Abstract
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, and 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 hyperkähler 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.
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1 Introduction
1.1 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 \(\overline{M}_{g,n}(X,\beta )\) of genus g degree \(\beta \in H_2(X,\mathbb {Z})\) stable maps:
Here \(\textrm{ev}_i :{\overline{M}}_{g,n}(X, \beta )\rightarrow X\) is the evaluation map at the ith marking, \(\psi _i\) is the ith cotangent line class, and \(\gamma _i \in H^{*}(X,\mathbb {Q})\) are cohomology classes. Since \({\overline{M}}_{g,n}(X, \beta )\) is a Deligne–Mumford stack, Gromov–Witten invariants are in general rational numbers, even if all \(\gamma _i\) 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) \(\mathbb {Z}\)-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 \(\sigma \in H^0(X,\Omega ^2_X)\). 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 classFootnote 1:
We are interested here in integer-valued invariants, which underlie the (reduced) Gromov–Witten invariants (0.1) of the holomorphic symplectic 4-fold X. In genus 0, all Gromov–Witten invariants can be reconstructed from the primary invariants, i.e. the integrals (0.1) where all \(k_i=0\). Our proposal for the genus 0 primary invariants is as follows:
Definition 0.1
(Definition 1.5). For any \(\gamma _1, \ldots , \gamma _n \in H^{*}(X,\mathbb {Z})\), we define the genus 0 Gopakumar–Vafa invariant \(n_{0, \beta }(\gamma _1, \ldots , \gamma _n) \in \mathbb {Q}\) recursively by:
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 \(\gamma \in H^4(X,{\mathbb {Z}})\). Because curves in imprimitive curve classes are very difficult to control in an ideal geometry (see Sect. 1.4) we will restrict us to a primitive curve class (i.e. where \(\beta \) is not a multiple of a class in \(H_2(X,\mathbb {Z})\)).
Definition 0.2
(Definition 1.6). Assume that \(\beta \in H_2(X,\mathbb {Z})\) is primitive. For any \(\gamma \in H^4(X, {\mathbb {Z}})\), we define the genus 1 Gopakumar–Vafa invariant \(n_{1,\beta }(\gamma )\in {\mathbb {Q}} \) by
In genus 2, the situation is even more complicated and attracting. In fact, the appearance of genus 2 invariants is a new phenomenon that is not available on ordinary Calabi–Yau 4-folds and Calabi–Yau 5-folds. By the virtual dimension of (0.2), one expects a finite number of isolated genus 2 curves. The genus 2 Gopakumar–Vafa invariant should be a count of these curves.
Definition 0.3
(Definition 1.7). Assume that \(\beta \in H_2(X,\mathbb {Z})\) is primitive. We define the genus 2 Gopakumar–Vafa invariant \(n_{2,\beta }\in {\mathbb {Q}}\) by
where \(N_{\textrm{nodal},\beta }\in {\mathbb {Q}}\) is the virtual count of rational nodal curves as defined in Eq. (1.4).
Our first main conjecture is about the integrality of these definitions:
Conjecture 0.4
(Conjecture 1.9). With the notations as above, we have
The definitions above are found via computations in an ‘ideal’ geometry where we assume that algebraic curves behave in the expected way, see §1.4, §1.5.Footnote 2 We justify Conjecture 0.4 in such an ideal case, which takes the whole §1.6, §1.7.
1.2 GV/\(\mathop {\textrm{DT}}\nolimits _4\) correspondence
The second main theme of this paper is to give a sheaf theoretic interpretation of Gopakumar–Vafa invariants. This is motivated by the parallel work of [CMT18, CT20a] on ordinary Calabi–Yau 4-folds.
Let \(M_\beta \) be the moduli scheme of one dimensional stable sheaves F on X with \(\mathop {\textrm{ch}}\nolimits _3(F)=\beta \), \(\chi (F)=1\). By [KiP, Sav], the ordinary \(\mathop {\textrm{DT}}\nolimits _4\) virtual class [BJ, OT] (see also [CL14]) of \(M_{\beta }\) vanishes. By Kiem-Park’s cosection localization [KiP], we instead have a (reduced) virtual class
As usual, the virtual class depends on a choice of orientation [CGJ, CL17]. More precisely, for each connected component of \(M_{\beta }\), there are two choices of orientation which affect the virtual class by a sign (component-wise). To define descendent invariants, consider the insertion operators:
where \({\mathbb {F}}_{\textrm{norm}}\) is the normalized universal sheaf, i.e. \( \det (\pi _{M*}{\mathbb {F}}_{\textrm{norm}})\cong \mathcal {O}_{M_\beta }\). Here a universal sheaf exists because of the condition \(\chi (F)=1\), see [HL, Thm. 4.5]. As in Gromov–Witten theory, for any \(\gamma _1, \ldots , \gamma _n \in H^{*}(X,{\mathbb {Z}})\) and \(k_1,\ldots ,k_n \in {\mathbb {Z}}_{\geqslant 0}\), we define \(\mathop {\textrm{DT}}\nolimits _4\) invariants:
Here is the second main conjecture of this paper, which gives a sheaf theoretic interpretation of our Gopakumar–Vafa invariants.
Conjecture 0.5
(Conjecture 2.2). For certain choice of orientation, the following holds.
When \(\beta \) is an effective curve class,
When \(\beta \) is a primitive curve class,
When \(\beta \) 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.
1.3 Verification of conjectures I: \(K3\times K3\)
Let \(X=S\times T\) be the product of two K3 surfaces. When the curve class \(\beta \in H_2(S \times T, \mathbb {Z})\) 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
By Behrend’s product formula [B99] (see Eq. (5.1)), we can easily compute all Gromov–Witten invariants and determine the Gopakumar–Vafa invariants as follows.
Theorem 0.6
(Proposition 5.1). Let \(\gamma ,\gamma '\in H^{4}(X)\), \(\alpha \in H^6(X)\) and let
be their Künneth decompositions. Then we have
If \(\beta \) is primitive, we have
where
with Eisenstein series:
In particular, Conjecture (0.4) holds for \(X=S\times T\).
On the Donaldson–Thomas side, a main result of this paper is the explicit computation of all \(\mathop {\textrm{DT}}\nolimits _4\) invariants of \(X = S \times T\) 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 \(X = S \times T\) and all effective curve classes \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\).
Here, since the moduli space \(M_{\beta }\) is connected, there are precisely two choices of orientation. We pick the one specified in Eq. (5.10) (invariants for the other differ only by an overall sign).
Contrary to the case of Gromov–Witten invariants, the computation of \(\mathop {\textrm{DT}}\nolimits _4\) invariants on \(S \times T\) is highly non-trivial. In Theorem 5.7, we first identify the virtual class explicitly. This expresses the \(\mathop {\textrm{DT}}\nolimits _4\) 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 \(k \geqslant 0\),
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.
1.4 Verification of conjectures II: \(T^*{\mathbb {P}}^2\)
Let \(T^{*} \mathbb {P}^2\) be the total space of the cotangent bundle on \(\mathbb {P}^2\), which is holomorphic symplectic. Let \(H \in H^2(T^{*} \mathbb {P}^2)\) be the pullback of the hyperplane class and use the identification \(H_2(T^{*} \mathbb {P}^2, \mathbb {Z}) \equiv \mathbb {Z}\) 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.
Proposition 0.9
(Corollary 6.2).
In particular, Conjecture 0.4 holds for \(T^{*} \mathbb {P}^2\).
On the sheaf side, we can compute \(\mathop {\textrm{DT}}\nolimits _4\) invariants for small degree curve classes.
Proposition 0.10
(Proposition 6.5). For certain choice of orientation, we have
In particular, Conjecture 0.5i holds for all \(d\leqslant 3\), and Conjecture 0.5 (ii), (iii) hold.
1.5 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 7.8and 7.11). Assume Conjectures A and C of [O22b]. Then the genus 1 and 2 part of Conjecture 0.4 hold for all primitive curve classes \(\beta \in H_2(S^{[2]},{\mathbb {Z}})\) satisfying \((\beta ,\beta )\leqslant 100\), where \((-,-)\) is the Beauville–Bogomolov–Fujiki pairing as in §7.2.
1.6 A Yau–Zaslow type formula on \(K3^{[2]}\)
A hyperkähler variety is of \(K3^{[2]}\)-type if it is deformation-equivalent to the Hilbert scheme of 2 points of a K3 surface S. Given a primitive curve class \(\beta \in H_2(S^{[2]},\mathbb {Z})\), consider the very general deformation \((X,\beta ')\) of a pair \((S^{[2]},\beta )\), where \(\beta \) 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 \(H_2(X,\mathbb {Z})\) are generated by \(\beta '\). In particular \(\beta '\) is irreducible. In this case, it is natural to expect that curves in \((X,\beta ')\) 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 \(\beta \in H_2(X,\mathbb {Z})\), the genus 2 Gopakumar–Vafa invariant \(n_{2,\beta }\) is the coefficient determined by \(\beta \) (see Definition 7.1) of the quasi-Jacobi form
where the functions \(\Theta ,\Delta , \wp , E_i\) are defined in §7.1.
In genus 1, it is convenient to encode the invariants in the genus 1 Gopakumar–Vafa class
which is defined by
where \(n_{1,\beta }(\gamma )\) is given in Definition 0.2. In an ideal geometry, \(n_{1,\beta }\) is the class of the surface swept out by elliptic curves in class \(\beta \). 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 Tables 1 and 2 below. Since the deformation class of a pair \((X,\beta )\) where \(\beta \) is a primitive curve class, only depends on the square \((\beta ,\beta )\) (see [O21a]), the Gopakumar–Vafa invariants only depend on \((\beta ,\beta )\).
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 \(\beta \). In the case \((\beta ,\beta )=-5/2\), any curve in class \(\beta \) is a line in a Lagrangian \(\mathbb {P}^2 \subset X\), 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 \((\beta ,\beta )=-1/2\) corresponds to the exeptional curve class on \(K3^{[2]}\) (the class of the exceptional curve of the Hilbert–Chow morphism \(K3^{[2]} \rightarrow \textrm{Sym}^{2}(K3)\)), and again there are no higher genus curves. The case \((\beta ,\beta )=-2\) is similar, see [HT]. The first time we see elliptic curves is in case \((\beta ,\beta )=0\), which corresponds to the fiber class of a Lagrangian fibration \(X \rightarrow \mathbb {P}^2\). Elliptic curves appear here in fibers over the discriminant. The case \((\beta ,\beta )=3/2\) corresponds to a very general Fano variety of lines on a cubic 4-fold, with \(\beta \) 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 \((\beta ,\beta )=2\) are the double covers of EPW sextics [O06]. The first time we should see isolated smooth genus 2 curves is the case \((\beta ,\beta )=11/2\), 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.
1.7 Appendix
In the appendix §A, we discuss several cases where we can extend the above GW/GV/\(\mathop {\textrm{DT}}\nolimits _4\) correspondence to imprimitive curve classes.
1.8 Notation and convention
All varieties and schemes are defined over \({\mathbb {C}}\). For a morphism \(\pi :X \rightarrow Y\) of schemes and objects \(\mathcal {F}, \mathcal {G}\in \mathrm {D^{b}(Coh(\textit{X\,}))}\) we will use
A class \(\beta \in H_2(X,{\mathbb {Z}})\) is called effective if there exists a non-empty curve \(C \subset X\) with class \([C] = \beta \). An effective class \(\beta \) 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 \(\sigma \in H^0(X,\Omega ^2_X)\). A holomorphic symplectic variety is irreducible hyperkähler if X is simply connected and \(H^0(X, \Omega _X^2)\) is generated by a symplectic form. A K3 surface is an irreducible hyperkähler variety of dimension 2.
2 Gopakumar–Vafa Invariants
Let X be a holomorphic symplectic 4-fold with symplectic form \(\sigma \in H^0(X,\Omega ^2_X)\).
In this section we first recall the definition of (reduced) Gromov–Witten invariants, and then give our definition of Gopakumar–Vafa invariants. In Sect. 1.4, we justify the definition by working in an ideal geometry of curves.
2.1 Gromov–Witten invariants
Let \({\overline{M}}_{g, n}(X, \beta )\) be the moduli space of n-pointed genus g stable maps to X representing the non-zero curve class \(\beta \in H_2(X,\mathbb {Z})\). The moduli space \(\overline{M}_{g,n}(X,\beta )\) admits a perfect obstruction theory [BF, LT]. By the construction of [MP13, §2.2] the symplectic form \(\sigma \) induces an everywhere surjective cosection of the obstruction sheaf. Concretely, the (standard) obstruction space for a stable map \(f: (C,p_1, \ldots , p_n) \rightarrow X\) is
and the cosection \(\textrm{Obs}_{f} \rightarrow \mathbb {C}\) is the map induced by the surjection
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:
In this paper we will always work with the reduced virtual fundamental class which we will hence simply denote by \([ - ]^{\textrm{vir}}\).
Given cohomology classes \(\gamma _i \in H^{*}(X)\) and integers \(k_i \geqslant 0\) the (reduced) Gromov–Witten invariants of X in class \(\beta \) are defined by
where \(\textrm{ev}_i :{\overline{M}}_{g,n}(X, \beta )\rightarrow X\) is the evaluation map at the ith marking and \(\psi _i\) is the ith cotangent line class. By the properties of the reduced virtual class, the integral (1.1) is invariant under deformations of the pair \((X,\beta )\) with preserve the Hodge type of the class \(\beta \). We call the invariant (1.1) a primary Gromov–Witten invariant if all the \(k_i\) are zero.
2.2 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 \(d:= D\cdot \beta \ne 0\). Then
Proof
By the divisor equation (e.g. [CK, pp. 305])
On the other hand, by rewriting \(\psi _1\) in terms of boundary divisors and using the splitting formula for reduced virtual classes as in [MPT, §7.3] one gets
\(\square \)
Lemma 1.2
For any \(\gamma \in H^4(X)\), we have: \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta } = \big \langle \tau _2(1) \tau _0(\gamma ) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta }. \)
Proof
Arguing as in Lemma 1.1 we can express both sides in terms of primary Gromov–Witten invariants, which yields the result. \(\quad \square \)
Lemma 1.3
\(\big \langle \tau _3(1) \big \rangle _{0,\beta }^{\mathop {\textrm{GW}}\nolimits } = \big \langle \tau _2(1) \tau _2(1) \big \rangle _{0,\beta }^{\mathop {\textrm{GW}}\nolimits }\).
Proof
Let \(D \in H^2(X)\) such that \(d:= D \cdot \beta \ne 0\). 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:
\(\square \)
Lemma 1.4
Assume that all fibers of the universal curve \(p: \mathcal {C}\rightarrow \overline{M}_{0,0}(X,\beta )\) are isomorphic to \(\mathbb {P}^1\). Let \(\pi : \overline{M}_{0,1}(X,\beta ) \rightarrow \overline{M}_{0,0}(X,\beta )\) be the forgetful morphism. Then
In particular, with \(f: \mathcal {C}\rightarrow X\) the universal map, we have
Proof
Let \({\tilde{p}}: \mathcal {C}_1 \rightarrow \overline{M}_{0,1}(X,\beta )\) be the universal curve and let \(s: \overline{M}_{0,1}(X,\beta ) \rightarrow \mathcal {C}_1\) be the universal section. By definition, we have
Recall that we have \(\mathcal {C}\cong \overline{M}_{0,1}(X,\beta )\). Moreover, since \(\mathcal {C}\rightarrow \overline{M}_{0,0}(X,\beta )\) parametrizes only smooth curves, we have
Under this isomorphism, the section s is identified with the diagonal morphism. We have the fiber diagram
Hence since \({\tilde{\pi }} \circ s = \text {id}\), we have
The second part follows since
\(\square \)
2.3 Definition of GV invariants
We consider the definition of Gopakumar–Vafa invariants.
In genus 0, by [BL, MP13], reduced Gromov–Witten invariants of X are equal to the (ordinary) Gromov–Witten invariants in fiber classes of the twistor space \(\mathcal {X}\rightarrow \mathbb {P}^1\) associated to the symplectic form \(\sigma \) (alternatively, we can view X embedded in a suitable 1-parameter family of holomorphic symplectic 4-folds such that the corresponding classifying map is transverse to the Noether–Lefschetz divisor defined by \(\beta \)). The definition of genus 0 Gopakumar–Vafa invariants for Calabi–Yau 5-folds proposed by Pandharipande and Zinger in [PZ, Eqn. (0.2)] hence can be viewed as a definition for genus 0 Gopakumar–Vafa invariants of X as follows:
Definition 1.5
For any \(\gamma _1, \ldots , \gamma _n \in H^{*}(X,\mathbb {Z})\), we define the genus 0 Gopakumar–Vafa invariant \(n_{0, \beta }(\gamma _1, \ldots , \gamma _n) \in \mathbb {Q}\) by
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 \((-1)^g \lambda _g\), 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 \(\beta \). Because curves in imprimitive curve classes are very difficult to control, we restrict hereby to the primitive case (i.e. to those \(\beta \) which are not a multiple in \(H_2(X,\mathbb {Z})\)). Consider the Chern classes of the tangent bundle of X:
Definition 1.6
Assume that \(\beta \in H_2(X,\mathbb {Z})\) is primitive. For any \(\gamma \in H^4(X, {\mathbb {Z}})\), we define the genus 1 Gopakumar–Vafa invariant \(n_{1,\beta }(\gamma )\in {\mathbb {Q}} \) by
Next we come to the genus 2 Gopakumar–Vafa invariants. Since the virtual dimension of the moduli space \(\overline{M}_{2,0}(X,\beta )\) is zero, GV invariants are defined without cohomological constraints. In other words, we expect that \(n_{2,\beta }\) should be given by the enumerative count of genus 2 curves in class \(\beta \). For the definition we require the following invariant introduced in [NO]:
where
-
\(\Delta _X \in H^8(X \times X)\) 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 \(N_{\textrm{nodal},\beta }\) is the expected number of rational nodal curves in class \(\beta \) [NO, Prop. 1.2]Footnote 3.
Definition 1.7
Assume that \(\beta \in H_2(X,\mathbb {Z})\) is primitive. We define the genus 2 Gopakumar–Vafa invariant \(n_{2,\beta }\in {\mathbb {Q}}\) by
Remark 1.8
For primitive \(\beta \in H_2(X,\mathbb {Z})\), 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.
Conjecture 1.9
(Integrality). We have
2.4 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 \(\overline{M}_{g,0}(X,\beta )\) 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 \(g \geqslant 3\).
We discuss now the expected behaviour of the curves in these families. We start with genus zero. Let \(p: \mathcal {C}^0_{\beta } \rightarrow S^0_{\beta }\) be a family of rational curves in class \(\beta \) over a smooth 2-dimensional surface \(S^0_{\beta }\), fiberwise embedded in X. Then we can have the following behaviour:
-
(i)
All the curves parametrized by \(S^0_{\beta }\) can be reducible.
Reason: Let \(\beta = \beta _1 + \beta _2\) and let \(\mathcal {C}^0_{\beta _i} \rightarrow S^0_{\beta _i}\) be a 2-dimensional family of rational curves in class \(\beta _i\). Let \(S^0_{\beta _1, \beta _2}\) be the preimage of the diagonal under the evaluation maps
$$\begin{aligned} j_1 \times j_2: \mathcal {C}^0_{\beta _1} \times \mathcal {C}^0_{\beta _2} \rightarrow X \times X. \end{aligned}$$Then \(S^0_{\beta _1, \beta _2}\) 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 \(\beta \).
-
(ii)
Given a generic curve \(\mathcal {C}^0_s:=p^{-1}(s) \subset X\) in the family, there exists another curve \(\mathcal {C}^0_{s'} \subset X\) in the family which meets it.
Reason: This follows by the same reasoning as in (i).
-
(iii)
For finitely many \(s \in S\), we expect the curve \(\mathcal {C}^0_s \subset X\) to be nodal.Footnote 4
Reason: The moduli space \(\overline{M}_{0,2}(X,\beta )\) is of expected dimension 4, and hence the preimage of the diagonal under \(\mathop {\textrm{ev}}\nolimits _1 \times \mathop {\textrm{ev}}\nolimits _2\) is of expected dimension 0.Footnote 5
-
(iv)
Even if all fibers of \(\mathcal {C}^0_{\beta } \rightarrow S_{\beta }\) are smooth \(\mathbb {P}^1\)’s, the natural morphism \(j: \mathcal {C}^0_{\beta } \rightarrow X\) is not necessarily an immersion.
(The differential \(dj: T_{\mathcal {C}^0_{\beta }} \rightarrow j^{*}(T_X)\) is expected to have a kernel in codimension \(\geqslant 2\).)
Similarly given a family \(p: \mathcal {C}^1_{\beta } \rightarrow S^1_{\beta }\) of elliptic curves in class \(\beta \) over a smooth 1-dimensional curve \(S^1_{\beta }\), fiberwise embedded in X, all the curves parametrized by \(S^1_{\beta }\) can be reducible. The argument is similar to (i) above, by considering the preimage of the diagonal under the evaluation maps
where \(\mathcal {C}^0_{\beta _1}\) (resp. \(\mathcal {C}^1_{\beta _2}\)) is a family of rational curves in class \(\beta _1\) (resp. elliptic curves in class \(\beta _2\)).
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 \(\overline{M}_{2,0}(X,\beta )\) 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 \(\beta \), it becomes increasingly difficult to control.
2.5 Ideal geometry: primitive case
We make the following additional assumptions:
-
X is irreducible hyperkähler,
-
the effective curve class \(\beta \in H_2(X,\mathbb {Z})\) 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 \((X,\beta )\) is deformation equivalent (through a deformation with keeps \(\beta \) of Hodge type) to a pair \((X', \beta ')\) where \(\beta ' \in H_2(X,\mathbb {Z})\) is irreducible. Hence we may without loss of generality make the following stronger assumptionFootnote 6:
-
the effective curve class \(\beta \in H_2(X,\mathbb {Z})\) is irreducible.
Under these assumptions our ideal geometry of curves simplifies to the following form:
-
(1)
The rational curves in X of class \(\beta \) 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 \(\mathcal {O}_{\mathbb {P}^1} \oplus \mathcal {O}_{\mathbb {P}^1} \oplus {\mathcal {O}}_{{\mathbb {P}}^{1}}(-2)\).
-
(2)
The arithmetic genus 1 curves in X of class \(\beta \) 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 \(L\oplus L^{-1}\oplus \mathcal {O}\), 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 \(g\geqslant 3\).
Example 1.10
Let \(Y \subset \mathbb {P}^5\) be a very general smooth cubic 4-fold and let \(F(Y) \subset \textrm{Gr}(2,6)\) 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 \(H_2(F(Y),\mathbb {Z})\) are of rank 1. Let \(\beta \) 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 \(\beta \) has been studied in [OSY, NO, GK]. The Chow variety of curves in class \(\beta \) is given by
where \(S \subset F(Y)\) is the smooth irreducible surface of lines of second type, and \(\Sigma \) is a smooth curve parametrizing genus 1 curves. There are precisely 3780 rational nodal curves corresponding to the intersection points \(S \cap \Sigma \), and all other rational curves are isomorphic to \(\mathbb {P}^1\). Moreover, there are no curves of genus \(\geqslant 2\) in class \(\beta \). We see that the curves in F(Y) of class \(\beta \) satisfy the requirements of the ideal geometry.
2.6 Justification: GV in genus 1
For a given class \(\gamma \in H^4(X,\mathbb {Z})\) let \(\Gamma \subset X\) be a generic topological cycle whose class is Poincaré dual to \(\gamma \). In an ideal geometry, the Gopakumar–Vafa invariant \(n_{1,\beta }(\gamma )\) should be the (enumerative) number \(n(\Gamma )\) of arithmetic genus 1 curves in X which are incident to \(\Gamma \). To derive an expression for it using Gromov–Witten invariants, we start with the genus 1 Gromov–Witten invariant:
where \(\beta \in H_2(X,\mathbb {Z})\) is primitive. Assuming the ideal geometry of Sect. 1.5 we will analyze the contributions from genus 0 and genus 1 stable maps to it (there are no contributions from genus 2 curves since they never meet the cycle \(\Gamma \), and also because genus 1-curves don’t map with non-zero degree to genus 2 curves). We show that the contribution from genus 1 curves is precisely \(n(\Gamma )\). This will yield the expression for \(n_{1,\beta }(\gamma )\).
2.6.1 Contribution from genus one curves
Let \(p:{\mathcal {C}}^1_\beta \rightarrow S^1_\beta \) be a 1-dimensional family of elliptic curves of class \(\beta \) as in Sect. 1.5, and let \(j: {\mathcal {C}}^1_\beta \rightarrow X\) be the evaluation map.
Since \(\Gamma \) (which represents \(\gamma \in H^4(X,\mathbb {Z})\)) is chosen generic, it intersects \(\mathcal {C}^1_\beta \) in precisely \((j_{*}[\mathcal {C}^1_\beta ] \cdot \gamma )\) many points. Following Sect. 1.5, we assume that the incident curves are smooth elliptic curves E with normal bundle \(N_{E/X} = L \oplus L^{-1} \oplus \mathcal {O}\). We find the contribution of this family to the invariant (1.5) is
where \(\omega \in H^2(E,\mathbb {Z})\) is the class of a point and the trivial factor \(H^1(E,N_{E/X}) = H^1(E,\mathcal {O}_E) = \mathbb {C}\) in the obstruction sheaf does not appear because we used the reduced virtual fundamental class.
2.6.2 Contribution from genus zero curves
Let \(p: \mathcal {C}^0_\beta \rightarrow S^0_\beta \) be a 2-dimensional family of embedded rational curves of class \(\beta \) in X parametrized by a smooth surface \(S^0_{\beta }\). The generic fiber of p is isomorphic to \(\mathbb {P}^1\) but over finitely many points we can have a rational nodal curve. The insertion \(\Gamma \) intersects the divisor \(\mathcal {C}^0_{\beta }\) in a curve that we can assume maps to a curve in \(S^0_{\beta }\). In particular, it avoids the singular fibers. For simplicity we may hence assume that there are no nodal fibers. Moreover, we can assume that this is the only family of curves in class \(\beta \). We will compute the contribution of this family to the genus 1 GW invariant (1.5).
Under these assumptions, for any genus 1 degree \(\beta \) stable map \(f: C \rightarrow X\), the source curve splits canonically as \(C \cong E \cup \mathbb {P}^1\), where E is an elliptic curve glued to \(\mathbb {P}^1\) at one point p. The map f is of degree 0 on E, and of degree \(\beta \) on \(\mathbb {P}^1\). (Because by our assumption, \(\beta \) is irreducible, all curves in \(\mathcal {C}^0_{\beta }\) are isomorphic to \(\mathbb {P}^1\), and these are the only curves in class \(\beta \), there is no map \(E \rightarrow X\) with degree \(\beta \) and every element in \(\overline{M}_{0,1}(X,\beta )\) is of the form \(f: (\mathbb {P}^1, p) \rightarrow X\).) Hence
By comparing the obstruction theories on the level of virtual classes, we get
where \([ - ]_{d}\) denotes taking the degree d part, \(\mathbb {E}\rightarrow \overline{M}_{g}\) is the Hodge bundle over the moduli space of curves (having fiber \(H^0(C,\omega _C)\) over a point [C]) with first Chern class \(\lambda _1 = c_1(\mathbb {E}) \in A^1(\overline{M}_{1,1})\), and \(\psi _1\) is the usual psi class on the moduli space \(\overline{M}_{0,1}(X,\beta )\). In the last line we have used that the dimension of \(\overline{M}_{0,k}(X,\beta )\) 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 \(\pi :\overline{M}_{1,1}(X,\beta ) \rightarrow \overline{M}_{1,0}(X,\beta )\) which at the same time is the universal curve over the moduli space. In particular, we have a decomposition
where \(\mathcal {E}\rightarrow \overline{M}_{1,1}\) and \(\mathcal {P}\rightarrow \overline{M}_{0,1}(X,\beta )\) are the universal curves. Since \(\pi \) is flat of relative dimension 1, we have
where \(a: \mathcal {E}\rightarrow \overline{M}_{1,1}(X,\beta )\) and \(b: \mathcal {P}\rightarrow \overline{M}_{1,1}(X,\beta )\) are the natural inclusions. We find
where for the second summand the \(\gamma \) is pulled back along the evaluation map \(\mathop {\textrm{ev}}\nolimits _1: \overline{M}_{0,1}(X,\beta ) \rightarrow X\) (since the map is constant on the elliptic curve).
In the second term in Eq. (1.7), the integrand over the factor \(\overline{M}_{1,2}\) is pulled back from \(\overline{M}_{1,1}\); hence this term vanishes. We conclude that Eq. (1.7) is equal to:
Here in the second step, we used that \(\pi ^{*}(\psi _1) = \psi _1 - s_{*}(1)\), where \(s: \overline{M}_{0,1}(X,\beta ) \rightarrow \overline{M}_{0,2}(X,\beta )\) is the section, so that \(\pi ^{*}(\psi _1^2) = \psi _1^2 - s_{*}(\psi _1)\), and in the last step we used Lemma 1.2.
2.6.3 Conclusion
By the discussion above we have obtained that in the ideal geometry we have
Since \(n(\Gamma )\) is the Gopakuma-Vafa invariant \(n_{1,\beta }(\gamma )\) in the ideal geometry, this ends the justification for both Definition 1.6 and integrality of genus 1 invariants in Conjecture 1.9.
2.7 Justification: GV in genus 2
In the ideal geometry of Sect. 1.5, the genus two Gopakumar–Vafa invariant \(n_{2,\beta }\) should be the (enumerative) number of genus 2 curves in the irreducible curve class \(\beta \). We hence make the ansatz
where the dots stand for the contributions from curves of genus \(\leqslant 1\). In this section we derive an expression for these lower genus contributions.
2.7.1 Contribution from genus one curves
We consider first the contributions from a 1-dimensional family of elliptic curves \({\mathcal {C}}^1_\beta \rightarrow S^1_\beta \) parametrized by a smooth curve \(S^1_{\beta }\), 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 \({\mathcal {C}}^1_{\beta }\) parametrizes all curves in class \(\beta \) (so there are no rational or genus 2 curves). We compute the invariant \(\big \langle \varnothing \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{2,\beta }\) in this geometry.
Under the above assumption we have the isomorphism
and with an argument parallel to Sect. 1.6.2, the virtual class is:
where \(\psi _1\) is the cotangent line class on \(\overline{M}_{1,1}(X,\beta )\) and \(\lambda _1 \in H^2(\overline{M}_{1,1})\), 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 \(\beta \), so that we have \(\psi _1 = \tau ^{*}(\psi _1)\), where \(\tau : \overline{M}_{1,1}(X,\beta ) \rightarrow \overline{M}_{1,1}\) is the forgetful morphism to the moduli space of stable curves, and therefore \(\psi _1^2=0\). We conclude that
In total hence we see that the family \({\mathcal {C}}^1_\beta \rightarrow S^1_\beta \) contributes \(-\frac{1}{24} n_{1,\beta }(c_2(X))\) to the integral (1.8).
Assume more generally that there are both rational and elliptic curves in class \(\beta \), but still no nodal rational curves. Then by the discussion in Sect. 1.6 and the above computation we have that \(-\frac{1}{24} n_{1,\beta }(c_2(X))\) is precisely the contribution from the elliptic curves to (1.8). Hence this contribution remains valid also in the presence of rational curves.
2.7.2 Contribution from genus zero curves
Let \(p: \mathcal {C}^0_\beta \rightarrow S^0_\beta \) be a family of degree \(\beta \) embedded rational curves in X parametrized by a smooth surface \(S^0_{\beta }\). We assume that all rational curves parametrized by \(S^0_{\beta }\) are smooth and that these are all the curves in class \(\beta \) (in particular, there are no curves of genus 1 or 2). Since \(\beta \) is irreducible, all of them are isomorphic to \(\mathbb {P}^1\).
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 \(\mathcal {C}^0_\beta \rightarrow S^0_\beta \). In particular, we have a diagram:
where \(C \rightarrow M\) is the universal curve over the moduli space (for both \(\overline{M}_{2,0}(X,\beta )\) and \(\overline{M}_{2,0}(\mathcal {C}^0_{\beta }/S^0_{\beta }, 1)\)), \({\tilde{f}}: C \rightarrow \mathcal {C}^0_{\beta }\) is the universal map of \(\overline{M}_{2,0}(\mathcal {C}^0_{\beta }/S^0_{\beta }, 1)\), and q is the structure morphism to the base. By definition the middle square is fibered. The moduli space \(\overline{M}_{2,0}(\mathcal {C}^0_{\beta }/S^0_{\beta }, 1)\) carries naturally a virtual fundamental class which we denote by
We also denote the reduced virtual fundamental class of \(\overline{M}_{2,0}(X,\beta )\) by
Since \(f_{\beta }\) is fiberwise an embedding we have the subbundle \(T_p \subset f_{\beta }^{*}(T_X)\). Let
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:
Lemma 1.12
We have
Proof
By the cohomology and base change theorem we have
Hence we find that
where in the second equality we used that N is locally free, and in the fourth equality we used flat base change. For the last step we used that \(S^0_{\beta } = \overline{M}_{0,0}(X,\beta )\) is smooth with tangent bundle given by \(p_{*}N\) (which at each point \(s \in S^0_\beta \) has fiber \(H^0(\mathcal {C}^0_{\beta ,s}, N_{\mathcal {C}^0_{\beta ,s}/X})\)).
\(\square \)
Lemma 1.13
We have the exact sequence:
and \(R^1 {\tilde{p}}_{*}( {\tilde{q}}^{*} N) = \mathcal {O}_{M}\).
Proof
The first statement is just an application of the Leray-Serre spectral sequence for the composition \(\pi = {\tilde{p}} \circ \rho \). Note that we have used (1.9) for the third term. For the second statement, we have by flat base change that:
By the existence of a global cosection, we have a surjection \(R^1 p_{*} N \rightarrow \mathcal {O}_{S^0_{\beta }}\). 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 \(\mathcal {O}_{S^0_{\beta }}\). \(\quad \square \)
Proof of Proposition 1.11
The ‘standard’ virtual tangent bundleFootnote 7 of \(\overline{M}_{2,0}(X,\beta )\) relative to the Artin stack of prestable curves \({\mathfrak {M}}_2\) is by definition given by
where \(f = f_{\beta } \circ {\tilde{f}}: C \rightarrow X\) is the universal map. The reduced virtual tangent bundle is defined to be the cone:
where \(\textsf{sr}_{\sigma }\) is the semi-regularity map associated to the symplectic form \(\sigma \), see [MP13, MPT]. The inclusion \(T_p \subset f_{\beta }^{*}(T_X)\) induces a natural distinguished triangle:
where the third term is defined as the cone of the first map. By Lemma 1.13 and since the restriction of \(\textsf{sr}_{\sigma }\) to \({\tilde{p}}_{*}\big ( (R^1 \rho _{*} \mathcal {O}_C) \otimes {\tilde{q}}^{*}N \big )\) vanishes, we have
Similarly, the virtual tangent bundle of the perfect obstruction theory of \(\overline{M}_{2,0}(\mathcal {C}^0_{\beta }/S^0_{\beta })\) fits into the distinguished triangle
By Lemma 1.12 there exists a natural morphism
which induces an isomorphism in degree 0 cohomology. This morphism induces a morphism from the complex (1.12) to the complex (1.10), and combining with Eq. (1.11), we obtain the distinguished triangle:
The claim now follows from the excess intersection formula. \(\quad \square \)
The moduli space M decomposes naturally as the union
where
and \(\mathbb {Z}_2\) acts by interchanging the two factors of \(\overline{M}_{1,1}\) and switching the markings on \(\overline{M}_{0,2}(\mathcal {C}^0_{\beta }/S^0_{\beta },1)\). The class \([ M ]^{\textrm{rel}}\) is of dimension 6, but the dimensions of \(M_1\) and \(M_2\) are 7 and 6 respectively. In particular, there exists some class \(\alpha \in A_{6}(M_1)\) such that
where \(\xi _i: M_i \rightarrow M\) are the natural (gluing) morphisms.Footnote 8 By Proposition 1.11, we find that:
These two terms are analyzed as follows:
Lemma 1.14
We have the vanishing
Proof
Let \(C \rightarrow M\) be the universal curve as before, and let \(C' \rightarrow M_1\) be its pull back along \(\xi _1: M_1 \rightarrow M\). There exists a natural decomposition \(C' = R \cup _q Z\) where R is the pullback of the universal curve over \(\overline{M}_{0,1}(X,\beta )\) and Z is the pullback of the universal curve from \(\overline{M}_{2,1}\). The curves R and Z are glued along the marked points \(v: M_1 \rightarrow C\). In particular, we have the diagram
where \(x=\rho ' \circ v\) is the image of the gluing point. Applying \(\rho '_{*}\) to the normalization exact sequence
shows that
where \(\textrm{pr}_1: M_1 \rightarrow \overline{M}_{2,1}\) is the projection and \(\mathbb {E}\rightarrow \overline{M}_{2,1}\) is the Hodge bundle (pulled back to the product). We obtain that:
where \({\tilde{\mathop {\textrm{ev}}\nolimits }}_1 = {\tilde{q}} \circ {\tilde{\xi }}_1 \circ x: M_1 \rightarrow \mathcal {C}^0_{\beta }\) is the evaluation map.
Using the defining exact sequence \(0 \rightarrow T_p \rightarrow f_{\beta }^{*}(T_X) \rightarrow N \rightarrow 0\) and that \({\tilde{\mathop {\textrm{ev}}\nolimits }}_1^{*}(T_p)\) is isomorpic to the cotangent line bundle of \(\overline{M}_{0,1}(\mathcal {C}^0_{\beta }/S^0_{\beta },1)\) at the marking, i.e. \({\tilde{\mathop {\textrm{ev}}\nolimits }}_1^{*}(T_p) \cong \mathbb {L}_{p_1}^{\vee }\), we obtain the exact sequence
where \(\mathop {\textrm{ev}}\nolimits _1: \overline{M}_{0,1}(\mathcal {C}^0_{\beta }/S^0_{\beta },1) \cong \overline{M}_{0,1}(X,\beta ) \rightarrow X\) 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 Lemma 1.15 below. Now a straightforward computation (using that \(\overline{M}_{0,1}(X,\beta )\) is of dimension 3 and the Mumford relation, and which may be performed by a computer program) shows that this degree 6 component vanishes.
\(\square \)
Lemma 1.15
We have
Proof
Let \(c_i = \mathop {\textrm{ev}}\nolimits ^{*} c_i(T_X)\). We have \(c_4=c_2^2=0\). By the splitting principle, we may assume that \(\mathbb {E}^{\vee } = L_a \oplus L_b\) where \(c_1(L_a) = 1+a\) and \(c_1(L_b) = 1+b\). Then
where we used the Mumford relation
\(\square \)
Lemma 1.16
Proof
Let \(C' \rightarrow M_2\) be the pullback of the universal curve \(C \rightarrow M\) to \(M_2\). We have a decomposition \(C' = R \cup E_1 \cup E_2\), 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 \(\rho ': C' \rightarrow \xi _2^{*} q^{*} \mathcal {C}^0_{\beta }\). We have
where \(\mathbb {E}_i = \textrm{pr}_i^{*}(\mathbb {E})\) 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 have
Then with \(\pi _i: \overline{M}_{0,2}(\mathcal {C}^0_{\beta }/S^0_{\beta },1) \rightarrow \overline{M}_{0,1}(\mathcal {C}^0_{\beta }/S^0_{\beta },1)\) the morphism that forgets all but the ith marking we have that
(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
For \(i=1,2\) and \((\lambda , \psi , \mathop {\textrm{ev}}\nolimits ):= (c_1(\mathbb {E}_i), \pi _i^{*}(\psi _i), \mathop {\textrm{ev}}\nolimits _i)\), we have
where \((\cdots )\) are terms that are not multiples of \(\lambda \).
Using this and Eq. (1.6), the term (1.14) becomes:
On \(\overline{M}_{0,2}(X,\beta )\) we have
where \(s: \overline{M}_{0,1}(X,\beta ) \rightarrow \overline{M}_{0,2}(X,\beta )\) is the canonical section, and therefore
Applying Lemma 1.2, we find that:
With a similar reasoning, using Lemma 1.3, we also get that:
Inserting both these vanishings into Eq. (1.15) concludes the claim. \(\quad \square \)
Inserting the two lemmata above into Eq. (1.13), the whole computation collpases into the following simple evaluation:
We hence conclude that the family \(\mathcal {C}^0_\beta \rightarrow S^0_\beta \) of rational curves with only smooth fibers contributes \(\big \langle \tau _0(c_2(X)) \tau _0(c_2(X)) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta } / (2 \cdot 24^2)\) to the Gopakumar–Vafa invariant \(n_{2,\beta }\).
2.7.3 Conclusion and contribution from nodal rational curves
Consider an ideal geometry of curves as in Sect. 1.5 without any additional assumptions. We expect the contributions from genus 0 and genus 1 curves to the invariant \(\big \langle \varnothing \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{2,\beta }\) to be as discussed above, plus a correction term coming from the nodal rational curves. This correction term should be local, and hence a multiple of the expected number of nodal rational curves \(N_{\text {nodal},\beta }\). We hence make the ansatz:
for a constant \(a \in \mathbb {Q}\) independent of \((X,\beta )\).
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 \(\beta \in H_2(X,\mathbb {Z})\) be the minimal effective curve class. As we will see in Sect. 7 we have the evaluations (assuming the conjectural holomorphic anomaly equation):
Moreover, by [NO, Thm. 1.3], we have
Since there are no genus 2 curves on X in class \(\beta \) (see [NO]) we set
Inserting this into Eq. (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.
3 Donaldson–Thomas Invariants
For a holomorphic symplectic 4-fold, we define (reduced) Donaldson–Thomas invariants (\(\mathop {\textrm{DT}}\nolimits _4\) 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.
3.1 Definitions
Let \(M_\beta \) be the moduli scheme of one dimensional stable sheaves F on X with \([F]=\beta \), \(\chi (F)=1\). Such moduli spaces are independent of the choice of polarization (e.g. [CMT18, Rmk. 1.2]) and are used in [CMT18, CT20a] to give sheaf theoretic interpretation of Gopakumar–Vafa type invariants of ordinary Calabi–Yau 4-folds [KP]. We also refer to [CMT19, CT19, CT20b, CT20c] for related conjectures and computations, which build on the works of virtual class constructions [BJ, OT] (see also [CL14]).
Parallel to Gromov–Witten theory, the ordinary virtual class of \(M_\beta \) vanishes [KiP, Sav]. For a choice of ample divisor H, one can define a reduced virtual class due to Kiem-Park [KiP, Def. 8.7, Lem. 9.4]Footnote 9:
depending on the choice of orientation [CGJ, CL17]. When the moduli space is smooth, the obstruction sheaf is a bundle, the reduced virtual class is the Poincaré dual of the reduced half Euler class of the obstruction bundle as recalled in Definition 5.4.
To define descendent invariants, we need insertions:
where \({\mathbb {F}}_{\textrm{norm}}\) is the normalized universal sheaf, i.e. \( \det (\pi _{M*}{\mathbb {F}}_{\textrm{norm}})\cong \mathcal {O}_{M_\beta }\) (ref. [CT20a, §1.4]).
Definition 2.1
For any \(\gamma _1, \ldots , \gamma _n \in H^{*}(X)\) and \(k_i \in {\mathbb {Z}}_{\geqslant 0}\) the \(\mathop {\textrm{DT}}\nolimits _4\) invariants are defined by
3.2 Conjectures
As in [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 \(\beta \) is an effective curve class,
When \(\beta \) is a primitive curve class,
When \(\beta \) is a primitive curve class,
By Lemma 1.1, \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta }\) can be deduced by \(g=0\) primary Gromov–Witten invariants. Therefore these formulae determine all genus Gopakumar–Vafa invariants from primary and descendent \(\mathop {\textrm{DT}}\nolimits _4\) invariants, which give a sheaf theoretic interpretation for them.
Remark 2.3
The way we write down Conjecture 2.2 (iii) is indirect. By [COT22, App. A], the LHS of (iii) equals to stable pair invariant \(P_{-1,\beta }\) which is conjecturally the same as genus 2 Gopakumar–Vafa invariants [COT22, Conj. 1.10]. We believe there is also a formula relating \(\big \langle \tau _2(\theta ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) \((\theta \in H^{2}(X,{\mathbb {Z}}))\) to genus 2 Gopakumar–Vafa invariants, which we haven’t found so far.
Remark 2.4
Our conjecture implicitly includes the independence of \(\mathop {\textrm{DT}}\nolimits _4\) invariants on the choice of ample divisor in defining reduced virtual classes (2.1).
3.3 Justification: primary \(\mathop {\textrm{DT}}\nolimits _4\) invariants
In the following two sections, we justify our conjecture in the case of ideal geometry of Sect. 1.5.
For Conjecture 2.2 (i), we consider the case \(\gamma _1,\gamma _2\in H^4(X,{\mathbb {Z}})\) for simplicity. These two 4-cycles (generically) cut out finite number of smooth rational curves and miss high genus curves.
As in [CMT18, §1.4], any one dimensional stable sheaf F with \([F]=\beta \) is \(\mathcal {O}_C\) for some rational curve C. Their moduli space \(M_\beta \) is identified with the moduli space \(S^0_{\beta }\) of rational curves. Since the normal bundle of a generic curve C satisfies \(N_{C/X}\cong \mathcal {O}_{{\mathbb {P}}^1}(-2,0,0)\), we have
The reduced part of the ‘half’ of \(\mathop {\textrm{Ext}}\nolimits ^2(F,F)\) is zero, therefore
for some choice of orientation. After imposing the primary insertion, we have
where \(p:{\mathcal {C}}^0_{\beta }\rightarrow S^0_{\beta }\) is the total space of rational curve family (RCF) of class \(\beta \) and \(f: {\mathcal {C}}^0_{\beta }\rightarrow X\) is the evaluation map. Therefore Conjecture 2.2 (i) is confirmed in this ideal setting as the right hand side of the equation counts rational curves of class \(\beta \) incident to cycles dual to \(\gamma _1\) and \(\gamma _2\) in the ideal geometry, which is the genus 0 Gopakumar–Vafa invariant.
3.4 Justification: descendent \(\mathop {\textrm{DT}}\nolimits _4\) invariants
For Conjecture 2.2 (ii), as we put the incident condition with one 4-cycle \(\gamma \) in \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) 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 \(\beta \), we have an embedding \(i:{\mathcal {C}}^0_\beta \hookrightarrow S^0_{\beta }\times X\) fitting into the diagram:
By Grothendieck-Riemann-Roch (GRR) formula, we have
Obviously \({\mathbb {F}}_{\textrm{norm}}=i_{*}\mathcal {O}_{{\mathcal {C}}^0_\beta }\), and therefore
where \(\omega _p\) is the relative cotangent bundle of p.
Combining with Eq. (2.5), we see RCF in class \(\beta \) contributes to \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) by
As \(\beta \) 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 \(\gamma \in H^4(X)\)). By Lemma 1.4, the RHS of Eq. (2.8) is equal to \(-\frac{1}{2}\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta }\). This justifies the first term in the RHS of Conjecture 2.2 (ii).
(2) Next we consider the contribution from ECF. Let \(p:{\mathcal {C}}^1_{\beta }\rightarrow S^1_{\beta }\) be the total space of ECF of class \(\beta \) and \(j: {\mathcal {C}}^1_{\beta }\rightarrow X\) be the evaluation map. The insertion \(\gamma \in H^4(X)\) (generically) intersects \({\mathcal {C}}^1_{\beta }\) in a finite number of points. We may assume \({\mathcal {C}}^1_{\beta }=E\times S^1_{\beta }\) is the product, p is the projection and j is an embedding in our computations. We further assume E is smooth with normal bundle \(L\oplus L^{-1}\oplus \mathcal {O}\) for a generic degree zero line bundle L on E.
Lemma 2.5
Let \(p:{\mathcal {C}}^1_{\beta }\rightarrow S^1_{\beta }\) be a one dimensional family of smooth elliptic curves E on X with normal bundle \(N_{E/X}=L\oplus L^{-1}\oplus \mathcal {O}\) for a generic \(L\in \mathop {\textrm{Pic}}\nolimits ^0(E)\). 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 \(\textrm{Tot}_E(L\oplus L^{-1})\) 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. \(\quad \square \)
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 \(\chi =1\). Note that \(M_{1,1}(E)\cong E\). A family version of such isomorphism gives
A similar calculation as Eqs. (2.3), (2.4) shows that the virtual class satisfies
for certain choice of orientation.
Next we compute the descendent insertion. In the following diagram:
a universal one dimensional sheaf \({\mathbb {F}}\) can be chosen as
where we treat \(\Delta _{{\mathcal {C}}^1_{\beta }}\) as a divisor of \({\bar{p}}^*(\Delta _{S^1_{\beta }})\) via
It is straightforward to check that \({\mathbb {F}}\) is normalized.
Below, we use notations from the following diagram
The GRR formula gives
Therefore, we have
where the last equality is because \(\dim _{{\mathbb {C}}}{\mathcal {C}}^1_{\beta }=2\) and \(c_1({\mathcal {C}}^1_{\beta })\cdot j^*\gamma \in H^6({\mathcal {C}}^1_{\beta })=0\).
From the exact sequence in \(\mathop {\textrm{Coh}}\nolimits ({\mathcal {C}}^1_{\beta }\times {\mathcal {C}}^1_{\beta })\):
we obtain
where \(\Delta _{S^1_{\beta }}: S^1_{\beta }\rightarrow S^1_{\beta }\times S^1_{\beta }\) denotes the diagonal embedding and we use GRR formula for the map \(\Delta _{S^1_{\beta }}\) in the last equation.
Combining Eqs. (2.13), (2.14), we obtain
where we note that \({\bar{p}}^*(\Delta _{S^1_{\beta }})_*(c_1(S^1_{\beta }))\) is some multiple of the fiber class of \({\bar{p}}\), so the first term in above vanishes. Therefore, ECF of class \(\beta \) contributes to \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) by
which gives exactly the genus 1 GV invariant \(n_{1,\beta }(\gamma )\) for primitive \(\beta \) as they are (virtually) enumerating elliptic curves of class \(\beta \) incident to the cycle dual to \(\gamma \).
Remark 2.6
For a general curve class \(\beta \) and any \(k\geqslant 1\) such that \(k|\beta \), one can similarly show that any elliptic curve family \({\mathcal {C}}^1_{\beta /k}\) of class \(\beta /k\) contributes to \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) by
Therefore, all elliptic curve families contribute to \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{DT}}\nolimits _4}_{\beta }\) by \(\sum _{k|\beta }n_{1,\beta /k}(\gamma )\).
4 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.
4.1 Setting
Let X be a holomorphic symplectic 4-fold with symplectic form \(\sigma \in H^0(X,\Omega _X^2)\). Consider a family \(p: \mathcal {C}\rightarrow S\) of embedded rational curves in the irreducible curve class \(\beta \in H_2(X,\mathbb {Z})\) parametrized by a smooth surface S.
We make the following assumptions:
-
(i)
All fibers of p are non-singular (isomorphic to \(\mathbb {P}^1\)).
-
(ii)
The evaluation map \(j: \mathcal {C}\rightarrow X\) is a (global) embedding.
-
(iii)
All curves in class \(d \beta \) for all \(d \geqslant 1\) are unions of curves of the family \(\mathcal {C}\rightarrow S\).
Let \(\sigma \in H^0(X,\Omega _X^2)\) be the holomorphic symplectic form. Since the pullback \(j^{*}(\sigma ) \in H^0(\mathcal {C}, \Omega _{\mathcal {C}}^2)\) vanishes on \(T_{p}\), there exists a 2-form \(\alpha \in H^0(S, \Omega _S^2)\) such that
If \(\alpha \) vanishes at a point \(s \in S\), then for every point x in the fiber \(\mathcal {C}_s:= p^{-1}(s)\) the form \(\sigma \) vanishes on the image of \(T_{\mathcal {C},x} \rightarrow T_{X,j(x)}\). Since \(\sigma _{j(x)}\) 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 \(\alpha \) 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 \(\sigma ' = \sigma |_{\mathcal {C}} \in H^0(\mathcal {C}, j^{*} \Omega _X^2)\) is non-degenerate; so the vanishing \(\sigma '(T_{p}, T_{\mathcal {C}}) = 0\) implies that we have an isomorphism
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 \(\Delta \) is the diagonal embedding and \(p: D\rightarrow S\) is a \({\mathbb {P}}^1\)-bundle. The pair \((S^{[2]}, \beta := j_{*}[ D_s ] )\) satisfies the assumptions (i-iii) for the family \(D \rightarrow S\).
4.2 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 \(\gamma _1, \ldots , \gamma _n \in H^{*}(X)\), we have
Proof
By condition (iii) the evaluation map factors as
Since \(\overline{M}_{0,n}(X,d\beta )\) is of virtual dimension \(2+n = \dim ( \mathcal {C}\times _S \cdots \times _S \mathcal {C})\) we have
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 \(\pi _n\) and \(\pi \) are the projections to the nth and the first \((n-1)\)-factors respectively, and p is the structure morphism. We obtain:
where we used that \(\pi _{*} \pi _n^{*}(j^{*} \gamma _n) = p^{*} p_{*}(j^{*}\gamma _n)\) and then induction in the last step. The claim follows by putting these two statements together. \(\quad \square \)
In genus 1 and 2, we have:
Lemma 3.3
For any \(\gamma \in H^4(X,\mathbb {Z})\) and \(d \geqslant 1\), we have
Proof
Under our assumptions we have an isomorphism of moduli spaces
where \(\overline{M}_{1,1}(\mathcal {C}, d F)\) is the moduli space of stable maps to the (total space of) \(\mathcal {C}\) of degree d times the fiber class F, and \(\overline{M}_{1,1}(\mathcal {C}/S, d)\) is the moduli space of stable maps into fibers of \(\mathcal {C}\rightarrow S\). By comparing the perfect-obstruction theories of the first two moduli spaces one finds that:
where the fiber of the bundle \(\mathcal {V}\) at a point \([f: \Sigma \rightarrow \mathcal {C},p_1] \in \overline{M}_{1,1}(\mathcal {C}, dF)\) is the kernel of the semiregularity map \(H^1(\Sigma , f^{*}(N_{\mathcal {C}/X})) \rightarrow H^1(\Sigma , \omega _{\Sigma }) = \mathbb {C}\).
Similarly, the virtual classes of the latter two moduli spaces are related by
Since S is symplectic, we have: \(e( \mathbb {E}^{\vee } \otimes p^{*}T_S) = c_2(T_S) - \lambda _1 c_1(T_S) = c_2(T_S)\). Hence
where in the last step we used that \(j^{*}(\gamma )\, p^{*} c_2(T_S) = 0 \in H^{*}(\mathcal {C})\) for dimension reasons.
The case of genus 2 is similar (using the Mumford relation (??)). \(\quad \square \)
We also have the following vanishing:
Lemma 3.4
For any \(\gamma \in H^4(X,\mathbb {Q})\) and \(d \geqslant 1\) we have:
Proof
Consider the invariant \(\big \langle \tau _1(\gamma ) \big \rangle ^{\mathop {\textrm{GW}}\nolimits }_{0,\beta }\). By Lemma 1.4, we have
Applying Lemma 1.1 to the divisor \([ \mathcal {C}] \in H^2(X,\mathbb {Z})\) which satisfies \([\mathcal {C}] \cdot \beta = -2\), we also have:
Since \(N_{\mathcal {C}/X} \cong T_{p}^{\vee } = \omega _p\), we have
Comparing Eqs. (3.1) and (3.2), we conclude with the help of Lemma 3.2 that:
The pair of short exact sequences
shows that
and hence
Inserting into Eq. (3.3), we find
By Lemma 3.2 this implies the claim (for all \(d \geqslant 1\)). \(\quad \square \)
We will also require the following evaluation.
Lemma 3.5
For any \(\gamma \in H^4(X,\mathbb {Q})\), we have
Proof
By Lemma 1.1 applied to \(D = [ \mathcal {C}] \in H^2(X,\mathbb {Z})\) (which satisfies \(D \cdot \beta = -2\)) we have:
By Eq. (3.3) and Lemma 3.2 the first term vanishes. And by Lemma 3.2 again we get:
\(\square \)
4.3 Gopakumar–Vafa invariants
We compute all \(g\geqslant 1\) Gopakumar–Vafa invariants in the setting (i-iii).
Lemma 3.6
For any \(\gamma \in H^4(X,\mathbb {Z})\), we have
Proof
By Lemmata 3.3 and 3.4 and the definition of Gopakumar–Vafa invariants it suffices to show that \(N_{\textrm{nodal},\beta }\) vanishes. Since \(\overline{M}_{0,2}(X,\beta ) = \mathcal {C}\times _S \mathcal {C}\) we have
To evaluate the first term we use that the preimage of the diagonal under \(j \times j: \mathcal {C}\times _S \mathcal {C}\rightarrow X \times X\) is equal to \(\mathcal {C}\) and that the refined intersection has an excess bundle which is an extension of \(T_S\) and \(T_p^{\vee }\). For the second term we use Eq. (3.4) and that by Lemma 1.4 we have \(\psi _1 = -c_1(T_p)\) under the isomorphism \(\overline{M}_{0,1}(X,\beta ) \cong \mathcal {C}\). With this the above becomes:
\(\square \)
4.4 \(\mathop {\textrm{DT}}\nolimits _4\) invariants
Lemma 3.7
In the setting (i-iii), for certain choice of orientation, we have
Moreover, all \(\mathop {\textrm{DT}}\nolimits _4\) invariants vanish in curve class \(d \beta \) for \(d>1\).
Proof
The computation is essentially done in §2.4. By [CMT18, Lem. 2.2], any one dimensional stable sheaf in class \(d\beta \) is scheme theoretically supported on a fiber of \(p: \mathcal {C}\rightarrow S\). Therefore
And their virtual classes satisfy
Under the isomorphism (3.6) and the commutative diagram
the normalized universal family \({\mathbb {F}}_{\textrm{norm}}\) is \(i_*\mathcal {O}_{\mathcal {C}}\). By Grothendieck-Riemann-Roch formula,
Therefore
Combining with Eqs. (3.7), (3.8), we are done. \(\quad \square \)
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.
5 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 Sect. 5 to compute \(\mathop {\textrm{DT}}\nolimits _4\) invariants on the product of K3 surfaces, though they are interesting in their own right.
5.1 Fujiki constants
The second cohomology \(H^2(M, \mathbb {Z})\) of an irreducible hyperkähler variety carries a integral non-degenerate quadratic form
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 \(\alpha \in H^{4j}(M,{\mathbb {C}})\) is of type (2j, 2j) on all small deformation of M. Then there exists a unique constant \(C(\alpha )\in {\mathbb {C}}\) depending only on \(\alpha \) and called the Fujiki constant of \(\alpha \) such that for all \(\beta \in H^2(M,\mathbb {C})\) 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.
For \(k \geqslant 2\) even, we define the classical Eisenstein series
where \(B_k\) are Bernoulli numbers, i.e. \(B_2=\frac{1}{6}\), \(B_4=-\frac{1}{30}\), \(\cdots \). For example, we have
Theorem 4.2
Let S be a K3 surface. For any \(k \geqslant 0\),
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.
Proof
Let \(L \in \mathop {\textrm{Pic}}\nolimits (S)\) be a line bundle on an arbitrary surface S. Consider the series
where we let \(L_n = \det ( L^{[n]} ) \otimes \det (\mathcal {O}_S^{[n]})^{-1}\). Since the integrand is multiplicative, by [NW, Prop 3] (which immediately follows from [EGL]), there exists power series \({\textsf{A}},{\textsf{B}},{\textsf{C}},{\textsf{D}}\) in q such that for any surface S and line bundle L, we have
The Göttsche formula
shows then that \({\textsf{C}}=0\) and provides an explicit expression for \({\textsf{D}}\). Hence we have
Replacing L by \(L^{\otimes t}\) for \(t \in \mathbb {Z}\) shows that
Since both sides are power series with coefficients which are polynomials in t, and the equality holds for all \(t \in \mathbb {Z}\), we find that Eq. (4.3) also holds for t, a formal variable. We write \(\Phi _{S,L}(t)\) for the series (4.3). We argue now in two steps.
Step 1: Specialization to K3 surfaces. Let S be a K3 surface. Since \(S^{[n]}\) is holomorphic symplectic, its odd Chern classes vanish. Together with Eq. (4.1) and \({\textsf{q}}(L_n) = c_1(L)^2\) this gives
On the other hand, we have
By taking the \(t^{2k}\) coefficient we obtain that
Step 2: Specialization to abelian surfaces. Let A be an abelian surface with a line bundle \(L \in \mathop {\textrm{Pic}}\nolimits (A)\) such that \(c_1(L)^2 \ne 0\). Let \(\sigma : A^{[n]} \rightarrow A\) be the sum map, and let
be the generalized Kummer variety of dimension \(2n-2\). We have the fiber diagram
In particular \(\nu \) is étale (of degree \(n^4\)), which implies that
Since the Chern classes of an abelian surface vanish, we obtain
Moreover one has (see [NW, Eqn. (2)]) that
We obtain that
Using that \(e(\textrm{Kum}_{n-1}(A)) = n^3 \sum _{d|n} d\) (ref. [GS]) and Eq. (4.3), we conclude that
where \([-]_{t^2}\) denotes the coefficient of \(t^2\) term. Hence
Combining with Eq. (4.4), we are done. \(\quad \square \)
For completeness we also state the Fujiki constants of Chern classes of the second known infinite family of hyperkähler varieties, the generalized Kummer varieties.
Proposition 4.3
For any \(k \geqslant 0\) and abelian surface A, we have
Proof
Using the universality (4.3) and the value of \({\textsf{A}}(q)\) we computed above, one concludes that for any line bundle L on A, we have:
Using \({\textsf{q}}(L_n|_{\textrm{Kum}_{n-1}(A)}) = c_1(L)^2\) we conclude the claim. \(\quad \square \)
Remark 4.4
It is remarkable that all Fujiki constants of \(c_k(T_X)\) for \(X \in \{ S^{[n]}, \textrm{Kum}_{n}(A)\}\) 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 \(n \leqslant 10)\). 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].
5.2 Descendent integrals on the Hilbert scheme
We now turn to integrals over descendents on Hilbert schemes, which are defined for \(\alpha \in H^{*}(S)\) and \(d \geqslant 0\) by
where \(\pi _{\mathop {\textrm{Hilb}}\nolimits }, \pi _S\) are projections from \(S^{[n]} \times S\) to the factors. We prove the following evaluations:
Proposition 4.5
Let \(\textsf{p}\in H^4(S)\) be the point class. Then
Proof
This is a special case of [QS], but we can give a direct argument. For any surface S and K-theory class \(x \in K(S)\) with \(\mathop {\textrm{ch}}\nolimits _0(x) = \mathop {\textrm{ch}}\nolimits _1(x) = 0\) consider the series
By [EGL] and since we know the answer for \(x=0\), there exists a series \({\textsf{A}}(q)\) such that
Setting \(x = t \mathcal {O}_{\textsf{p}}\), we in fact get the equality of
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 \(\nu ^{*} \mathop {\textrm{ch}}\nolimits _2( \mathcal {O}_{\textsf{p}}^{[n]} )|_{A \times pt} = n \textsf{p}\). (To see the last statement, consider the diagram
Let \(\mathcal {Z}\subset A^{[n]} \times A\) be the universal subscheme, and let \(\mathcal {Z}_{\textrm{Kum}} \subset \textrm{Kum}_{n-1}(A) \times A\) be its restriction to the Kummer. Inside \(A \times \textrm{Kum}_{n-1} \rightarrow A\), we have an equality of subschemes:
where \(m_{13}\) is the addition map on the outer factors. Restricting to \(A \times pt \times A\) we find that \(m_{13}|_{A \times pt \times A}^{*}(n \textsf{p}) = n \Delta _{A}\). Then the claim follows from the definition). We hence obtain that
Combining with Eq. (4.5), we are done. \(\quad \square \)
Proposition 4.6
where \(G_k\) is given in (4.2).
Proof
Recall that for any hyperkähler variety X, the Looijenga–Lunts–Verbitsky Lie algebra \(\mathfrak {g}(X)\) is isomorphic to \(\textrm{so}(H^2(X,\mathbb {Q}) \oplus U_{\mathbb {Q}})\), where \(U = \left( {\begin{array}{c}0\ 1\\ 1\ 0\end{array}}\right) \) is the hyperbolic lattice [LL, Ver95, Ver96]. The degree 0 part of the Lie algebra splits as \(\mathfrak {g}_0(X) = \mathbb {Q}h \oplus \mathfrak {so}(H^2(X,\mathbb {Q}))\) where h is the degree grading operator. Looijenga and Lunts show that for the natural action of \(\mathfrak {g}(X)\) on cohomology, the subliealgebra \(\mathfrak {so}(H^2(X,\mathbb {Q}))\) acts by derivations. In other words, if \(\mathfrak {t}\subset \mathfrak {so}(H^2(X))\) is a maximal Cartan, we have a decomposition
which is multiplicative, i.e. \(V_{\lambda } \cdot V_{\mu } \subset V_{\lambda + \mu }\). Here \(\lambda \) runs over all weights of the torus and \(V_{\lambda }\) is the corresponding eigenspace.
For a Hilbert scheme, let \(\delta = c_1( \mathcal {O}_S^{[n]})\) and recall the natural decomposition
We consider the subliealgebra \(\mathfrak {so}(H^2(S)) \subset \mathfrak {so}(H^2(X,\mathbb {Q}))\) and for a Cartan \(\mathfrak {t}' \subset \mathfrak {so}(H^2(S))\) 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 \(\mathfrak {t}'\) such that D lies in a non-zero eigenspace of the action of \(\mathfrak {so}(H^2(S))\) on \(H^2(S,\mathbb {Q})\). Since the map \(\gamma \mapsto \pi _{\mathop {\textrm{Hilb}}\nolimits *}\left( \mathop {\textrm{ch}}\nolimits _d(\mathcal {O}_{\mathcal {Z}}) \pi _S^{*}(\gamma ) \right) \) is equivariant with respect to the action of \(\mathfrak {so}(H^2(S))\) on \(H^{*}(S^{[d]})\) and \(H^{*}(S)\) respectively, we conclude that also
is of non-trivial weight with respect to \(\mathfrak {t}'\), i.e. lies in \(V_{\mu }\) for \(\mu \ne 0\).
By multiplicativity of the decomposition, it follows that the integrand \(c_{2d-2}(S^{[d]}) \cdot \mathfrak {G}_3(D)\) 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 \(W \in H^2(S,\mathbb {Q})\) we have
In the notation of [NOY, Eqn. (36)] consider the element \(h_{F \delta } = F \wedge \delta \) in \(\mathfrak {so}(H^2(X,\mathbb {Q}))\) for some \(F \in H^2(S, \mathbb {Q})\). 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 \(\frac{d}{dt}|_{t=0}\) of Eq. (4.6), we find that:
Note that by Theorem 4.2, we have
Moreover, by considering \(\log \)-derivative, one has
Therefore we find as desired
where we used the following Ramanujan differential equation [BGHZ, pp. 49, Prop. 15]
\(\square \)
5.3 Descendent integrals on moduli spaces of 1-dimensional sheaves
Let \(\beta \in H_2(S,\mathbb {Z})\) be an effective curve class and let \(M_{S,\beta }\) be the moduli space of one dimensional stable sheaves F on S with \([F]=\beta \) and \(\chi (F)=1\). By a result of Mukai [M], \(M_{S,\beta }\) is a smooth projective holomorphic symplectic variety of dimension \(\beta ^2+2\). Let \(\mathbb {F}\) be the normalized universal family, i.e. which satisfies \(\det \textbf{R}\pi _{M*} \mathbb {F}= \mathcal {O}_{M_{S,\beta }}\). For \(\alpha \in H^{*}(S)\), we define the descendents
We have the following evaluations:
Proposition 4.7
Let \(\beta \in H_2(S,\mathbb {Z})\) be an effective curve class. For the point class \(\textsf{p}\in H^4(S)\) and \(D \in H^2(S)\), we have
where \(N_{1}(l)\), \(N'(l)\) for all \(l \in \mathbb {Z}\) are defined by the generating series
5.4 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 \(\Lambda = H^{*}(S,\mathbb {Z})\) endowed with the Mukai pairing
where, if we decompose an element \(x \in \Lambda \) according to degree as (r, D, n), we write \(x^{\vee } = (r,-D,n)\). Given a sheaf or a complex of sheaves E on S, its Mukai vector is defined by
Let M(v) be a proper smooth moduli space of stable sheaves on S with Mukai vector \(v \in \Lambda \) (where stability is with respect to some fixed polarization). We assume that there exists a universal family \(\mathbb {F}\) on \(M(v) \times S\). If it does not exist, everything below can be made to work by working with the Chern character \(\mathop {\textrm{ch}}\nolimits (\mathbb {F})\) of a quasi-universal family, see [M08] or [O22a]. Let \(\pi _M, \pi _S\) be the projections to M(v) and S. One has the Mukai morphism \(\theta _{\mathbb {F}}: \Lambda \rightarrow H^2(M(v))\) defined by
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
where on the right we consider the Beauville–Bogomolov–Fujiki form. Define the universal class
which is independent of the choice of universal family \(\mathbb {F}\). For \(x \in \Lambda \), consider the normalized descendents:
and let \(B_k(x) = [ B(x) ]_{\deg =2k}\) its degree 2k component.
Example 4.8
For \(v=(1,0,1-d)\), the moduli space becomes the punctual Hilbert scheme: \(M(v) = S^{[n]}\). Then we have
where we let \(\delta = \pi _{*} \mathop {\textrm{ch}}\nolimits _3( \mathcal {O}_{\mathcal {Z}} )\) (so that \(-2 \delta \) is the class of the locus of non-reduced subschemes).
We define the standard descendents on the Hilbert scheme by
where \(\alpha \in H^{*}(S)\). One obtains that
For a divisor \(D \in H^2(S)\) one finds
And for the unit,
Example 4.9
Let \(\beta \in \mathop {\textrm{Pic}}\nolimits (S)\) be an effective class of square \(\beta \cdot \beta = 2d-2\). For the Mukai vector \(v=(0,\beta ,1)\) the moduli space is \(M(v) = M_{S,\beta }\). Let \(\mathbb {F}\) be the normalized universal family, i.e. which satisfies \(\det \textbf{R}\pi _{M*} \mathbb {F}= \mathcal {O}\). For \(\alpha \in H^{*}(S)\), we define as before the descendents
By the normalization condition and GRR, we have:
Moreover, by a direct computation, one also has:
Using the vanishing \(c_1( \textbf{R}\pi _{M*} \mathbb {F}) = 0\) again yields
This shows
By rewriting the B’s in terms of the \(\sigma \)’s using the formulae above and then inverting the relation, we obtain for all \(D \in H^2(S)\) by a straightforward calculation the following:
and
For later, we also record some pairings with respect to the Beauville–Bogomolov–Fujiki form:
Lemma 4.10
Proof
By Eq. (4.11) and since (4.10) is an isometry and moreover \(B_1( - )|_{v^{\perp }} = \theta _{\mathbb {F}}|_{v^{\perp }}\) we have
Similarly,
The last one uses again Eq. (4.11). \(\quad \square \)
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 \(g: \Lambda \otimes \mathbb {Q}\rightarrow \Lambda \otimes \mathbb {Q}\). 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 \(g: \Lambda \otimes \mathbb {Q}\rightarrow \Lambda \otimes \mathbb {Q}\) by
for all \(D \in H^2(S,\mathbb {Z})\). 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 \(k_i \geqslant 0\), \(\alpha _i \in H^{*}(S)\) and any polynomial P,
5.5 Proof of Proposition 4.7
Let \(\beta \in H_2(S,\mathbb {Z})\) be an effective curve class with \(\beta ^2 = 2d-2\). 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 \(M(q) = \prod _{n \geqslant 1} (1-q^n)^{-24}\). 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
6 Product of K3 Surfaces
In this section, we consider the product of two K3 surfaces S and T:
If the curve class \(\beta \in H_2(S \times T, \mathbb {Z})\) 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.Footnote 10 Because of that we always take \(\beta \) in the image of the natural inclusion
where \(\iota : S \times \{ t \} \hookrightarrow X\) is the inclusion of a fiber. In §5.1, we first discuss the computations of GW/GV invariants. Then we completely determine all \(\mathop {\textrm{DT}}\nolimits _4\) invariants. By comparing them, we prove Conjecture 2.2 for \(X=S\times T\).
6.1 Gromov–Witten invariants
For \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\), 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.
In genus 0, one defines BPS numbers \(n_{0,\beta }(S)\) by the multiple cover formula
By the Yau–Zaslow formula proven by Klemm, Maulik, Pandharipande and Scheidegger [KMPS], the invariant \(n_{0,\beta }(S)\) only depends on the square \(\beta ^2\). By the evaluation for primitive curve classes due to Bryan and Leung [BL], one then has
where
In genus 1, by Pandharipande-Yin [PY, pp. 12, (8)], we have the multiple cover formula
where \(N_{1}(l)\) is defined as in (4.9) of the last section, that is
We remark that although genus 2 Gromov–Witten invariants are zero (5.1), the corresponding Gopakumar–Vafa invariants are nontrivial (Proposition 5.1).
6.2 Gopakumar–Vafa invariants
Let \(\gamma ,\gamma '\in H^{4}(X)\) be cohomology classes and
be their decompositions under the Künneth isomorphism:
Fix also a curve class
Proposition 5.1
For any effective curve class \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\), we have
If \(\beta \) is primitive, we have
where \(N_1(l)\) is defined as in (4.9) and
In particular, Conjecture 1.9 holds for \(X=S\times T\).
Proof
By the divisor equation, we have
The genus 0 formula hence follows from Eq. (5.2) and the Yau–Zaslow formula (5.3). In genus 1, the product formula (5.1) and Eq. (5.5) imply that for any effective class \(\beta \in H_2(S,\mathbb {Z})\) we have:
Moreover, by the first part we have
Hence taking these formulae for primitive \(\beta \) yields the result.
For the genus 2 Gopakumar–Vafa invariant, let \(\beta \) be primitive. Observe that we have
The nodal invariant is computed as follows:
If \(\beta _h\) is a primitive curve class of square \(\beta _h^2 = 2\,h-2\), we conclude:
where we used \(\Delta (q) = q \prod _{n \geqslant 1} (1-q^n)^{24}\) and the identity (ref. Eq. (4.7)):
Using the definition of \(n_{2,\beta }\), we conclude that:
This is exactly the desired result. \(\quad \square \)
We will also need the following later (in the appendix):
Lemma 5.2
For any effective curve class \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\), we have
Proof
Consider a divisor \(D=\textrm{pr}_1^{*}(\alpha ) \in H^2(X)\) with \(d:=\alpha \cdot \beta \ne 0\). By Lemma 1.1 and Eq. (5.1) we have
By Eq. (5.2) and the Yau–Zaslow formula (5.3), we obtain the claim. \(\quad \square \)
6.3 \(\mathrm {DT_4}\) virtual classes
The moduli space \(M_{\beta }\) of one dimensional stable sheaves on X satisfies (e.g. [CMT18, Lem. 2.2]):
where \(M_{S,\beta }\) is the moduli space of one dimensional stable sheaves F on S with \([F]=\beta \) and \(\chi (F)=1\). By a result of Mukai [M], \(M_{S,\beta }\) is a smooth projective holomorphic symplectic variety of dimension \(\beta ^2+2\). In order to determine the \(\mathop {\textrm{DT}}\nolimits _4\) virtual class of \(M_{\beta }\), we first recall:
Definition 5.3
([Sw, Ex. 16.52, pp. 410], [EG, Lem. 5]). Let E be a \(\textrm{SO}(2n,{\mathbb {C}})\)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M. Denote \(E_+\) to be its positive real form Footnote 11. The half Euler class of (E, Q) is
where the sign depends on the choice of orientation of \(E_+\).
Definition 5.4
([EG, KiP, Def. 8.7]). Let E be a \(\textrm{SO}(2n,{\mathbb {C}})\)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M. An isotropic cosection of (E, Q) is a map
such that the composition
is zero. If \(\phi \) is furthermore surjective, we define the (reduced) half Euler class:
as the half Euler class of the isotropic reduction. Here \({\bar{Q}}\) denotes the induced non-degenerate symmetric bilinear form on \((\phi ^{\vee }\mathcal {O}_M)^{\perp }/(\phi ^{\vee }\mathcal {O}_M)\).
We show reduced half Euler classes are independent of the choice of surjective isotropic cosection.
Lemma 5.5
Let E be a \(\textrm{SO}(2n,{\mathbb {C}})\)-bundle with a non-degenerate symmetric bilinear form Q on a connected scheme M and
be a surjective isotropic cosection. Then we can write the positive real form \(E_+\) of E as
such that
Moreover, it is independent of the choice of surjective cosection.
In particular, when \(E=\mathcal {O}^{\oplus 2} \oplus V\) such that \(Q=\begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} \oplus Q|_{V}\), we have
Proof
Let \(E_-:=\sqrt{-1}\cdot E_+\), then \(E=E_+\oplus E_-\). Since \(\phi \) is surjective, \(\phi ^{\vee }\) determines a trivial subbundle \(\mathcal {O}_M\) of E. In the diagram:
for \(v\in \phi ^{\vee }(\mathcal {O}_M)\), write \(v=v_++v_-\) based on above decomposition. The isotropic condition gives
If \(v_+=0\), then \(Q(v_-,v_-)=0\) which implies \(v_-=0\) as Q on \(E_-\) is negative definite. Therefore the composition \(\pi _{\pm }\circ \phi ^{\vee }\) determines a trivial subbundle \(\underline{{\mathbb {R}}}\subset E_\pm \).
We write \((\phi ^{\vee }\mathcal {O}_M)^{\perp }=V_+\oplus V_-\) for \(V_{\pm }=E_{\pm }\bigcap \,(\phi ^{\vee }\mathcal {O}_M)^{\perp }\), which fits in the diagram
Then \(\mathop {\textrm{rank}}\nolimits _{{\mathbb {R}}}V_++\mathop {\textrm{rank}}\nolimits _{{\mathbb {R}}}V_-=4n-2\) and \(\mathop {\textrm{rank}}\nolimits _{{\mathbb {R}}}V_{\pm }\leqslant \mathop {\textrm{rank}}\nolimits _{{\mathbb {R}}}E_{\pm }\). As \((\phi ^{\vee }\mathcal {O}_M)^{\perp }/(\phi ^{\vee }\mathcal {O}_M)\) has an induced non-degenerate symmetric bilinear form, so
Let \(\mathcal {E}_+:=V_+/\underline{{\mathbb {R}}}\), by the metric \(Q|_{V_+}\) on \(V_+\), we may write
Under the identification \(E^{\vee }{\mathop {\cong }\limits ^{Q}} E\), we have
Therefore \(E/(\phi ^{\vee }\mathcal {O}_M)^{\perp }{\mathop {\cong }\limits ^{\phi }}\mathcal {O}_M\). By using the metric \(Q|_{E_+}\) on \(E_+\), we may write
Combining with Eq. (5.9), we have
By definition, the reduced half Euler class is the Euler class of \(\mathcal {E}_+\).
Given two surjective cosections \(\phi _1, \phi _2\), if \(\phi ^{\vee }_1\mathcal {O}_M=\phi ^{\vee }_2\mathcal {O}_M\), then the bundle \(\mathcal {E}_+\) they determine are the same, so are the reduced half Euler classes. If \(\phi ^{\vee }_1\mathcal {O}_M\ne \phi ^{\vee }_2\mathcal {O}_M\), we divide into two cases: (1) when \(\phi ^{\vee }_2\mathcal {O}_M\subseteq (\phi ^{\vee }_1\mathcal {O}_M)^{\perp }\) (which automatically implies \(\phi ^{\vee }_1\mathcal {O}_M\subseteq (\phi ^{\vee }_2\mathcal {O}_M)^{\perp }\)), it is easy to see the corresponding \(\mathcal {E}_+\) has a trivial subbundle \(\underline{{\mathbb {R}}}\), so both reduced half Euler classes vanish. (2) when \(\phi ^{\vee }_2\mathcal {O}_M\nsubseteq (\phi ^{\vee }_1\mathcal {O}_M)^{\perp }\) (hence also \(\phi ^{\vee }_1\mathcal {O}_M\nsubseteq (\phi ^{\vee }_2\mathcal {O}_M)^{\perp }\)), then we have
Taking quotient by \(\phi ^{\vee }_1\mathcal {O}_M\oplus \phi ^{\vee }_2\mathcal {O}_M\), we obtain
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 \(E=\mathcal {O}^{\oplus 2} \oplus V\) such that \(Q=\begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} \oplus Q|_{V}\) follows from this. \(\quad \square \)
Recall a \(\textrm{Sp}(2r,{\mathbb {C}})\)-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.
Lemma 5.6
Let \((V_1,\omega _1)\), \((V_2,\omega _2)\) be a \(\textrm{Sp}(2r,{\mathbb {C}})\) (resp. \(\textrm{Sp}(2,{\mathbb {C}})\)-bundle) on a connected scheme M. Then
defines a \(SO(4r,{\mathbb {C}})\)-bundle whose half Euler class satisfies
Proof
Consider the universal \(\textrm{Sp}(2r,{\mathbb {C}})\)-bundle \({\mathcal {V}}_1\) (resp. \(\textrm{Sp}(2,{\mathbb {C}})\)-bundle \({\mathcal {V}}_2\)) on the classifying space \(\textrm{BSp}(2r,{\mathbb {C}})\) (resp. \(\textrm{BSp}(2,{\mathbb {C}})\)). Their tensor product gives a \(\textrm{SO}(4r,{\mathbb {C}})\)-bundle on \(\textrm{BSp}(2r,{\mathbb {C}})\times \textrm{BSp}(2,{\mathbb {C}})\), whose half Euler class is denoted by \(e^{\frac{1}{2}}({\mathcal {V}}_1\otimes {\mathcal {V}}_2)\).
By the property of half Euler class (e.g. [EG, Prop. 2]):
where we use the fact that the odd Chern classes of \({\mathcal {V}}_i\) vanish in the second equality. Note that above expression is the same as the square of \(c_{2r}({\mathcal {V}}_1)-c_{2r-2}({\mathcal {V}}_1)c_2({\mathcal {V}}_2)\). Since \(H^*(\textrm{BSp}(2r,{\mathbb {C}})\times \textrm{BSp}(2,{\mathbb {C}}))\) is the tensor product of two polynomial rings (e.g. [Sw, Thm. 16.10]), hence it is an integral domain. Therefore
Since this construction is universal, we are done. \(\quad \square \)
Finally, we can determine the (reduced) virtual class of \(M_{\beta }\).
Theorem 5.7
For certain choice of orientation, we have
Proof
Under the isomorphism (5.8):
a universal family \({\mathbb {F}}\) of \(M_{\beta }\) satisfies
where \({\mathbb {F}}_S\) is a universal sheaf of \(M_{S,\beta }\) and \(\Delta _T\) denotes the diagonal in \(T\times T\).
Then the obstruction sheaf of \(M_{\beta }\):
is a vector bundle with quadratic form given by Serre duality pairing. The two trivial subbundles \(\mathcal {E}xt_{\pi _{M_{S,\beta }}}^2({\mathbb {F}}_S,{\mathbb {F}}_S)\), and \(\mathcal {E}xt_{\pi _{M_{S,\beta }}}^0({\mathbb {F}}_S,{\mathbb {F}}_S)\boxtimes \wedge ^2T_T\) are Serre dual to each other. By Lemmata 5.5, 5.6, we are done. \(\quad \square \)
6.4 \(\mathop {\textrm{DT}}\nolimits _4\) invariants and proof of conjectures
In this section, we determine all \(\mathop {\textrm{DT}}\nolimits _4\) invariants of \(S \times T\). Let \(\gamma ,\gamma '\in H^{4}(X)\) be cohomology classes and decompose them as
according to the Künneth decomposition:
Fix also a divisor class
and a curve class
Theorem 5.8
Let \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\) be any effective curve class. With respect to the choice of orientation (5.10), we have
where \(N_0(l)\), \(N_1(l)\) are defined in Eqs. (5.4), (4.9) respectively and
The above theorem immediately implies the following:
Corollary 5.9
Conjecture 2.2 holds for the product \(X = S \times T\) and \(\beta \in H_2(S,{\mathbb {Z}})\subseteq H_2(X,{\mathbb {Z}})\).
Proof
This follows by inspection using Theorem 5.8 on \(\mathop {\textrm{DT}}\nolimits _4\) invariants and Proposition 5.1 and Lemma 5.2 for the GV/GW invariants respectively. \(\quad \square \)
Another remarkable consequence of Theorem 5.8 is that all \(\mathop {\textrm{DT}}\nolimits _4\) invariants of \(S \times T\) depend upon the curve class \(\beta \) only via the square \(\beta ^2\) and not the divisibility. More precisely, given pairs \((S,\beta )\) and \((S',\beta ')\) of a K3 surface and an effective curve class such that \(\beta ^2 = \beta '^2\), let
be any real isometry such that \(\varphi (\beta ) = \beta '\). Extend \(\varphi \) to the full cohomology by setting \(\varphi (1) = 1\) and \(\varphi (\textsf{p}_S) = \textsf{p}_{S'}\) where \(\textsf{p}_S \in H^4(S,\mathbb {Z})\) is the point class.
Corollary 5.10
With respect to the choice of orientation (5.10), we have
for any \(\gamma _i \in H^{*}(X)\) and \(k_i \geqslant 0\).
This raises the question whether a similar independence of the divisibility holds for Donaldson–Thomas invariants of holomorphic symplectic 4-folds more generally.
6.5 Proof of Theorem 5.8
We split the proof in two parts.
Proof of Theorem 5.8 Part (i), (v)
We begin with part (v). For the diagonal embedding \(\Delta : T\rightarrow T\times T\), by GRR, we have
By Eqs. (5.11) and (5.12), the primary insertion becomes
where \(\pi _S\), \(\pi _{M_{S,\beta }}\) are projections to each factor of \(S\times M_{S,\beta }\). Therefore
where “others” lie in \(H^2(M_{S,\beta })\otimes H^2(T)\). Combining with Theorem 5.7, we get
There exists a Hilbert–Chow map
to the linear system \(|\beta |\) and \(\mathop {\textrm{ch}}\nolimits _1({\mathbb {F}}_S)=(\text {id}_S\times \textrm{HC})^*[{\mathcal {C}}]\), where \({\mathcal {C}}\) is the universal curve of the linear system:
Since \([{\mathcal {C}}]=p^*\beta +q^*h\) for the hyperplane class h of \(|\beta |\), we have
By Theorem 4.1, we have
Therefore Eq. (5.13) becomes
Finally, since \(M_{S,\beta }\) is deformation equivalent to \(S^{[d]}\) (\(\beta ^2=2d-2\)) (e.g. [Y09, Cor. 3.5, pp. 136]), they have the same Euler numbers:
which is given by \(N_0(\beta ^2/2)\) due to Göttsche [G90]. This proves (v).
For (i), we similarly have
\(\square \)
Proof of Theorem 5.8 Parts (ii-iv) and (vi-viii)
We first express the \(\mathop {\textrm{DT}}\nolimits _4\) descendent invariants as integrals on \(M_{S,\beta }\). Let \({\mathbb {F}}_S^{\textrm{norm}}\) be the normalized universal sheaf on \(M_{S,\beta }\times S\), i.e.
where \(\pi _{M_{S,\beta }}: M_{S,\beta }\times S\rightarrow M_{S,\beta }\) is the projection. (Note that in the proof of Parts (i), (v) of Theorem 5.8, we do not have to take the normalized universal sheaf, since we have only used operators on \({\mathbb {F}}_S\) invariant under \({\mathbb {F}}_S \mapsto {\mathbb {F}}_S^\mathrm{{norm}}\), e.g. \(\mathop {\textrm{ch}}\nolimits _1({\mathbb {F}}_S)=\mathop {\textrm{ch}}\nolimits _1({\mathbb {F}}_S^\mathrm{{norm}})\). For the remaining parts of the theorem, we need to take the normalized one.) By Eq. (5.11), we have
Hence the family \({\mathbb {F}}\) is normalized if and only if \({\mathbb {F}}_S\) is so. By (5.12), we obtain
By base change to a point, we have
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 \(M_{S,\beta }\) and \(S^{[d]}\) are deformation equivalent they share the same Fujiki constants:
where \(\beta ^2 = 2d-2\). This implies the claim. Finally for (vii) and (viii), we similarly find:
\(\square \)
7 Cotangent Bundle of \(\mathbb {P}^2\)
We consider the geometry \(X = T^{*} \mathbb {P}^2\). There is a natural identification of curve classes:
where \(\ell \subset \mathbb {P}^2\) is a line.
7.1 GW and GV invariants
Let \(H \in H^2(T^{*} \mathbb {P}^2)\) be the pullback of hyperplane class. We identify \(H_2(T^{*} \mathbb {P}^2, \mathbb {Z}) \equiv \mathbb {Z}\) by its degree against H.
Proposition 6.1
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.
\(\square \)
Based on Definition 1.5, 1.6, 1.7, we then obtain the following:
Corollary 6.2
In particular, Conjecture 1.9 holds for \(T^{*} \mathbb {P}^2\).
Proof
In genus 0 and 1, this follows from a direct calculation using the definition and that
In genus 2, it remains to determine the nodal invariant \(N_{\textrm{nodal}, \beta }\). In \(H^{*}(\mathbb {P}^2 \times \mathbb {P}^2)\), we have
Using Lemma 1.1 and Eq. (1.2) we find that
The vanishing \(n_{2,1} =0\) follows now from a direct calculation. \(\quad \square \)
7.2 \(\mathop {\textrm{DT}}\nolimits _4\) invariants
Let \(M_{T^*{\mathbb {P}}^2,d}\) (resp. \(M_{{\mathbb {P}}^2,d}\)) be the moduli scheme of compactly supported one dimensional stable sheaves F on \(T^*{\mathbb {P}}^2\) (resp. \({\mathbb {P}}^2\)) with \([F]=d[\ell ]\) (\(d\geqslant 1\)) and \(\chi (F)=1\).
Lemma 6.3
Let \(\iota : {\mathbb {P}}^2\rightarrow T^*{\mathbb {P}}^2\) be the zero section. Then the pushforward map
is an isomorphism.
Proof
The map \(\iota _{*}\) is obviously injective. We show that \(\iota _{*}\) is also surjective. As \(T^{*}{\mathbb {P}}^2\) admits a birational contraction \(T^{*}{\mathbb {P}}^2 \rightarrow Y\) which contracts the zero section \({\mathbb {P}}^2 \hookrightarrow T^{*}{\mathbb {P}}^2\) to \(0 \in Y\) and Y is affine, any one dimensional sheaf on \(T^{*}{\mathbb {P}}^2\) is set theoretically supported on the zero section. It is enough to show that any one dimensional stable sheaf F on \(T^{*}{\mathbb {P}}^2\) is scheme theoretically supported on the zero section.
Recall the following fact as stated in [CMT18, Lem. 2.2]: let \(g :Z \rightarrow T\) be a morphism of \({\mathbb {C}}\)-schemes, and take a closed point \(t \in T\). Let \(Z_t \subset Z\) be the scheme theoretic fiber of g at t. Suppose that \(F \in \mathop {\textrm{Coh}}\nolimits (Z)\) is set theoretically supported on \(Z_t\) and satisfies \(\mathop {\textrm{End}}\nolimits (F)={\mathbb {C}}\). Then F is scheme theoretically supported on \(Z_t\).
It should be well-known (and easy) that the scheme theoretic fiber of \(T^{*}{\mathbb {P}}^2 \rightarrow Y\) at \(0 \in Y\) is the reduced zero section, then surjectivity of \(\iota _{*}\) follows from the above fact. As we cannot find its reference, we give another argument here. Consider the closed embedding \(T^{*}{\mathbb {P}}^2 \subset {\mathcal {O}}_{{\mathbb {P}}^2}(-1)^{\oplus 3}\) induced by the Euler sequence on \({\mathbb {P}}^2\). Note that \({\mathcal {O}}_{{\mathbb {P}}^2}(-1)^{\oplus 3}\) is an open subscheme of \([{\mathbb {C}}^6/{\mathbb {C}}^{*}]\), where \({\mathbb {C}}^{*}\) on \({\mathbb {C}}^6\) by
and corresponds to \((x_1, x_2, x_3) \ne (0, 0, 0)\). The stack \([{\mathbb {C}}^6/{\mathbb {C}}^{*}]\) admits a good moduli space
One can easily calculates that the scheme theoretic fiber of the above morphism restricted to \((x_1, x_2, x_3) \ne (0, 0, 0)\) is \((y_1=y_2=y_3=0)\). It follows that the scheme theoretic fiber of \(T^{*}{\mathbb {P}}^2 \subset \mathcal {O}_{{\mathbb {P}}^2}(-1)^{\oplus 3} \rightarrow T\) at \(0 \in T\) is the reduced zero section \({\mathbb {P}}^2\). As T is affine, any one dimensional stable sheaf is set theoretically supported on the (scheme theoretic) fiber of \(0 \in T\). Using the above fact, it is also scheme theoretically supported on it. Therefore \(\iota _{*}\) is surjective.
Since \(M_{{\mathbb {P}}^2, d}\) is smooth and \(\iota _{*}\) is bijective on closed points, it remains to show that \(\iota _{*}\) induces an isomorphisms on tangent spaces. For a one dimensional stable sheaf F on \({\mathbb {P}}^2\), the tangent space of \(M_{T^{*}{\mathbb {P}}^2, d}\) at \(\iota _{*}F\) is
By the Euler sequence and stability, we have
Therefore \(\iota _{*}\) induce an isomorphism of tangent spaces. \(\quad \square \)
Then the following result is straightforward.
Lemma 6.4
Under the isomorphism (6.1), we have
Here \(\textrm{PD}\) denotes the Poincaré dual, \(e_{\textrm{red}}^{\frac{1}{2}}\) is the reduced half Euler class as in Definition 5.4, \({\mathbb {F}}\) denotes a universal sheaf of \(M_{{\mathbb {P}}^2,d}\) and \(\pi _M: M_{{\mathbb {P}}^2,d}\times {\mathbb {P}}^2 \rightarrow M_{{\mathbb {P}}^2,d}\) is the projection.
Proposition 6.5
For certain choice of orientation, we have
In particular, for \(X=T^*{\mathbb {P}}^2\), we have
Proof
We present the proof of \(d=2\) case (the \(d=1\) case follows similarly). The support map
is an isomorphism. The normalized universal sheaf satisfies \({\mathbb {F}}_{\textrm{norm}}=\mathcal {O}_{{\mathcal {C}}}\) for the universal (1, 2)-divisor \({\mathcal {C}}\hookrightarrow {\mathbb {P}}^5\times {\mathbb {P}}^2\). Let \(\pi _M: M_{{\mathbb {P}}^2,2}\times {\mathbb {P}}^2\rightarrow M_{{\mathbb {P}}^2,2}\) be the projection. Bott’s formula implies
Therefore, we have
By Grothendieck–Verdier duality, it is easy to see
is a maximal isotropic subbundle of \(\textbf{R}{\mathcal {H}}om_{\pi _M}(\mathcal {O}_{{\mathcal {C}}},\mathcal {O}_{{\mathcal {C}}}\boxtimes T^*{\mathbb {P}}^2)[1]\). Hence the reduced virtual class satisfies
Let \(h\in H^2({\mathbb {P}}^5)\) 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 factorFootnote 12 which implies the vanishing of (reduced) invariants. \(\quad \square \)
8 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)
-
(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.
8.1 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 \(\mathbb {C}\times \mathbb {H}\rightarrow \mathbb {C}\) in \((z,\tau )\) where \(y = e^{2 \pi i (z+1/2)}\) and \(q = e^{2 \pi i \tau }\). 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 \(\frac{d}{dG_2} F(y,q)\), see [vIOP].
8.2 Curve classes
Since \(X:=S^{[2]}\) is irreducible hyperkähler, recall from Sect. 4.1 the integral, even, non-degenerate Beauville–Bogomolov–Fujiki form
Since \({\textsf{q}}\) is non-degenerate, we obtain an inclusion of finite index
where we write \(( -, - )\) for the induced inner product on \(H^2(X,\mathbb {Z})\). By extending \({\textsf{q}}\), we hence obtain a \(\mathbb {Q}\)-valued non-degenerate quadratic form
Given a class \(\beta \in H_2(X,\mathbb {Z})\), we write
for its dual with respect to the Beauville–Bogomolov–Fujiki form \((-, -)\). We have
Let also
be the inverse of the Beauville–Bogomolov–Fujiki form, i.e. the image of \({\textsf{q}} \in H^{2}(X)^{*} \otimes H^{2}(X)^{*}\) under the natural isomorphism \(H^2(X,\mathbb {Q})^{*} \cong H^2(X,\mathbb {Q})\) induced by \({\textsf{q}}\).
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 \(z \mapsto z+\tau \) (in generators this means it is independent of \(\frac{1}{\Theta } y\frac{d}{d y} \Theta \); we will only encounter such kind here).
For any class \(\beta \in H_2(X,\mathbb {Z})\), the \(\beta \)-coefficient of F(y, q),
is defined to be the coefficient of \(q^d y^k\) for any \(d, k \in \mathbb {Z}\) such that \((\beta ,\beta ) = 2d-k^2/2\).
Remark 7.2
The choice of d, k is not unique, but the coefficient \(F_{\beta }\) is independent of the choice by the elliptic transformation law of Jacobi forms [EZ].
8.3 Gromov–Witten invariants
Consider the quasi-Jacobi forms:
The first coefficients read:
The following completely determines all primary genus zero Gromov–Witten invariants of \(X=S^{[2]}\) in primitive curve classes (see [O21c] for the imprimitive case):
Theorem 7.3
([O18, O21a]) Let \(\beta \in H_2(X,\mathbb {Z})\) be a primitive curve class. We have
as well as:
Modulo conjectures we have the following evaluation of genus 1, 2 Gromov–Witten invariants:
Theorem 7.4
Assume Conjectures A and C of [O22b]. Then for any primitive curve class \(\beta \in H_2(X,\mathbb {Z})\), in genus 1, we have:
where
In genus 2, we have
where
The first coefficients of \(\mathcal {A}\) and \(\mathcal {B}\) and I are as follows:
8.4 Proof of Theorem 7.4: Holomorphic anomaly equations
The global Torelli theorem for hyperkähler varieties implies that the Hilbert scheme \(S^{[2]}\) has a large monodromy group, we refer to [M11] for an introduction. In our case, as [OSY, §2.7] or [O21a] the monodromy implies that for a primitive curve class, we have
for some constants \(\mathcal {A}_{\beta }, \mathcal {B}_{\beta }, I_{\beta } \in \mathbb {Q}\) which only depend on the square \((\beta , \beta )\) of the class.
To determine these constants, we can work with an elliptic K3 surface \(S \rightarrow \mathbb {P}^1\) with section. The Hilbert scheme in this case has an induced Lagrangian fibration \(S^{[2]} \rightarrow \mathbb {P}^2\) with section. Let B, F be the section and fiber class of S respectively, and let \(A \in H_2(S^{[2]},\mathbb {Z})\) be the class of the locus of non-reduced subschemes supported at a single point. There exists a natural isomorphism
given by the Nakajima basis [O18, §0.2]. For \(h \geqslant 0\) and \(k \in \mathbb {Z}\), we consider the classes
which are of square
The set of these squares contains all possible squares of curve classes \(\beta \in H_2(X,\mathbb {Z})\), we see that any \((X,\beta )\) can be deformed to \((S^{[2]}, \beta _{h,k})\) for some h, k. We form the generating series
The \(\textsf{F}_{g}\)’s are conjectured to be quasi-Jacobi forms and that their formal derivatives \(\frac{d}{dG_2} \textsf{F}_g\) are determined by a holomorphic anomaly equation [O22b].
Below we will freely use the language of Nakajima operators
for all \(i \in \mathbb {Z}\) and \(\alpha \in H^{*}(S)\), where we follow the conventions of [NOY]. Given \(\gamma _1, \ldots , \gamma _k \in H^{*}(S)\) and \(n_1, \ldots , n_k \geqslant 1\), we will write
where the unit \(1 \in H^{*}(S^{[0]})\) is also sometimes called the vacuum.
Proof of Theorem 7.4: Genus 1 case
By [O22b, Conj. C], we have for any \(\gamma \in H^4(X)\) the following holomorphic-anomaly equation:
where \(W = B+F\) and
which is viewed here both as a morphism \(H^{*}(S^{[2]}) \rightarrow H^{*}(S^{[2]})\) and by Poincaré duality in the last line as a class in \(H^{*}(S^{[2]} \times S^{[2]})\) (we let \(\mathfrak {q}_i'\) denote the Nakajima operator acting on the second factor of \(H^{*}(S^{[n]}) \otimes H^{*}(S^{[m]})\)). Moreover, \(\textsf{F}_g(\lambda _1; \ldots )\) stands for the (obvious) generating series where we integrate also over the tautological class \(\lambda _1\), see [O22b].
We consider the invariant \(\textsf{F}_1( \mathfrak {q}_1(F)^2 1)\). Using the holomorphic anomaly equation above, the known results in genus 0 (Theorem 7.3) and the discussion in [O18, OP] on how to reduce the series \(\textsf{F}_1(\lambda _1;... )\) to genus 0 invariants, we have:
Integrating with respect to \(G_2\) yields
where \(\wp ^2\) does not appear, because it would yield the only pole on the left hand side (contradicting Conjecture A of [O22b] or also monodromy invariance). By Proposition 6.1 and since every curve in class \(B-A\) is a line in \(\mathbb {P}^2 \cong B^{[2]} \subset S^{[2]}\) (see e.g. [O18, Lemma 6] for the geometry of curves in class \(B+hF+kA\)), we have
Solving for a one finds \(a = 1/96\), and hence
Similarly, we have
where we used that \(U(c_2(X)) = 30 \mathfrak {q}_1(F) \mathfrak {q}_1(1) 1\). This yields
where, since there are no poles on the left hand side, the poles in \((D_z \wp )^2\) and \(\wp ^3\) 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:
Solving with these conditions for a and b, we obtain
Finally, by Lemma 7.5 below and the definition of \(\mathcal {A}_{\beta }, \mathcal {B}_{\beta }\) in (7.1), the functions
satisfy:
This proves the claim by solving for \(\mathcal {A}\) and \(\mathcal {B}\).
We remark that determining \(\textsf{F}_{1}( \mathfrak {q}_1(F)^2 1 )\) 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 \(\beta \in \{ B, B+F+A \}\)). For \(\textsf{F}_1(c_2(X))\) the system is likewise overdetermined: we only used 2 of the 3 available constraints. \(\quad \square \)
Proof of Theorem 7.4: Genus 2 case
Using Lemma 7.5 below, the standard intersections
and the genus 1 part of Theorem 7.4, the holomorphic anomaly equation of [O22b] reads:
Integration with respect to \(G_2\) yields
for some \(a,b \in \mathbb {C}\). Here we used that \(F_2\) is determined up to the functions \(\wp ^4, \wp (y \frac{d}{dy} \wp )^2, \wp ^2 E_4, E_4\) and that the poles in the first of these functions have to cancel which replaces them with a \(E_6 \wp \) term and then that \(\wp ^2 E_4\) can also not appear because of holomorphicity. Finally, using the following evaluations (ref. Proposition 6.1, Lemma 3.3):
yields that \(a=-13/18432\) and \(b = -1/96\) and thus
This implies the result by monodromy invariance (we even have one more condition to spare, namely the vanishing of \(\big \langle \varnothing \big \rangle _{2,B}^{\mathop {\textrm{GW}}\nolimits }\)). \(\quad \square \)
Lemma 7.5
Let \({\widetilde{\beta }}_{d,k} = W + dF + kA\) and let \(h_{d,k} = {\widetilde{\beta }}_{d,k}^{\vee }\) be the dual. Then
Proof
Let \(\delta = c_1( \mathcal {O}_S^{[2]}) = -\frac{1}{2} \Delta _{S^{[2]}}\) and \(D(\alpha ) = \mathfrak {q}_1(\alpha ) \mathfrak {q}_1(1)1\). We have
This yields, for example
The other cases are similar (use that \(\mathfrak {q}_1(W) \mathfrak {q}_1(F) \cdot \delta ^2 = \mathfrak {q}_1(\textsf{p}) \mathfrak {q}_1(1) \cdot \delta ^2 = -1\)). For the last expression we use the Fujiki constant \(C(c_2) = 30\). \(\quad \square \)
8.5 Genus 1 Gopakumar–Vafa invariants
A hyperkähler variety X is of \(K3^{[2]}\)-type if it is deformation equivalent to the Hilbert scheme \(S^{[2]}\) for a K3 surface S. For any primitive curve class \(\beta \in H_2(X,\mathbb {Z})\), we define the genus 1 Gopakumar–Vafa class
by
where \(n_{1,\beta }(\gamma )\) is given in Definition 1.6. In an ideal geometry (ref. §1.5), \(n_{1,\beta }\) is the class of the surface swept out by the elliptic curves in class \(\beta \).
Our discussion above leads to the following formula. Define
and recall the series \(\mathcal {A}, \mathcal {B}\) from Theorem 7.4.
Theorem 7.6
Assume Conjectures A and C of [O22b]. For any hyperkähler variety X of \(K3^{[2]}\) type and for any primitive curve class \(\beta \in H_2(X,\mathbb {Z})\), we have
where \(a_{\beta } = \mathcal {A}_{\beta } + \mathcal {A}'_{\beta }\) and \(b_{\beta } = \mathcal {B}_{\beta } + \mathcal {B}'_{\beta }\).
Proof
Since the Chern class \(c_2(X)\) is monodromy invariant, we can write
for some \(\mathcal {A}'_{\beta }, \mathcal {B}'_{\beta }\). Using Theorem 7.3, one computes that these are precisely the \(\beta \)-coefficients of the functions \(\mathcal {A}', \mathcal {B}'\) defined above. The claim now follows from Theorem 7.4 and the definition of genus 1 Gopakumar–Vafa invariants. \(\square \)
The integrality conjecture for Gopakumar–Vafa invariants (Conjecture 1.9) would imply that \(n_{1,\beta } \in H^4(X,\mathbb {Q})\) is an integral class. We give the following criterion:
Lemma 7.7
\(n_{1,\beta }\) is integral, i.e. lies in \(H^4(X, \mathbb {Z})\) if and only if the following holds:
-
(i)
If \((\beta ,\beta ) \in 2 \mathbb {Z}\), then \(a_{\beta }\) is an even integer and \(3 b_{\beta } \in \mathbb {Z}.\)
-
(ii)
If \((\beta , \beta ) = 2d - \frac{1}{2}\), then \(a_{\beta }\), \(24 b_{\beta }\), \(\frac{1}{8} a_{\beta } - 3 b_{\beta }\) all lie in \(\mathbb {Z}\).
Proof
Using deformation invariance (e.g. [O21a, Cor. 2]), we may work with \(X=S^{[2]}\) for an elliptic K3 surface S with \(\mathop {\textrm{Pic}}\nolimits (S)\) generated by the class of a section B and the fiber class F, Moreover, we can use the curve class
for \(d \geqslant -1\) and \(k \in \{ 0, 1 \}\). With the notation of Lemma 7.5, we then have:
By the main result of [Nova], a basis for the Hodge classes
is given by the 7 classes
The class \(n_{1,\beta }\) has the following expansion in this integral basis:
If \((\beta , \beta ) \in \mathbb {Z}\) then \(k=0\), so integrality of \(n_{1,\beta }\) impies (by the first summand) that \(a_{\beta } \in 2\mathbb {Z}\) and by the last summand that \(3 b_{\beta } \in \mathbb {Z}\), and this is clearly sufficient. If \((\beta , \beta ) = 2d-\frac{1}{2}\), we have \(k=1\), which gives \(a_{\beta }, 24 b_{\beta }, \frac{1}{8} a_{\beta } - 3 b_{\beta } \in \mathbb {Z}\) and this is clearly sufficient. \(\square \)
The criterion of the lemma can be easily checked using a computer program. We obtain:
Corollary 7.8
Under the assumptions of Theorem 7.6, \(n_{1,\beta }\) is integral for all \((\beta ,\beta ) \leqslant 100\).
Example 7.9
(A real life example) Let \(F(Y) \subset \textrm{Gr}(2,6)\) be the Fano variety of lines on a very general cubic 4-fold \(Y \subset \mathbb {P}^5\). Let \(\mathcal {U}\subset \mathcal {O}_{\textrm{Gr}}^{\otimes 6}\) be the universal subbundle on \(\textrm{Gr}(2,6)\) and set
The unique primitive curve class is \(\beta = \frac{1}{2} g^{\vee }\) and is of square \((\beta ,\beta )=3/2\) since \((g,g)=6\). The basic geometry of these classes is discussed in [Ot], in particular we have
Theorem 7.6 implies that the surface in F(Y) swept out by elliptic curves in class \(\beta \) has class:
This is indeed integral and effective (the surface of lines meeting a given line is \(\frac{1}{3} (g^2 - c))\).
8.6 Genus 2 Gopakumar–Vafa invariants
Since we can control now all Gromov–Witten invariants for \(S^{[2]}\) in arbitrary genus (for primitive classes), it is also straightforward to compute genus 2 Gopakumar–Vafa invariants (see [NO] for the computation of the nodal invariants):
Theorem 7.10
Assume Conjectures A and C of [O22b]. For any hyperkähler variety X of \(K3^{[2]}\) type and for any primitive curve class \(\beta \in H_2(X,\mathbb {Z})\), we have
where
Using a computer program, we immediately obtain:
Corollary 7.11
Under the assumptions of Theorem 7.10, \(n_{2,\beta }\) is integral for all \((\beta ,\beta ) \leqslant 138\).
8.7 Genus 0 Gopakumar–Vafa invariants
For completeness, we also give a proof of the integrality of genus 0 Gopakumar–Vafa invariants discussed in the introduction.
Proof of Theorem 0.11
Inverting the definition of genus 0 Gopakumar–Vafa invariants, we have
where \(\mu (k)\) is the Möbius function. Consider also the “BPS invariants" introduced in [O21a]:
Then it is straightforward to show that (see [O21a, Def. 1] for the notation \([-]\)):
-
(i)
If \(\left[ \frac{\beta }{\div (\beta )} \right] = 0\), then \(n_{0,\beta }(\gamma _1, \ldots , \gamma _n) = {\widetilde{n}}_{0,\beta }(\gamma _1, \ldots , \gamma _n)\),
-
(ii)
If \(\left[ \frac{\beta }{\div (\beta )} \right] = 1\), then
$$\begin{aligned}{} & {} n_{0,\beta }(\gamma _1, \ldots , \gamma _n) \\{} & {} = {\left\{ \begin{array}{ll} {\widetilde{n}}_{0,\beta }(\gamma _1, \ldots , \gamma _n) &{} \text {if } \div (\beta ) \text { is odd or } 4 | \div (\beta ), \\ {\widetilde{n}}_{0,\beta }(\gamma _1, \ldots , \gamma _n) - {\widetilde{n}}_{0,\beta /2}(\gamma _1, \ldots , \gamma _n) &{} \text {if } \div (\beta ) \text { is even but} \div (\beta /2) \text { is odd}. \end{array}\right. } \end{aligned}$$
Hence it suffices to show that \({\widetilde{n}}_{0,\beta }(\gamma )\) is integral for any effective curve class \(\beta \in H_2(X,\mathbb {Z})\). As conjectured in [O21a] and proven in [O21c], the invariant \({\widetilde{n}}_{0,\beta }(\gamma )\) only depends on
Hence we may assume that \(\beta \) is primitive. But here the result follows since for a very general pair \((X,\beta )\), where X is a hyperkähler variety of \(K3^{[2]}\)-type, it is well-known that \(\overline{M}_{0,1}(X,\beta )\) is an algebraic space (there are no non-trivial automorphisms) of expected dimension (e.g. [OSY, §1.1]), therefore
is integral (the same argument also shows the integrality of \(n_{0,\beta }(\gamma _1,\ldots ,\gamma _n)\) for arbitrary number of markings if \(\beta \) is primitive). \(\quad \square \)
Notes
We will only work with the reduced virtual class in this paper, hence we will denote it simply by \([ - ]^{\textrm{vir}}\).
A naive model for (ii,iii) would be for the image of \(\mathcal {C}_\beta ^0\) in X to be \((S \times \mathbb {P}^1) / ((s,0) \sim (gs, \infty ))\), where \(g \in \mathop {\textrm{Aut}}\nolimits (S)\) is an automorphism with a finite number of fixed points (which lead to nodal curves).
This is related to what is called the double point number.
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 \(\beta = (\beta _1, \beta _2)\) with both \(\beta _i\) non-zero effective, any deformation that keeps \(\beta \) Hodge, will keep both \(\beta _i\) Hodge. In particular \(\beta \) stays reducible under deformations.
If \(E^{\bullet } \rightarrow \mathbb {L}_M\) is a perfect obstruction theory, then the associated virtual tangent bundle is \(T^{\textrm{vir}}_M:= (E^{\bullet })^{\vee }\).
The naïve splitting of the virtual class \([ \overline{M}_{2,0}(X,\beta )]^{\textrm{vir}}\) as the sum
$$\begin{aligned}{} & {} [ \overline{M}_{2,1} \times \overline{M}_{0,1}(X,\beta ) ] \left[ \frac{ c( \mathbb {E}^{\vee } \otimes \mathop {\textrm{ev}}\nolimits _1^{*}(T_X)) }{1 - \psi _1 - \psi _1'} \right] _{7} + \frac{1}{2} [\overline{M}_{1,1}^2\\{} & {} \quad \times \overline{M}_{0,2}(X,\beta )]^{\text {vir}} \left[ \frac{ c( \mathbb {E}_a^{\vee } \otimes \mathop {\textrm{ev}}\nolimits _1^{*}(T_X)) }{1 - \lambda _{a,1} - \psi _1} \frac{ c( \mathbb {E}_b^{\vee } \otimes \mathop {\textrm{ev}}\nolimits _2^{*}(T_X)) }{1 - \lambda _{b,1} - \psi _2} \right] _{6} \end{aligned}$$does not hold. The long detour to the relative virtual class \([M]^{\textrm{rel}}\) is necessary to decompose the virtual class!
Their construction gives a Chow cycle element, here we use its image in Borel-Moore homology under cycle map for convenience.
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.
This means a real half dimensional subbundle such that Q is real and positive definite on it. By homotopy equivalence \(\textrm{SO}(m,{\mathbb {C}})\sim \textrm{SO}(m,{\mathbb {R}})\), it exists and is unique up to isomorphisms.
We thank Sergej Monavari for his observation and help on this.
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Acknowledgements
We thank Chen Jiang, Young-Hoon Kiem, Sergej Monavari, Rahul Pandharipande and Hyeonjun Park for helpful discussions. A special thanks goes to Luca Battistella for several useful discussions about the splitting of the Gromov–Witten virtual class in genus 2 and to Sergej Monavari for his help on a localization computation. We are grateful to the referee for many insightful comments and suggestions which improve the exposition of this paper. Y. C. is partially supported by RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), World Premier International Research Center Initiative (WPI), MEXT, Japan, JSPS KAKENHI Grant Number JP19K23397 and Royal Society Newton International Fellowships Alumni 2020 and 2021. G.O. is partially supported by Deutsche Forschungsgemeinschaft (DFG) - OB 512/1-1. Y. T. is partially supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research grant (No. 19H01779) from MEXT, Japan.
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Appendix A. Imprimitive Curve Classes
Appendix A. Imprimitive Curve Classes
Let X be a holomorphic symplectic 4-fold. We consider here the Gromov–Witten, Gopakumar–Vafa and \(\mathop {\textrm{DT}}\nolimits _4\) invariants of X in a (possibly imprimitive) curve class \(\beta \in H_2(X,\mathbb {Z})\). As discussed in Sect. 1.4, the ideal geometry of curves in this case is very difficult to control. Moreover, there are only very few geometries where both the GW and \(\mathop {\textrm{DT}}\nolimits _4\) invariants can be completely computed, and these geometries do not reflect the general structure of the GW/GV/\(\mathop {\textrm{DT}}\nolimits _4\) invariants. As such, the general definition of Gopakumar–Vafa invariants \(n_{g,\beta }\) for \(g>0\) and imprimitive \(\beta \) is not clear at this point. Nevertheless, in this section we define genus 1 Gopakumar–Vafa invariants for imprimtive curve classes in the two geometries where all the invariants can be controlled, and then prove a GV/\(\mathop {\textrm{DT}}\nolimits _4\) relation.
1.1 A.1. Genus 1 Gopakumar–Vafa invariants
There are two geometries where we know all GW and \(\mathop {\textrm{DT}}\nolimits _4\) invariants:
-
(i)
The Embedded Rational Curve family of Sect. 3
-
(ii)
The product of two K3 surfaces \(S \times T\) for all curve classes which lie in \(H_2(S,\mathbb {Z})\).
These geometries are special because all primary GW and \(\mathop {\textrm{DT}}\nolimits _4\) invariants with insertion \(c_2(T_X)\) vanish. This implies that the genus 1 Gopakumar–Vafa invariants do not have any contributions from genus 0 curves. Based on a computation in the ideal geometry following Sect. 1.6.1, one expects that
where \(\sigma (k):=\sum _{l|k}l\) and \((\cdots )\) stands for contributions from genus 0 curves. This suggests that for the geometries (i) and (ii), there should be no contributions in genus 0. Hence we make the following adhoc definition in this case:
Definition A.1
Let X be a holomorphic symplectic 4-fold and \(\beta \in H_2(X,\mathbb {Z})\) be an effective curve class of type (i) or (ii) above. For any \(\gamma \in H^4(X,\mathbb {Z})\), we define \(n_{1, \beta }(\gamma )\) by:
We also introduce the following:
Definition A.2
For any \(\gamma \in H^4(X, \mathbb {Z})\), we define \(n_{0, \beta }(\gamma ;\psi )\in {\mathbb {Q}}\) by the multiple cover formula:
We then can prove the following generalization of the genus one part of Conjecture 2.2:
Proposition A.3
Let X be a holomorphic symplectic 4-fold, and \(\beta \in H_2(X,\mathbb {Z})\) be an effective curve class of type (i) or (ii) above. For certain choice of orientation, we have
Remark A.4
The second part in the RHS of the above equality is also consistent with the ideal geometry computation (see Remark 2.6).
1.2 A.2. Proof of Proposition A.3: embedded rational curve family
Let X be a holomorphic symplectic 4-fold and let \(\beta \in H_2(X,\mathbb {Z})\) be a curve class which satisfy conditions (i-iii) of Sect. 3. Then by Lemma 3.3, all genus 1 GV invariants \(n_{1,\beta }(\gamma )\) vanish. Moreover, by Lemma 3.5 and with the notation of that section, we have
Similarly, by Lemma 3.7, for certain choice of orientation we have
This implies the claim. \(\quad \square \)
1.3 A.3. Proof of Proposition A.3: \(K3\times K3\)
Let \(X = S \times T\) and \(\beta \in H_2(S,\mathbb {Z})\) be an effective curve class. Consider a cohomology class \(\gamma \in H^4(X,\mathbb {Z})\) with Künneth decomposition
The claim follows from the following two lemmata:
Lemma A.5
\(n_{0, \beta }(\gamma ;\psi )=-2 A_1\cdot N_{0}\left( \frac{\beta ^2}{2}\right) \).
Proof
This follows from Lemma 5.2 and the definition. \(\quad \square \)
Lemma A.6
We have
Proof
Recall that by (5.7) we have
Hence by Eq. (A.2), \(n_{1,\beta }(\gamma )\)’s are the unique (recursively defined) integers which satisfy the relation
We show that the integers \(n_{1,\beta }(\gamma )\) defined by Eq. (A.4) satisfy the relation (A.5). This then completes the proof. Indeed, using Eq. (A.4), the right hand side of Eq. (A.5) becomes
which is precisely the left hand side of Eq. (A.5). Hence Eq. (A.4) holds. \(\quad \square \)
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. \(\quad \square \)
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Cao, Y., Oberdieck, G. & Toda, Y. Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds. Commun. Math. Phys. 405, 26 (2024). https://doi.org/10.1007/s00220-023-04882-8
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DOI: https://doi.org/10.1007/s00220-023-04882-8