument here. Consider the closed embedding T * P 2 ⊂ O P 2 (−1) ⊕3 induced by the Euler sequence on P 2 . Note that O P 2 (−1) ⊕3 is an open subscheme of [C 6 /C * ], where C * on C 6 by t(x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) = (tx 1 , tx 2 , tx 3 , t −1 y 1 , t −1 y 2 , t −1 y 3 ), and corresponds to (x 1 , x 2 , x 3 ) = (0, 0, 0). The stack [C 6 /C * ] admits a good moduli space One can easily calculates that the scheme theoretic fiber of the above morphism restricted to (x 1 , x 2 , x 3 ) = (0, 0, 0) is (y 1 = y 2 = y 3 = 0). It follows that the scheme theoretic fiber of T * P 2 ⊂ O P 2 (−1) ⊕3 → T at 0 ∈ T is the reduced zero section P 2 . As T is affine, any one dimensional stable sheaf is set theoretically supported on the (scheme theoretic) fiber of 0 ∈ T . Using the above fact, it is also scheme theoretically supported on it. Therefore ι * is surjective.
Since M P 2 ,d is smooth and ι * is bijective on closed points, it remains to show that ι * induces an isomorphisms on tangent spaces. For a one dimensional stable sheaf F on P 2 , the tangent space of M T * P 2 ,d at ι * F is Ext 1 T * P 2 (ι * F, ι * F ) ∼ = Ext 1 P 2 (F, F ) ⊕ Hom(F, F ⊗ T * P 2 ). By the Euler sequence and stability, we have Therefore ι * induce an isomorphism of tangent spaces.
Then the following result is straightforward.
Lemma 6.4. Under the isomorphism (6.1), we have Here PD denotes the Poincaré dual, e 1 2 red is the reduced half Euler class as in Definition 5.4, F denotes a universal sheaf of M P 2 ,d and π M : M P 2 ,d × P 2 → M P 2 ,d is the projection.
Let h ∈ H 2 (P 5 ) denote the hyperplane class. It is straightforward to check By integration again the virtual class, we have the desired result for d = 2 case. The d = 3 case can be computed by a torus localization as in [CKM19,CKM20]. One sees that for any torus fixed point, the reduced obstruction space has a trivial factor 12 which implies the vanishing of (reduced) invariants. 7. Hilbert scheme of two points on a K3 surface Let S be a K3 surface. There are three fundamental conjectures which govern the Gromov-Witten invariants of the Hilbert scheme of points S [n] : (i) Multiple cover conjecture (proposed in [O21a], and proven partially in [O21c]) which expresses Gromov-Witten invariants for imprimitive curve classes as an explicit linear combination of primitive invariants, (ii) Quasi-Jacobi form property (proposed in [O18, O22b]), (iii) Holomorphic anomaly equation (proposed in [O22b], see also [O21b] for a progress report). For the Hilbert scheme of two points S [2] these conjectures have been established in genus 0 by [O18, O21c, O22b]. Together with [O18] they yield a complete evaluation of all genus 0 Gromov-Witten invariants of S [2] , that is for all curve classes and all insertions. We consider here the case of genus 1 and genus 2 Gromov-Witten invariants of S [2] for primitive curve classes. The strategy is to assume both the quasi-Jacobi form property (ii) and the holomorphic anomaly equation (iii). Under this assumption, the natural generating series of genus 1 and 2 Gromov-Witten invariants are given in terms of Jacobi forms and are determined up to finitely many coefficients. Using our earlier computations in ideal geometries we are able to uniquely fix these finitely many coefficients. Modulo the above conjectures, this leads to a complete evaluation of Gopakumar-Vafa invariants for S [2] in all genera. 7.1. Quasi-Jacobi forms. To state the result we will work with quasi-Jacobi forms. We refer to [Lib, vIOP] for an introduction to quasi-Jacobi forms, and to [O18,App. B] for the variable conventions that we follow here. We work here entirely on the level of (q, y)-series. We need the following series: Sometimes it will also be convenient to use the following alternative convention of Eisenstein series: The algebra of quasi-Jacobi forms is then the subring of consisting of all series which define holomorphic functions C × H → C in (z, τ ) where y = e 2πi(z+1/2) and q = e 2πiτ . A key fact is that the generator G 2 (q) is algebraically independent in the algebra of quasi-Jacobi forms from the other generators. Hence for any quasi-Jacobi form F (y, q) we can speak of its 'holomorphic anomaly', which is defined by d dG2 F (y, q), see [vIOP].
We will also require the following definition: Definition 7.1. Let F (y, q) be a quasi-Jacobi form of index 1 which satisfies the transformation law of Jacobi forms for the elliptic transformation z → z + τ (in generators this means it is independent of 1 Θ y d dy Θ; we will only encounter such kind here). For any class β ∈ H 2 (X, Z), the β-coefficient of F (y, q), is defined to be the coefficient of q d y k for any d, k ∈ Z such that (β, β) = 2d − k 2 /2.
To determine these constants, we can work with an elliptic K3 surface S → P 1 with section. The Hilbert scheme in this case has an induced Lagrangian fibration S [2] → P 2 with section. Let B, F be the section and fiber class of S respectively, and let A ∈ H 2 (S [2] , Z) be the class of the locus of non-reduced subschemes supported at a single point. There exists a natural isomorphism H 2 (S [2] , Z) = H 2 (S, Z) ⊕ ZA given by the Nakajima basis [O18, §0.2]. For h 0 and k ∈ Z, we consider the classes β h,k = B + hF + kA, which are of square (β h,k , β h,k ) = 2h − 2 − k 2 2 .
The F g 's are conjectured to be quasi-Jacobi forms and that their formal derivatives d dG2 F g are determined by a holomorphic anomaly equation [O22b].
Solving for a one finds a = 1/96, and hence F 1 (q 1 (F ) 2 1) = Θ 2 ∆ where we used that U (c 2 (X)) = 30q 1 (F )q 1 (1)1. This yields F 1 (c 2 (X)) = Θ 2 ∆ where, since there are no poles on the left hand side, the poles in (D z ℘) 2 and ℘ 3 cancel and give the Eisenstein series E 6 . By Proposition 6.1 and since the pair (S [2] , B + F + A) is deformation equivalent to (S [2] , A) and we have seen in Lemma 3.3 that the genus 1 invariants vanishing in this case, we have:  This proves the claim by solving for A and B. We remark that determining F 1 (q 1 (F ) 2 1) only required a single geometric constraint, namely the computation for class B−A. However, the formula also matches the vanishings obtained from computations in the ideal geometry (which applies to classes β ∈ {B, B +F +A}). For F 1 (c 2 (X)) the system is likewise overdetermined: we only used 2 of the 3 available constraints.
Proof of Theorem 7.4: Genus 2 case. Using Lemma 7.5 below, the standard intersections c 2 (X) · q 1 (p)q 1 (1)1 = 27, c 2 (X) · q 1 (W )q 1 (F )1 = 3 and the genus 1 part of Theorem 7.4, the holomorphic anomaly equation of [O22b]  for some a, b ∈ C. Here we used that F 2 is determined up to the functions ℘ 4 , ℘(y d dy ℘) 2 , ℘ 2 E 4 , E 4 and that the poles in the first of these functions have to cancel which replaces them with a E 6 ℘ term and then that ℘ 2 E 4 can also not appear because of holomorphicity. Finally, using the following evaluations (ref. This implies the result by monodromy invariance (we even have one more condition to spare, namely the vanishing of ∅ GW 2,B ).
Proof. This follows from Lemma 5.2 and the definition. We conclude the proof of Proposition A.3: By Theorem 5.8, we have which is precisely the right hand side of (A.3) by the two lemmata above.