Gromov-Witten theory and Noether-Lefschetz theory for holomorphic-symplectic varieties
aa r X i v : . [ m a t h . AG ] F e b GROMOV-WITTEN THEORY ANDNOETHER-LEFSCHETZ THEORYFOR HOLOMORPHIC-SYMPLECTIC VARIETIES
GEORG OBERDIECK
Abstract.
We use Noether-Lefschetz theory to study the reduced Gro-mov–Witten invariants of a holomorphic-symplectic variety of K [ n ] -type. This yields strong evidence for a new conjectural formula that ex-presses Gromov-Witten invariants of this geometry for arbitrary classesin terms of primitive classes. The formula generalizes an earlier conjec-ture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. Thisreproves and extends a result of Debarre, Han, O’Grady and Voisin onHLS divisors on the moduli space of Debarre-Voisin fourfolds. Contents
0. Introduction 11. The monodromy in K [ n ] -type 82. The multiple cover conjecture 123. Noether-Lefschetz theory 184. Gromov–Witten theory and Noether-Lefschetz theory 325. Mirror symmetry 346. Results 38Appendix A. A multiple cover rule for abelian surfaces 41References 430. Introduction
K3 surfaces.
Gromov-Witten theory is the intersection theory of themoduli space M g,n ( X, β ) of stable maps to a target X in degree β ∈ H ( X, Z ). If X carries a holomorphic symplectic form, the virtual funda-mental class of the moduli space vanishes. Instead Gromov-Witten theoryis defined through the reduced virtual fundamental class[ M g,n ( X, β )] red ∈ A ∗ ( M g,n ( X, β )) . When working with reduced Gromov-Witten invariants, one observes inmany examples the following dichotomy:1. The invariants are notouriously difficult to compute, in particular ifthe class β is not primitive.2. The structure of the invariants is simpler than for general target X .That is, the invariants have additional non-geometric symmetriessuch as the independence (understood correctly) from the divisibilityof the curve class.Physicists would say that X has additional super-symmetry, which shouldexplain this phenomenon. A mathematical explanation is unfortunately verymuch missing so far.As an example, let us consider a K3 surface S and the Hodge integral R g,β = Z [ M g,n ( S,β )] red ( − g λ g . We can formally subtract multiple cover contributions from β by passing tothe BPS numbers: r g,β = X k | β k g − µ ( k ) R g,β/k where we have used the M¨obius function µ ( k ) = ( ( − ℓ if k = p · · · p ℓ for distinct primes p i . By deformation invariance, r g,β depends on ( S, β ) only through the divisi-bility m = div ( β ) and the square s = β · β . One writes r g,β = r g,m,s . The following remarkable result by Pandharipande and Thomas showsthat indeed the invariants r g,β do not depend on the divisibility. Theorem 1 ([47]) . For all g, m, s we have: r g,m,s = r g, ,s . The calculation of the primitive invariants r g, ,s is much easier comparedto the imprimitive case and was performed first in [37]. Several other proofsare available in the literature by now. Together with the theorem this yieldsa complete determination of r g,β , called the Katz-Klemm-Vafa formula [24].In [43] it was conjectured that more generally any Gromov-Witten in-variant of a K3 surface is independent of the divisibility, after subtractingmultiple covers. The divisibility 2 case was solved recently [1], but the gen-eral case remains a challenge.
ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 3
Holomorphic-symplectic varieties.
A smooth projective variety X is (irreducible) holomorphic-symplectic if it is simply connected and thespace of holomorphic 2-forms H ( X, Ω X ) is spanned by a symplectic form.These varieties can be viewed as higher-dimensional analogues of K3 sur-faces. For example, the cohomology H ( X, Z ) carries a canonical non-de-generate integer-valued quadratic form. The prime example of a holomorphic-symplectic variety is the Hilbert scheme of n points of a K3 surface and itsdeformations, which we call varieties of K [ n ] type.We conjecture in this paper that the Gromov-Witten theory of K [ n ] -typevarieties is independent of the divisibility of the curve class, made precisein the following sense: Let β ∈ H ( X, Z ) be an effective curve class, andconsider the Gromov-Witten class C g,N,β ( α ) = ev ∗ (cid:16) τ ∗ ( α ) ∩ [ M g,N ( X, β )] red (cid:17) ∈ H ∗ ( X N )where τ : M g,N ( X, β ) → M g,N is the forgetful morphism to the modulispace of curves and α ∈ H ∗ ( M g,N ) is a tautological class [15]. The classes C g,N,β ( α ) encode the full numerical Gromov-Witten theory of X .We formally subtract the multiple cover contributions from this class:(1) c g,N,β ( α ) = X k | β µ ( k ) k g − N − deg( α ) ( − [ β ]+[ β/k ] C g,N,β/k ( α )where we use that the residue of β with respect to the quadratic form[ β ] ∈ H ( X, Z ) /H ( X, Z )can be canonically identified up to multiplication by ± Z / (2 n − Z , see Section 1.5.Let X ′ be any variety of K [ n ] -type and let ϕ : H ( X, R ) → H ( X ′ , R )be any real isometry such that ϕ ( β ) ∈ H ( X ′ , Z ) is a primitive effectivecurve class satisfying ± [ ϕ ( β )] = ± [ β ] in Z / (2 n − Z . Extend ϕ to the full cohomology as a parallel transport lift (Section 1.6) ϕ : H ∗ ( X, R ) → H ∗ ( X ′ , R ) . The following is the main conjecture:
Conjecture A. c g,N,β ( α ) = ϕ − (cid:16) C g,N,ϕ ( β ) ( α ) (cid:17) The right hand side of the conjecture is given by the Gromov-Wittentheory for a primitive class. Hence the conjecture reduces calculations inimprimitive classes (which are hard) to those for primitive ones (which areeasier). A different but equivalent version of the conjecture is formulated
GEORG OBERDIECK in Section 2.3. The equivalent version shows that the above reduces for K3surfaces to the conjecture [43, Conj.C2].Up to the sign ( − [ β ]+[ β/k ] , (21) is precisely the formula that definesthe BPS numbers of a Calabi-Yau manifold in terms of Gromov-Witteninvariants. However, the appearence of the sign is a new feature, particularto the holomorphic-symplectic case. For example, it does not appear inthe definition of BPS numbers of Calabi-Yau 4-folds as given by Klemm-Pandharipande [26]. We expect a similar multiple cover formula to hold forall holomorphic-symplectic varieties. What stops us from formulating it isthat the precise term that generalizes the sign is not clear (aside from thatthere would be no evidence available at all).In the appendix we also formulate a multiple cover rule for abelian sur-faces, extending a proposal for abelian varieties in [7].0.3. Noether-Lefschetz theory.
There are three types of invariants asso-ciated to a 1-parameter family π : X → C of quasi-polarized holomorphic-symplectic varieties:(i) the Noether-Lefschetz numbers of π ,(ii) the Gromov-Witten invariants of X in fiber classes,(iii) the reduced Gromov-Witten invariants of a holomorphic-symplecticfiber of the family.We refer to Section 3 for the definition of a 1-parameter family of quasi-polarized holomorphic-symplectic varieties and its Noether-Lefschetz num-bers. By a result of Maulik and Pandharipande [36], there is a geometricrelation intertwining these three invariants. This relation (for a carefullyselected family π ) was used in [25] to prove Theorem 1 in genus 0, andthan later in [47] in the general case. Roughly, for a nice family the rela-tion becomes invertible, and reduces the problem to considering ordinaryGromov-Witten invariants of a K3-fibered threefold which then can be at-tacked by more standard methods.In this paper we follow the same strategy for holomorphic symplecticvarieties of K [ n ] -type. We first discuss the Maulik-Pandharipande relationin this case (Section 4), and then apply it in two cases:(i) A generic pencil of Fano varieties of a cubic fourfold [2] : X ⊂ Gr (2 , × P , π : X → P . (ii) A generic pencil of Debarre-Voisin varieties [14]: X ⊂ Gr (6 , × P , π : X → P . The examples are choosen such that the Gromov-Witten invariants of X can be computed using mirror symmetry [11]. For the family of Fanovarieties, the Noether-Lefschetz numbers have already been computed by ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 5
Li and Zhang [30]. The Gromov-Witten/Noether-Lefschetz relation for theFano family then yields an explicit infinite family of relations that need tohold for Conjecture A to be true. Using a computer we checked that theserelations are satisfied up to degree d ≤
38 (with respect to the Pl¨ucker po-larization). The relations involve curve classes of both high self-intersectionand high divisibility. Together with previous known cases we also obtain thefollowing:
Proposition 1.
In genus and K [2] -type Conjecture A holds for β = m · α where α is primitive whenever: • ( α, α ) ≤ • ( α, α ) = 3 / and m ∈ { , , } . Ideally we would like to apply our methods to other families. However,it is quite difficult to find appropriate 1-parameter families. They must be(a) a zero section of a homogeneous vectorbundle on the GIT quotient of avector space by a reductive group, and (b) their singular fibers must havemild singularieties. A promising candidate seemed to be a generic pencil ofIliev-Manivel fourfolds [23]
X ⊂ Gr (2 , × Gr (2 , × Gr (2 , × P , π : X → P but unfortunately the singular fibers appear to be too singular. Debarre-Voisin fourfolds.
For the generic pencil π : X → P ofDebarre-Voisin fourfolds the Noether-Lefschetz numbers have not yet beendetermined. Instead we use the known cases of Conjecture A and the mirrorsymmetry calculations for the total space X to obtain constraints for theNoether-Lefschetz numbers. By using the modularity of the generating seriesof Noether-Lefschetz numbers due to Borcherds and McGraw [5, 38] we canthen determine the full series.Consider the generating series of Noether-Lefschetz numbers of π as de-fined in Section 3.8.1, ϕ ( q ) = X D ≥ q D/ NL π ( D )where D runs over all squares modulo 11.Define the weight 1 , , E ( τ ) = 1 + 2 X n ≥ q n X d | n χ p (cid:18) nd (cid:19) , ∆ ( τ ) = η ( τ ) η (11 τ ) E ( τ ) = X n ≥ q n X d | n d χ p (cid:18) nd (cid:19) Another candidate is the family of Fano varieties of Pfaffian cubics, X ⊂ Gr (4 , × Gr (2 , GEORG OBERDIECK where η ( τ ) = q / Q n ≥ (1 − q n ) is the Dirichlet eta function, q = e πiτ and χ is the Dirichlet character given by the Legendre symbol (cid:0) · (cid:1) . Considerthe following weight 11 modular forms for Γ (11) and character χ : ϕ ( q ) = − E + 430 E E + 51999209 ∆ E − E + 491944409 ∆ E E + 248350 E E − E E − E E + 512435003 ∆ E + 133145254027 ∆ E E + 3490194409 E E = − q + 255420 q + 14793440 q + 262345260 q + . . .ϕ ( q ) = − E + 110 E E + 7227403993 ∆ E − E − E E − E E + 118940363 ∆ E + 56091803993 ∆ E E + 2920846011979 ∆ E E + 350033 ∆ E + 26109801089 E E = − q + 990 q + 5500 q + 11440 q + . . . Theorem 2.
Let π : X → P be a generic pencil of Debarre-Voisin fourfolds.Then the generating series of its Noether-Lefschetz numbers is: ϕ ( q ) = ϕ ( q ) + ϕ ( q ) = −
10 + 640 q + 990 q + 5500 q + 11440 q +21450 q + 198770 q + 510840 q + . . . Debarre-Voisin varieties are parametrized by a 20-dimensional projectiveirreducible GIT quotient M DV = P ( ∧ V ∨ ) // SL( V )where V is a vector space of dimension 10. The period map from thismoduli space to the moduli space of holomorphic-symplectic varieties p : M DV M H is birational [40] and regular on the open locus corresponding to smoothDebarre-Voisin varieties of dimension 4. When passing to the Baily-Borelcompactification M H this birational map can be resolved. An HLS divisor (for Hassett-Looijenga-Shah) in M H is the image of an exceptional divi-sor under this resolved map. These divisors reflect a difference between theGIT and the Baily-Borel compactification as they parametrize holomorphic-symplectic fourfolds of the same polarization type as a Debarre-Voisin four-fold, but for which the generic member is not a Debarre-Voisin fourfold. ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 7
Let e be a square modulo 11 and let C e ⊂ M H be the Noether-Lefschetz divisor of the first type of discriminant e , see Sec-tion 3.8.2 for the precise definition. Observe that there is a natural gap inthe modular form ϕ ( q ): ϕ ( q ) = −
10 + 0 · q + 0 · q + . . . + 0 · q | {z } gap +640 q + 990 q + 5500 q + . . . Translating from Heegner divisors to the irreducible divisors C e , this gapyields the following: Corollary 1.
The divisors C , C , C , C , C are HLS divisors of the modulispace of Debarre-Voisin fourfolds. The divisor C is not HLS. The fact that C , C , C , C are HLS gives an independent and mostlyformal proof to the main result of [12]. The only geometric input lies inunderstanding the geometry of the singular fibers (a result of V. Benedettiand J. Song [3]). The result that C is not a HLS divisor answers a questionof [12]. The fact that C is HLS seems to be new, and it would be interestingto understand the geometry of these loci, as done in [12] for the other cases.0.5. Convention. If γ ∈ H i ( X, Q ) is a cohomology class, we write deg( γ ) = i/ γ . For X holomorphic-symplecticwe identify Pic( X ) with its image in H ( X, Z ) under the map taking the firstChern class. Let Hilb n ( S ) be the Hilbert scheme of points of a K3 surface.Given α ∈ H ∗ ( S, Q ) and i > q i ( α ) : H ∗ ( Hilb n ( S )) → H ∗ ( Hilb n + i ( S ))be the i -th Nakajima operator [39] given by adding a i -fat point on a cyclewith class α ; we use the convention of [44]. We write A ∈ H ( Hilb n ( S ))for the class of a generic fiber of the singular locus of the Hilbert Chowmorphism, and we let − δ be te class of the diagonal. We identify H ( Hilb n ( S )) ≡ H ( S, Z ) ⊕ Z δ, H ( Hilb n ( S )) ≡ H ( S, Z ) ⊕ Z A using the Nakajima operators [41].0.6. Acknowledgements.
The idea to use Noether-Lefschetz theory forthe Gromov-Witten theory of K [2] -type varieties is due to E. Scheideggerand quite old [48]. I also owe a great debt to the beautiful paper on Noether-Lefschetz theory by D. Maulik and R. Pandharipande [36]. I further thank In fact, the gap determines the modular form (viewed as a vector-valued modularform) up to a constant.
GEORG OBERDIECK
T. Beckmann, J. Bryan, T. H. Buelles, O. Debarre, E. Markman, G. Mon-gardi, R. Mboro, and J. Song for useful comments. The author was fundedby the Deutsche Forschungsgemeinschaft (DFG) – OB 512/1-1.1.
The monodromy in K [ n ] -type Overview.
Let X be a (irreducible) holomorphic-symplectic variety.The lattice H ( X, Z ) is equipped with the integral and non-degenerateBeauville-Bogomolov-Fujiki quadratic form. We will also equipp H ∗ ( X, Z )with the usual Poincar´e pairing. Both pairings are extended to the C -valued cohomology groups by linearity. Let Mon ( X ) be the subgroup of O ( H ∗ ( X, Z )) generated by all monodromy operators, and let Mon ( X ) beits image in O ( H ( X, Z )).The goal of this section is to describe the monodromy group in the casethat X is of K [ n ] -type, and we will assume so from now on. The mainreferences for the sections are Markman’s papers [33, 32].1.2. Monodromy.
Let X be of K [ n ] -type. By work of Markman [33,Thm.1.3], [34, Lemma 2.1] we have that(2) Mon ( X ) ∼ = Mon ( X ) = e O + ( H ( X, Z ))where the first isomorphism is the restriction map and e O + ( H ( X, Z )) is thesubgroup of O ( H ( X, Z )) of orientation preserving lattice automorphismswhich act by ± The first isomorpism implies thatany parallel transport operator H ∗ ( X , Z ) → H ∗ ( X , Z ) between two K [ n ] -type varieties is uniquely determined by its restriction to H ( X , Z ).If g ∈ Mon ( X ), we let τ ( g ) ∈ {± } be the sign by which g acts on thediscriminant lattice. This defines a character τ : Mon ( X ) → Z . Zariski closure.
By [32, Lemma 4.11] if n ≥ Mon ( X ) in O ( H ∗ ( X, C )) is O ( H ( X, C )) × Z . The inclusion yields therepresentation(3) ρ : O ( H ( X, C )) × Z → O ( H ∗ ( X, C ))which acts by degree-preserving orthogonal ring isomorphism. There is anatural embedding e O + ( H ( X, Z )) → O ( H ( X, C )) × Z , g ( g, τ ( g ))under which ρ restricts to the monodromy representation. In case n ∈ { , } the Zariski closure is O ( H ( X, C )). In this case, we define the representation(3) by letting it act through O ( H ( X, C )). Let C = { x ∈ H ( X, R ) |h x, x i > } be the positive cone. Then C is homotopyequivalent to S . An automorphism is orientation preserving if it acts by +1 on H ( C ) = Z . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 9
The representation ρ is determined by the following properties: Property 0.
For any ( g, τ ) ∈ O ( H ( X, C )) × Z we have ρ ( g, τ ) | H ( X, C ) = g. Property 1.
The restriction of ρ to SO( H ( X, C )) × { } is the integratedaction of the Looijenga-Lunts-Verbitsky algebra [31, 49]. Property 2.
We have ρ (1 , −
1) = D ◦ ρ ( − id H ( X, C ) , , where D acts on H i ( X, C ) by multiplication by ( − i . Property 3.
Assume that X = Hilb n ( S ) and identify H ( X, Z ) with H ( S, Z ) ⊕ Z δ . Then the restriction of ρ to O ( H ( X, C )) δ × O ( H ( S, C ))) is the Zariski closure of the induced action onthe Hilbert scheme by the monodromy representation of S .In particular, the action is equivariant with respect to the Nakajima op-erators: For g ∈ O ( H ( X, C )) δ let ˜ g = g | H ( S, C )) ⊕ id H ( S, Z ) ⊕ H ( S, Z ) . Then ρ ( g, Y i q k i ( α i )1 ! = Y i q k i (˜ gα i )1 . Property 4.
Let P ψ : H ∗ ( X , Z ) → H ∗ ( X , Z ) be a parallel transportoperator with ψ = P ψ | H ( X , Z ) . Then P − ψ ◦ ρ ( g, τ ) ◦ P ψ = ρ ( ψ − gψ, τ ) . Property 1 follows by [32, Lemma 4.13]. The other properties also followfrom the results of [32]. Properties 1-3 determine the action ρ completelyin the Hilbert scheme case. Moreover by [44] this description is explicitin the Nakajima basis. The last condition extends this presentation thento arbitrary X . The parallel transport operator between different modulispaces of stable sheaves can also be described more explicitly [32].1.4. Parallel transport.
LetΛ = E ( − ⊕ ⊕ U be the Mukai lattice. For n ≥
2, any holomorphic-symplectic variety of K n ]-type is equipped with a canonical choice of a primitive embedding ι X : H ( X, Z ) → Λunique up to composition by an element by O (Λ), see [33, Cor.9.4]. Theorem 3. ( [33, Thm.9.8] ) An isometry ψ : H ( X , Z ) → H ( X , Z ) isthe restriction of a parallel transport operator if and only of if it is orienta-tion preserving and there exists an η ∈ O (Λ) such that η ◦ ι X = ι X ◦ ψ, Orientation preserving is here defined with respect to the canonical choiceof orientation of the positive cone of X i given by the real and imaginary partof the symplectic form and a K¨ahler class. If X = X the theorem reducesto the second isomorphism in (2).1.5. Curve classes.
By Eichler’s criterion [19, Lemma 3.5], Theorem 3yields a complete set of deformation invariants of curve classes in K [ n ] -type. To state the result we need the following constriction:The orthogonal complement L = ι X ( H ( X, Z )) ⊥ ⊂ Λis isomorphic to the lattice Z with intersection form (2 n − v ∈ L bea generator and consider the isomorphism of abelian groups(4) L ∨ /L ∼ = −→ Z / (2 n − Z determined by sending v/ (2 n −
2) to the residue class of 1. Since the gener-ators of L are ± v , (4) is canonical up to multiplication by 1.Since Λ is unimodular there exists a natural isomorphism ([21, Sec.14]) H ( X, Z ) ∨ /H ( X, Z ) ∼ = −→ L ∨ /L. If we use Poincar´e duality to identify H ( X, Z ) with H ( X, Z ) ∨ this yieldsthe residue map r X : H ( X, Z ) /H ( X, Z ) ∼ = −→ L ∨ /L ∼ = −→ Z / (2 n − Z . The map depends on the choice of the generator v and hence is unique upto multiplication by ± Definition 1.
The residue set of a class β ∈ H ( X, Z ) is defined by ± [ β ] = {± r X ([ β ]) } ⊂ Z / (2 n − Z if n ≥
2, and by ± [ β ] = 0 otherwise.Note that since r X is canonical up to sign, the residue set is independentof the choice of map r X . Remark . (i) Since parallel transport operators respect the embedding i X up to composing with an isomorphism of Λ, the residue set [ β ] is preservedunder deformation. (This is also reflected in the fact, that the monodromyacts by ± X = Hilb n ( S ) of a K3 surface, let A ∈ H ( X ) be the class of an exceptional curve, that is the class of a fiberof the Hilbert-Chow morphism Hilb n ( S ) → Sym n ( S ) over a generic point inthe singular locus. We have a natural identification H ( X, Z ) = H ( S, Z ) ⊕ Z A. ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 11
The morphism r X then sends (up to sign) the class [ A ] to 1 ∈ Z n − .(iii) In other words, we could have defined the residue class also by firstdeforming to the Hilbert scheme, and then taking the coefficient of A modulo2 n −
2. This is usually the practical way to compute the residue class. (cid:3)
Let β ∈ H ( X, Z ) be a class. The class β has then the following deforma-tion invariants:(i) the divisibility div ( β ) in H ( X, Z ),(ii) the Beauville–Bogomolov norm ( β, β ) ∈ Q , and(iii) the residue set ± h β div ( β ) i ∈ Z / (2 n − Z .The global Torelli theorem and Eichler’s criterion yields the following: Corollary 2.
Two pairs ( X, β ) and ( X ′ , β ′ ) of a K [ n ] -type variety and aclass in H ( X, Z ) which pairs positively with a K¨ahler class are deformationequivalent if and only if the invariants (i-iii) agree.Remark . By the global Torelli theorem, if β and β ′ are both of Hodgetype, the deformation in the corollary can be choosen such that the curveclass stays of Hodge type.1.6. Lifts of isometries of H . Let X , X be of K [ n ] -type, and let g : H ( X , C ) → H ( X , C )be an isometry. An operator e g : H ∗ ( X , C ) → H ∗ ( X , C ) is a paralleltransport lift of g if it is of the form e g = ρ ( g ◦ ψ − , ◦ P ψ for a parallel transport operator P ψ : H ∗ ( X , Z ) → H ∗ ( X , Z ) with restric-tion ψ = P | H ( X , Z ) . In particular, any parallel transport lift is a degree-preserving orthogonal ring isomorphism.Recall from Section 1.3, Property 2, the operator e D = ρ (id , −
1) = D ◦ ρ ( − id H ( X , Z ) , . Lemma 1.
A parallel transport lift of g is unique up to composition by e D .Proof. Consider two parallel transport lifts of g , e g i = γ ( g ◦ ψ − i ) ◦ P ψ i , i = 1 , P ψ , P ψ . We will show that e g = e g or e g = e D ◦ e g . Let γ ( h ) = ρ ( h, τ ( ψ ◦ ψ − ) = 1 then γ ( g ◦ ψ − ) ◦ P ψ ◦ P − ψ = γ ( gψ − ) ◦ γ ( ψ ◦ ψ − ) = γ ( gψ − ) . If τ ( ψ ◦ ψ − ) = − γ ( g ◦ ψ − ) ◦ P ψ ◦ P − ψ = γ ( gψ − ) ◦ D ◦ γ ( − ψ ◦ ψ − ) = e D ◦ γ ( gψ − ) (cid:3) The multiple cover conjecture
Overview.
Let X be a variety of K [ n ] -type and let β ∈ H ( X, Z )be an effective curve class. The moduli space M g,N ( X, β ) of N -markedgenus g stable maps to X in class β carries a reduced virtual fundamentalclass [ M g,N ( X, β )] red of dimension (2 n − − g ) + N + 1. Gromov-Witteninvariants of X are defined by pairing with this class:(5) (cid:10) α ; γ , . . . , γ n (cid:11) Xg,β := Z [ M g,n ( X,β )] red π ∗ ( α ) ∪ Y i ev ∗ i ( γ i ) , where ev i : M g,n ( X, β ) → X are the evaluation maps, τ : M g,n ( X, β ) → M g,n is the forgetful map, and α ∈ H ∗ ( M g,n ) is a tautological class [15].In this section we state a conjecture to express the invariants (5) for β anarbitrary class in terms of invariants where β is primitive.2.2. Invariance.
Let β ∈ H ( X, Z ) be an effective curve class, and let e O + ( H ( X, Z )) β ⊂ e O + ( H ( X, Z ))be the subgroup fixing β (either via the monodromy representation or equiv-alently, via the dual action on H ( X, Z ) ∨ ∼ = H ( X, Z ) under the Beauville-Bogomolov form). Applying Remark 2 and the deformation invariance ofthe reduced Gromov-Witten invariants, we find that D α ; γ , . . . , γ n E Xg,β = D α ; µ ( ϕ ) γ , . . . , µ ( ϕ ) γ n E Xg,β for all ϕ ∈ e O + ( H ( X, Z )) β .The image of e O + ( H ( X, Z )) β is Zariski dense in G β = ( O ( H ( X, C )) × Z / β := { g ∈ O ( H ( X, C )) × Z / Z | ρ ( g ) β = β } . It follows that for all g ∈ G β we have(6) D α ; γ , . . . , γ n E Xg,β = D α ; ρ ( g ) γ , . . . , ρ ( g ) γ n E Xg,β . Equivalently, the pushforward of the reduced virtual class lies in the invari-ant part of the diagonal G β action:ev ∗ (cid:16) τ ∗ ( α ) ∩ [ M g,n ( X, β )] red (cid:17) ∈ H ∗ ( X n , Q ) G β . The representation ρ restricted to { id } × Z acts trivially on H ( X, C ).Hence for X a K3 surface, we obtain the invariance of the Gromov-Witten ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 13 class under the group O ( H ( X, C )) β . This matches what was conjecturedin [43, Conj.C1] and then proven along the above lines in [9].2.3. Multiple-cover conjecture.
The main conjecture is the following:Let β ∈ H ( X, Z ) be any effective curve class. For every divisor k | β , let X k be a variety of K [ n ] type and let ϕ k : H ( X, R ) → H ( X ′ , Q )be a real isometry such that: • ϕ k ( β/k ) is a primitive curve class • ± [ ϕ k ( β/k )] = ± [ β/k ].We extend ϕ k as a parallel transport lift (Section 1.6) to the full cohomology: ϕ k : H ∗ ( X, R ) → H ∗ ( X ′ , R ) , By Section 2.6 below, pairs ( X k , ϕ k ) satisfying these properties can alwaysbe found. Conjecture B.
For any effective curve class β ∈ H ( X, Z ) we have D α ; γ , . . . , γ N E Xg,β = X k | β k g − N − deg( α ) ( − [ β ]+[ β/k ] D α ; ϕ k ( γ ) , . . . , ϕ k ( γ N ) E X k g,ϕ k ( β/k ) . The invariance property discussed in Section 2.2 and Property 4 of Sec-tion 1.3 imply that the right hand side of the conjecture is independent ofthe choice of ( X k , ϕ k ). Using that P k | a µ ( k ) = δ a , Conjecture B is alsoseen to be equivalent to Conjecture A of the introduction.The reduced Gromov–Witten invariants of X can only be non-zero if thedimension constraint(dim X − − g ) + N + 1 = deg( α ) + X i deg( γ i )is satisfied. Hence the conjecture can also be rewritten as: D α ; γ , . . . , γ n E Xg,β = X k | β k dim( X )( g − − P i deg( γ i ) ( − [ β ]+[ β/k ] D α ; ϕ k ( γ ) , . . . , ϕ k ( γ n ) E Xg,ϕ k ( β/k ) . Since for K3 surfaces the residue always vanishes, Conjecture B specializesfor K3 surfaces to the conjecture made in [43, Conj C2].
Remark . The condition that we ask of the residue, i.e. ± [ ϕ k ( β/k )] = ± [ β/k ] is necessary for the conjecture to hold. For example consider X of K [5] -type and two primitive classes β , β with ( β · β ) = ( β · β ) = 16but [ β ] = 0 and [ β ] = 4 in Z / Z . Then by [41] one hasev ∗ [ M , ( X, β )] red = 1464 β ∨ , ev ∗ [ M , ( X, β )] red = 480 β ∨ so an isometry taking β to β does not preserve Gromov-Witten invariants.2.4. Uniruled divisors.
An uniruled divisor D ⊂ X which is swept out bya rational curve in class β is a component of the image of ev : M , ( X, β ) → X . The virtual class of these uniruled divisors is given by the pushforwardev ∗ [ M , ( X, β )] red . For β primitive and ( X, β ) very general, the virtual classis closely related to the actual class [45].By monodromy invariance there exists N β ∈ Q such that(7) ev ∗ [ M , ( X, β )] red = N β · h where h = h β, −i ∈ H ( X, Q ) is the dual of β . Since by deformation invari-ance N β only depends on the divisibility m = div ( β ), the square s = ( β, β )and the residue r = [ β/ div ( β )] we write N β = N m,s,r . Conjecture B then says that(8) N m,s,r = X k | β k ( − mr + mk r N , sk , mrk The primitive numbers N ,s,r have been determined in [41]. Example 1.
Let X = Hilb ( S ), and let A ∈ H ( X, Z ) be the class of theexceptional curve. We haveev ∗ [ M , ( X, β )] red = ∆ Hilb ( S ) = − δ. Since A ∨ = − δ we see N , − , = 4. The multiple cover formula thenpredicts that for even ℓ ∈ Z ≥ we have N ℓA = − ℓ N , − , + 1( ℓ/ N , − , = − ℓ + 8 ℓ = 4 ℓ where we have used N , − , = 1. This matches the degree-scaling propertydiscussed in [46, 35]. (cid:3) Fourfolds.
We consider genus 0 Gromov-Witten invariants of a variety X of K [2] -type. By dimension considerations all genus 0 Gromov-Witteninvariants are determined by the 2-point class:ev ∗ [ M , ( X, β )] red ∈ H ∗ ( X × X ) . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 15
Following the arguments of [45, Sec.2] , for every effective β ∈ H ( X, Z )there exist constants F β , G β ∈ Q such that:ev ∗ [ M , ( X, β )] red = F β ( h ⊗ h ) + G β (cid:0) h ⊗ β + β ⊗ h + ( h ⊗ h ) · c BB (cid:1) + (cid:18) −
130 ( h ⊗ c + c ⊗ h ) + 1900 ( β, β ) c ⊗ c (cid:19) ( G β + ( β, β ) F β )where • h = ( β, − ) ∈ H ( X, Q ) is the dual of the curve class, • c = c ( X ) is the second Chern class, and • c BB ∈ H ( X ) ⊗ H ( X ) is the inverse of the Beauville-BogomolovFujiki form.In K [2] -type the residue r = [ β ] of a curve class is determined by s =( β, β ) via r = 2 s mod 2. So we can write F β = F m,s and G β = G m,s . Themultiple cover conjecture for K [2] -type in genus 0 is then equivalent to: F m,s = X k | m k ( − s + s/k ) F , sk G m,s = X k | m k ( − s + s/k ) G , sk . We can also define the BPS numbers f m,s = X k | m µ ( k ) k ( − s +2 s/k F mk , sk g m,s = X k | m µ ( k ) k ( − s +2 s/k G mk , sk . and arrive at: Lemma 2.
Conjecture B in K [2] -type and genus is equivalent to: ∀ m, s : f m,s = f ,s , g m,s = g ,s . The first few cases are known:
Proposition 2.
Conjecture B in K [2] -type and genus holds for all classes β such that β · β ≤ .Proof. The case β · β < β, β ) = 0 followsby reducing to Hilb ( P × E ) and using the series T in [41, Thm.9]. (cid:3) One uses that the class is monodromy invariant and a Lagrangian correspondence,and that for very general (
X, β ) the image of the Hodge classes under this correspondenceis annihilated by the symplectic form, see [45, Sec.1.3]. The reference treats only the caseof primitive β , but the imprimitive case follows likewise with minor modifications. The constant N β appearing in (7) is equal to G β in the K [2] -case. For later use we will also need to tollowing expression for 1-pointed de-scendence invariants:ev ∗ [ M , ( X, β )] = G β β ∨ ev ∗ (cid:16) ψ · [ M , ( X, β )] (cid:17) = 2 F β h −
115 ( G β + ( β, β ) F β ) c ( X )ev ∗ (cid:16) ψ · [ M , ( X, β )] (cid:17) = − F β ev ∗ (cid:16) ψ · [ M , ( X, β )] (cid:17) = 24 F β This follows by monodromy invariance and topological recursions. In par-ticular, to check the multiple cover formula in K [2] -type and genus 0 it isenough to consider 1-point descendent invariants. Remark . For convenience we recall the evaluation of f ,m,s and g ,m,s . Let ϑ ( q ) = X n ∈ Z q n , α ( q ) = X odd n> d | n dq n ,G ( q ) = −
124 + X n ≥ X d | n dq n , ∆( q ) = q Y n ≥ (1 − q n ) Then by [41] one has: X s f ,s ( − q ) s = 14 − ϑ ( q ) α ( q )∆( q ) X s g ,s ( − q ) s = 112 ϑ ( q ) + 4 α ( q ) + 24 G ( q ) ϑ ( q ) α ( q )∆( q ) . Hilbert schemes of elliptic K3 surfaces.
Let π : S → P be an elliptic K3 surface with a section. Let B, F ∈ H ( S, Z ) be the classof the section and a fiber respectively, and let W = B + F. We consider the Hilbert scheme X = Hilb n S and the generating series ofGromov-Witten invariants F g,m ( α ; γ , . . . , γ N ) = ∞ X d = − m X r ∈ Z h α ; γ , . . . , γ N i Hilb n Sg,mW + dF + rA q d ( − p ) r . We state a characterization of the multiple cover formula:Consider the basis of H ∗ ( S, R ) defined by B = { , p , W, F, e , . . . , e } , ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 17 where p is the class of a point and e , . . . , e is a basis of Q h W, F i ⊥ in H ( S, R ). An element γ ∈ H ∗ ( X, R ) is in the Nakajima basis with respectto B if it is of the form γ = Y i q k i ( α i )1 , α i ∈ B . Let w ( γ ) and f ( γ ) be the number of classes α i which are equal to W and F respectively, and define a modified degree grading deg by:deg( γ ) = deg( γ ) + w ( γ ) − f ( γ ) . For a series f = P d,r c ( d, r ) q d p r define the formal Hecke operator by T m,ℓ f = X d,r X k | ( m,d,r ) k ℓ − c (cid:18) mdk , rk (cid:19) q d p r . Lemma 3.
Conjecture B holds if and only if for all g, N, m, α and (deg , deg) -bihomogeneous classes γ i we have: (9) F g,m ( α ; γ , . . . , γ N ) = m P i deg( γ i ) − deg( γ i ) · T m,ℓ F g, ( α ; γ , . . . , γ N ) where ℓ = 2 n ( g −
1) + P i deg( γ i ) .Proof. Given the class β = mW + dF + rA and a divisor k | β , consider thereal isometry ϕ k : H ( X, R ) → H ( X, R ) defined by W km WF mk Fγ γ for all γ ⊥ W, F.
We extend this map to the full cohomology by ϕ k = ρ ( φ k ,
0) : H ∗ ( X, R ) → H ∗ ( X, R ). We then have ϕ k (cid:18) βk (cid:19) = W + dmk F + rk A. The Nakajima operators are equivariant with respect to the action of ϕ k and the isometry of H ∗ ( S, R ) given by e ϕ k = ϕ k | H ( S, R ) ⊕ id H ( S, R ) ⊕ H ( S, R ) , see Property 3 of Section 1.3.Let γ i ∈ H ∗ ( X, Q ) be elements in the Nakajima basis with respect to B .If Conjecture B holds, then its application with respect to ϕ k yields:(10) F g,m ( α ; γ , . . . , γ N ) = X d,r q d p r X k | ( m,d,r ) ( − r/k k g − N − deg( α ) × (cid:18) mk (cid:19)P i f ( γ i ) − w ( γ i ) h α ; γ , . . . , γ N i Hilb n g,W + mdk F + rk A . Using the dimension constraint, our modified degree function and the formalHecke operators this becomes(11) F g,m ( α ; γ , . . . , γ N ) = m P i deg( γ i ) − deg( γ i ) × X d,r q d p r X k | ( m,d,r ) ( − r/k k n ( g − − P i deg( γ i ) h α ; γ , . . . , γ N i Hilb n g,W + mdk F + rk A which is (9).Conversely, equality (9) implies Conjecture B since any pair ( X, β ) isdeformation equivalent to some (
Hilb n ( S ) , e β = mW + dF + kA ), and theright hand side of Conjecture B is independent of choices. (cid:3) We will reinterprete the lemma in terms of Jacobi forms in [42]. See also[1] for a parallel discussion in the case of K3 surfaces.3.
Noether-Lefschetz theory
Lattice polarized holomorphic-symplectic varieties.
Let V bethe abstract lattice, and let L ⊂ V be a primitive non-degenerate sublattice .An L -polarization of a holomorphic-symplectic variety X is a primitiveembedding j : L ֒ → Pic( X )such that • there exists an isometry ϕ : V ∼ = −→ H ( X, Z ) with ϕ | L = j , and • the image j ( L ) contains an ample class.We call the isometry ϕ as above a marking of ( X, j ). If the image j ( L )only contains a big and nef line bundle, we say that X is L -quasipolarized.Let M L be the moduli space of L -quasipolarized holomorphic-symplecticvarieties of a given fixed deformation type.The period domain associated to L ⊂ V is D L = { x ∈ P ( L ⊥ ⊗ C ) | h x, x i = 0 , h x, ¯ x i > } and has has two connected components. Let D + L be one of these components.Consider the subgroup Mon ( V ) ⊂ O ( V )which, for some choice of marking ϕ : V → H ( X, Z ) for some ( X, j ) defininga point in M L , can be identified with the monodromy group Mon ( X ) of X . Let
Mon ( V ) L be the subgroup of Mon ( V ) that acts trivially on L . Thenthe global Torelli theorem says that the period mappingPer : M L → D + L / Mon ( V ) L i.e. the quotient is torsion free. If the monodromy group
Mon ( V ) is normal in O ( V ), then it does not depend on thechoice of ( X, j ); this is known for all known examples of holomorphic-symplectic varieties.
ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 19 is surjective, restricts to an open embedding on the open locus of L -polarizedholomorphic symplectic varieties, and any fiber consists of birational holo-morphic symplectic varieties, see [33] for a survey and references.3.2. 1 -parameter families. Let L i , i = 1 , . . . , ℓ be an integral basis of L .Given a compact complex manifold X of dimension 2 n + 1, line bundles L , . . . , L ℓ ∈ Pic( X )and a morphism π : X → C to a smooth proper curve, following [25, 0.2.2],we call the tuple ( X , L , . . . , L ℓ , π ) a 1 -parameter family of L -quasipolarizedholomorphic-symplectic varieties if the following holds:(i) For every t ∈ C , the fiber ( X t , L i
7→ L i | X t ) is a L -quasipolarizedholomorphic-symplectic variety.(ii) There exists an vector h ∈ L which yields a quasi-polarization on allfibers of π simultaneously.Any 1-parameter family as above defines a morphism ι π : C → M L into themoduli space of L -quasipolarized holomorphic-symplectic varieties (of thedeformation type specified by a fiber).3.3. Noether-Lefschetz cycles.
Given primitive sublattices L ⊂ e L ⊂ V ,consider the open substack M ′ e L ⊂ M e L parametrizing pairs ( X, j : e L ֒ → Pic( X )) such that j ( L ) contains a quasi-polarization. There exists a natural proper morphism ι : M ′ e L → M L definedby restricting j to L . The Noether-Lefschetz cycle associated to e L is theclass of the reduced image of this map: NL e L = [ ι ( M ′ e L )] ∈ A c ( M L ) . The codimension c of the cycle is given by rank( e L ) − rank( L ). For c = 1 wecall NL e L a Noether-Lefschetz divisor of the first type.3.4. Heegner divisors.
We review the construction of Heegner divisors.Their relation to Noether-Lefschetz divisors will yield modularity results forintersection numbers with Noether-Lefschetz divisors.Consider the lattice M = L ⊥ ⊂ V and the subgroupΓ M = { g ∈ O + ( M ) | g acts trivially on M ∨ /M } where O + ( M ) stands for those automorphisms which preserve the orienta-tion, or equivalently, the component D + L . We consider the quotient D + L / Γ M . For every n ∈ Q < and γ ∈ M ∨ /M the associated Heegner divisor is: y n,γ = X v ∈ M ∨ v · v = n, [ v ]= γ v ⊥ / Γ M . For n = 0 we define y n,γ by the descent K of the tautological line bundle O ( −
1) on D L equipped with the natural Γ M -action. Concretely, we set y ,γ = ( c ( K ∗ ) if γ = 00 otherwise.In case n > y n,γ = 0 in all cases.Define the formal power series of Heegner divisorsΦ( q ) = X n ∈ Q ≤ X γ ∈ M ∨ /M y n,γ q − n e γ which is an element of Pic( D + L / Γ M )[[ q /N ]] ⊗ C [ M ∨ /M ], where e γ are theelements of the group ring C [ M ∨ /M ] indexed by γ and N is the smallestinteger for which M ∨ ( N ) is an even lattice.We recall the modularity result of Borcherds in the formulation of [36]: Theorem 4. ( [5, 38] ) The generating series Φ( q ) is the Fourier-expansion ofa modular form of weight rank( M ) / for the dual of the Weil representation ρ ∨ M of the metaplectic group Mp ( Z ) : Φ( q ) ∈ Pic( D + L / Γ M ) ⊗ Mod (Mp ( Z ) , rank( M ) / , ρ ∨ M ) . The modular forms for the dual of the Weil representations can be com-puted easily by a Sage program of Brandon Williams [51].3.5.
Noether-Lefschetz divisors of the second type.
The precise re-lationship between Noether-Lefschetz and Heegner divisors for arbitraryholomorphic-symplectic varieties is somewhat painful to state. For oncethe monodromy group
Mon ( X ) is not known in general, and even if it isknown it usually does not contain Γ M or is contained in Γ M . To simplifythe situation we from now on restrict to the case of K n ]-type for n ≥ V = E ( − ⊕ ⊕ U ⊕ (2 − n )and we fix an identification V ∨ /V = Z / (2 n − Z .We define the Noether-Lefschetz divisors of second type: NL s,d, ± r ∈ A ( M L )where d = ( d , . . . , d ℓ ) ∈ Z ℓ , s ∈ Q and r ∈ Z / (2 n − Z are given. Considerthe intersection matrix of the basis L i : a = (cid:0) a ij (cid:1) ℓi,j =1 = (cid:0) L i · L j (cid:1) ℓi,j =1 . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 21
We set ∆( s, d ) = det (cid:18) a d t d s (cid:19) = det a · · · a ℓ d ... ... ... a ℓ · · · a ℓℓ d ℓ d · · · d ℓ s Case: ∆( s, d ) = 0 . We define NL s,d, ± r = X L ⊂ ˜ L ⊂ V µ ( s, d, r | L ⊂ ˜ L ⊂ V ) · NL ˜ L where the sum runs over all isomorphism classes of primitive embeddings L ⊂ ˜ L ⊂ V with rank( ˜ L ) = ℓ + 1. The multiplicity µ ( s, d, r | L ⊂ ˜ L ⊂ V ) isthe number of elements β ∈ V ∨ which are contained in e L ⊗ Q and satisfy: β · L i = d i , β · β = s, ± [ β ] = ± r in Z / (2 n − . Here we have used the canonical embeddings V ∨ ⊂ V ⊗ Q and e L ⊗ Q ⊂ V ⊗ Q . Case: ∆( s, d ) = 0 . In this case any curve class with these invariants hasto lie in L ⊗ Q and is uniquely determined by the degree d . Hence we let β ∈ L ⊗ Q be the unique class so that β · L i = d i for all i . If β lies in V ∨ and has residue [ β ] = ± r we define NL s,d, ± r = c ( K ∨ ) , and we define NL s,d, ± r = 0 otherwise. Remark . Often the residue set ± [ β ] of a class β ∈ H ( X, Z ) is determinedby the degrees d i = β · L i . For example, if L contains a class ℓ such that h ℓ, H ( X, Z ) i = (2 n − Z , h ℓ, H ( X, Z ) i = Z we may define a natural isomorphism by cupping with ℓ : H ( X, Z ) /H ( X, Z ) ∼ = −→ Z / (2 n − Z , γ γ · ℓ. (Not every polarization is of that form, for example the case of double coversof EPW sextics.) In other cases the residue set is determined by the norm β · β , for example in K [2] -type. When the residue is determined by s and d we will drop it from the notation of Noether-Lefschetz divisors. (cid:3) For this construction it would be more natural to work with pairs of a holomorphic-symplectic varieties and a primitive embedding j : L → N ( X ) into the group of effective1-cycles N ( X ) ⊂ H ( X, Z ). If we then consider a rank 1 overlattice L ⊂ e L ⊂ N ( X ) wedefine the multiplicity µ as the number of β ∈ e L such that β · L i = d i , β · β = s , and[ β ] = ± r . The condition above is more cumbersome but equivalent to this definition. The class is given by P ℓi,j =1 d i ( a − ) ij L j . Heegner and Noether-Lefschetz divisors.
By the result of Mark-man,
Mon ( V ) ⊂ O ( V ) is the subgroup of orientation preserving isome-tries which act by ± id on the discriminant. Hence we have the inclusionΓ M ⊂ Mon ( V ) L of index 1 or 2. This yields the diagram: D + L / Γ M M L D + L / Mon ( V ) L . π Per where π is either an isomorphism or of degree 2.Let C ⊂ M L be a complete curve, and define the modular formΦ C ( q ) = h Φ( q ) , π ∗ [Per( C )] i . We write Φ C [ n, γ ] for the coefficient of q n e γ in the Fourier-expansion of Φ C .We will need also: e ∆( s, d ) := − · a ) det (cid:18) a d t d s (cid:19) . The following gives the main connection between the Noether-Lefschetz di-visors of the second type and the Heegner divisors.
Proposition 3.
There exists a canonically defined class γ ( s, d, r ) ∈ M ∨ /M (abbreviated also by γ ( r ) ) such that we have the following:(a) If π is an isomorphism, C · NL s,d, ± r = Φ C h e ∆( s, d ) , γ ( r ) i . (b) If π is of degree , C · NL s,d, ± r = Φ C h e ∆( s, d ) , γ ( r ) i if r = − r (cid:16) Φ C h e ∆( s, d ) , γ ( r ) i + Φ C h e ∆( s, d ) , γ ( − r ) i(cid:17) otherwise In K [2] -type, we have V ∨ /V = Z so that π is an isomorphism. More-over, the residue r of any β ∈ V ∨ is determined by its norm s = β · β . Henceomitting r from the notation we find: Corollary 3. In K [2] type, there exists a canonical class γ = γ ( d, s ) with C · NL s,d = Φ C [ e ∆( s, d ) , γ ] . Remark . In fact, in K [2] type, the proof below will imply the equality ofdivisors NL s,d = Φ[ e ∆( s, d ) , γ ] = y − ˜∆( s,d ) ,γ on M L , where we have omitted the pullback by the period map Per on theright hand side. ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 23
For the proof of Proposition 3 we will repeatedly use the following basiclinear algebra fact whose proof we skip.
Lemma 4.
Consider a R -vector space Λ with inner product h− , −i and aorthogonal decomposition L ⊕ M = Λ . Let L i be a basis of L with intersectionmatrix a ij = L i · L j . For β ∈ Λ with d i = β · L i , let v = β − P i,j d i a ij L j bethe projection of β onto M , where a ij are the entries of a − . Then we have h v, v i = 1det( a ) det (cid:18) a d t d h β, β i (cid:19) . where d = ( d , . . . , d ℓ ) . The main step in the proof of the proposition is given by the followinglemma: For fixed d = ( d , . . . , d ℓ ) , s and r ∈ Z n − consider the divisor on D L / Γ M given by NL s,d,r = X β β ⊥ / Γ M where the sum is over all classes β ∈ V ∨ such that(12) β · β = s, β · L i = d i , i = 1 , . . . , ℓ, and [ β ] = r ∈ V ∨ /V. Moreover, β ⊥ stands for the hyperplane in P ( V ) orthogonal to β intersectedwith the period domain D L . Lemma 5.
There exists a canonically defined class γ = γ ( s, d, r ) ∈ M ∨ /M such that NL s,d,r = y n,γ ∈ A ( D L / Γ M ) where n =
12 1det( a ) det (cid:0) a dd s (cid:1) .Proof. In view of the definition of both sides of the claimed equation it isenough to establish a bijection between(a) the set of classes β ∈ V ∨ satisfying (12), and(b) the set of classes v ∈ M ∨ satisfying v = a ) det (cid:0) a dd s (cid:1) and [ v ] = γ for an appropriately defined γ .Consider a primitive embedding V ⊂ Λ into the Mukai latticeΛ = E ( − ⊕ ⊕ U ⊕ . Let e, f be a symplectic basis of one summand of U . We choose the embed-ding such that V ⊥ = Z L where L = e + ( n − f . Since Λ is unimodular,there exists a canonical isomorphism(13) V ∨ /V ∼ = ( Z L ) ∨ / Z L and we may assume that under this isomorphism the class L / (2 n −
2) mod Z L corresponds to 1 ∈ Z / (2 n − Z . Step 1.
Let d ∈ Z be any integer such that d ≡ r modulo 2 n −
2, and let e s ∈ Z such that s = 12 n − (cid:18) n − d d e s (cid:19) ⇐⇒ e s = s + d n − . (We may assume such e s exists: Otherwise the set in (a) is empty, and bythe argument below also the set in (b)). Then we claim that there exists abijection between the set in (a) and(c) the set of classes e β ∈ Λ such that e β · L i = d i for all i = 0 , . . . , ℓ and e β · e β = e s . Proof of Step 1.
Given e β satisfying the conditions in (c) then β = e β − d n − L lies in V ∨ . Moreover, [ β ] is the class in V ∨ /V corresponding to d / (2 n − Z L ) ∨ / Z L under (13), hence [ β ] = r . Also β · L i = d i for i = 1 , . . . , ℓ .The equality β · β = s is by definition of e s and Lemma 4.Conversely, let δ = − e + ( n − f and observe that L · δ = 0 and L / (2 n −
2) + δ/ (2 n −
2) = f . Hence, if β satisfies (a) then β is an elementof d · δ n − + V and hence β β + d n − L ∈ Λdefines the required inverse. (cid:3)
We consider now the embedding M ⊂ Λ and the orthogonal complement b L = M ⊥ . Since Λ is unimodular, we have a the isomorphism b L ∨ / b L ∼ = M ∨ /M. We specify the class γ via this isomorphism. Concretely, we set γ := ℓ X i,j =0 d i a ij L j ∈ b L ∨ / b L where we let a ij denote the entries of the inverse of the extended intersectionmatrix ˆ a = ( L i · L j ) i,j =0 ,...,ℓ .If we replace d by d + (2 n − a = 1 / (2 n −
2) and a j = 0for j = 0, the expression P d i a ij L j gets replaced by the same expressionplus L . Hence the class γ only depends on s, ( d , . . . , d ℓ ) , r . Step 2.
There exists a bijection between the classes in (c) and (b).
ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 25
Proof of Step 2.
We have the bijection e β e β − r X i,j =0 d i a ij L j ∈ M ∨ . (cid:3) Combining Step 1 and 2 finished the proof of the lemma. (cid:3)
Proof of Proposition 3.
By definition we have: NL s,d ,...,d r , ± r = Per ∗ X β β ⊥ / Mon ( V ) L where the sum is over all β ∈ V ∨ such that β · β = s, β · L i = d i , [ β ] = ± r. Hence if π is an isomorphism, the result follows from this by Lemma 5.Hence assume now π is of degree 2, and let e C = Per( C ).If − r = r , then the morphism NL s,d,r → NL s,d, ± r given by restriction of π is of degree 2. Therefore C · NL s,d, ± r = 12 e C · π ∗ NL s,d,r = 12 π ∗ [ e C ] · NL s,d,r which then implies the claim by Lemma 5. If r = − r , then we have π ∗ NL s,d,r = NL s,d, ± r from which the result follows. (cid:3) Noether-Lefschetz numbers.
Let ( X , L , . . . , L ℓ , π ) be a 1-parameterfamily of L -quasipolarized holomorphic-symplectic varieties of K [ n ] -type.We have the associated classifying morphism ι π : C → M L . We define the Noether-Lefschetz numbers of the family by NL πs,d, ± r = Z C ι ∗ π NL πs,d, ± r . Intuitively, the Noether-Lefschetz numbers are the number of fibers of π forwhich there exists a Hodge class β with prescribed norm β · β = s , degree β · L i = d i and residue [ β ] = ± r .In K [2] -type we will also often write Φ π ( q ) = ι ∗ π Φ( q ).The families of holomorphic-symplectic varieties we will encounter in geo-metric constructions often come with mildly singular fibers. The definitionof Noether-Lefschetz numbers can be extended to these families as follows.Let π : X → C be a projective flat morphism to a smooth curve andlet L , . . . , L ℓ ∈ Pic( X ). We assume that over a non-empty open subset of C this defines a 1-parameter family of L -quasipolarized holomorphic-symplectic varieties of K [ n ] type. We also assume that around every sin-gular point the monodromy is finite. Then there exists a cover f : e C → C such that the pullback family f ∗ X → e C is bimeromorphic to a 1-parameterfamily of L -quasipolarized holomorphic-symplectic varieties of K [ n ] -type, e π : ˜ X → e C. See for example [27]. Concretely, around each basepoint of a singular fiber,after a cover that trivializes the monodromy, the rational map C M L can be extended. (In the examples we will consider, we can construct thecover ˜ C → C and the birational model ˜ X explicitly). We define the Noether-Lefschetz numbers of π by: NL πs,d, ± r := 1 k NL e πs,d, ± where k is the degree of the cover e C → C . Since the Noether-Lefschetzdivisors are pulled back from the separated period domain, the definition isindependent of the choice of cover.3.8. Example: Prime discriminant in K [2] -type. Let V = E ( − ⊕ ⊕ U ⊕ ⊕ Z δ, δ = − K [2] -lattice and consider a primitive vector H ∈ V satisfying • H · H = 2 p for a prime p with p ≡ • h H, V i = 2 Z .Equivalently, H/ V ∨ and has norm p/
2. ByEichler’s criterion [19, Lemma 3.5] there exists a unique O ( V ) orbit of vectors H satisfying these condition. To be concrete we choose H = 2 (cid:18) e ′ + p + 14 f ′ (cid:19) + δ where e ′ , f ′ is a basis of one of the summands U . In V one then has M = H ⊥ ∼ = E ( − ⊕ ⊕ U ⊕ ⊕ − − − − p +12 ! , a lattice of discriminant group Z /p Z .We consider H -quasipolarized holomorphic-symplectic varieties X of K [2] -type. Examples are the Fano varieties of lines ( p = 3) or the Debarre-Voisinfourfolds ( p = 11), see below. For these varieties the Borcherds modularforms and the relationship between between Noether-Lefschetz divisors offirst and second type can be described very explicitly. ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 27
The Borcherds modular forms.
Consider the series of Noether-Lefschetznumbers of second type Φ π ( q ) = X γ Φ πγ ( q ) e γ for a 1-parameter family π of holomorphic-symplectic varieties of this polar-ization type. This is a modular form of weight 11 for the Weil representationon M ∨ /M . The space of such forms is easily computed through [51] andthe first values are given in the following table. p Table 1.
The dimension of the space of modular forms ofweight 11 for the Weilrepresentation associated to M .If we write y , y for the standard basis of the lattice − − − − p +12 ! , thenthe discrimimant of M is generated by y ′ = 1 p (2 y − y )which has norm y ′ · y ′ = − /p . Hence for any element v of M ∨ , written as v = w + ky ′ ∈ M ∨ , w ∈ M, k ∈ Z , we have − pv = k modulo p . In particular, this determines [ v ] ∈ Z p upto multiplication by ±
1. Thus for any v ∈ M ∨ we see that:(i) D := − p v · v is a square modulo p , and(ii) r = [ v ] is determined from D via r ≡ D mod p , up to multiplicationby ± y n,γ = y n, − γ , the coefficient q n e γ ofΦ( q ) is thus determined by n alone. It is hence enough to consider(14) ϕ π ( q ) = 12 Φ π ( q ) + 12 X γ ∈ M ∨ /M Φ πγ ( q ) . Let χ p be the Dirichlet character given by the Legendre symbol (cid:16) · p (cid:17) . Proposition 4.
The series Φ π ( q ) and P γ ∈ M ∨ /M Φ πγ ( q p ) are modular formsof weight and character χ p for the congruence subgroup Γ ( p ) .Proof. The modularity of the first series is well-known [4]. The second isone direction of the Bruinier-Bundschuh isomorphism [6]. (cid:3)
The generators of the ring of modular forms for the character χ p is easilycomputable (see e.g. [4, Sec.12]) which yields explicit formulas for ϕ π . Oneexample for Fano varieties can be found in [30]. We will consider the caseof Debarre-Voisin fourfolds below.Finally, the Noether-Lefschetz numbers of the family are given by: NL πs,d = ϕ π (cid:20) − p det (cid:18) p dd s (cid:19)(cid:21) . Noether-Lefschetz divisors the first type.
The relationship betweenNoether-Lefschetz divisors of the first and second type is not so easy to statein general. However, here the situation simplifies. For any w ∈ H ⊥ ⊂ V weconsider the intersection of w ⊥ with the period domain D H , D w ⊥ = { x ∈ D H |h x, w i = 0 } . The image of this divisor under the quotient map D H → D H / Γ M defines anirreducible divisor that by a result of Debarre and Macr`ı [13] only dependson the discriminant − e := disc( w ⊥ ⊂ M ) . Moreover, e is a square modulo p . We write C e for this divisor.The relationship between Noether-Lefschetz divisors of first and secondtype is given as follows: Proposition 5.
Let D ≥ be a square modulo p , and let α ∈ Z /p Z suchthat α ≡ D mod p . The associated Heegner divisor y − D/p,α , denoted alsoby NL ( D ) , is given by NL ( D ) = X a ≥ ,k ∈{ ,..., ⌊ p ⌋} e = pa + k ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) c ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) c = De , kc ≡ α mod p (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) C e In particular, we have NL ( D ) = ( C D + . . . if D = 0 mod 112 C D + . . . if D = 0 mod 11where . . . stands for terms C e with e < D . This shows that Noether-Lefschetz divisors of the first type from the Heegner divisors determine eachothers. If D is square free, then NL ( D ) and C D agree up to a constant. Proof.
For any positive e = pa + k with k ∈ { , . . . , ⌊ p ⌋} and a ≥ K e ⊂ V containing H and such that disc( K ⊥ e ) = − e . Thelattice is unique up to an automorphism of V that fixes H [13]. Fix s ∈ Z with 2 s ≡ d ≥ D = − det (cid:0) p dd s (cid:1) = ( d − ps ). Thenby Proposition 3, Remark 6 and the definition we have NL ( D ) = NL s,d = X e µ ( K e , s, d ) C e ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 29 where the multiplicity is given by(15) µ ( K e , s, d ) = |{ β ∈ K e ⊗ Q | β ∈ V ∨ , β · β = s, β · H = d }| . It remains to calculate the multiplicty. We first embed V into the Mukailattice Λ as the orthogonal of e + f such that δ = − e + f . Here e, f is asymplectic basis of a not previously used copy of U . One finds that b L = ( M ⊥ ⊂ Λ) ∼ = p +12 ! which has the integral basis x = e + f, x = e ′ + p + 14 f ′ + f. Let us next choose K e = Z H ⊕ Z ( kf ′ + e ′′ − a f ′′ )where e ′′ , f ′′ is a symplectic basis of a third copy of U . The saturation of K e ⊕ Z ( e + f ) inside Λ is then given by f K e ∼ = p +12 k k − a where the lattice is generated by x , x and x = kf ′ + e ′′ − a f ′′ .We follow the recipe of the proof of Lemma 5, that is we compare themultiplicity (15) with a simpler multiplicity for ˜ K e . If s ∈ Z , then for D to be an integer, we must have d even. Then as in Lemma 5 one gets: µ ( K e , s, d ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) β ∈ f K e (cid:12)(cid:12)(cid:12)(cid:12) β · x = 0 , β · x = d , β · β = s (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) If s + ∈ Z , then d is odd and µ ( K e , s, d ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) β ∈ f K e (cid:12)(cid:12)(cid:12)(cid:12) β · x = 1 , β · x = d + 12 , β · β = s + 12 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) The result follows from this by a direct calculation. For exposition weevaluate the multiplicity in the first case. Using that β · x = 0, any element β ∈ ˜ K e as on the right hand side is given by β = a ( − x + 2 x ) + cx e . Let ˜ d = d/
2. The condition β · x = ˜ d yields ap + kc = ˜ d which can besolved if and only if kc ≡ ˜ d mod p , in which case a = ( ˜ d − kc ) /p . Insertingthis expression into β · β yields c e = ˜ d − p s = −
14 (2 ps − d ) = D. Finally, D = ˜ d mod p , and hence if α = D mod p , then α = ± ˜ d . If α ≡ p , then the result follows. In the other case, among c ∈ {± p D/e } there is precisely one solution to kc ≡ ˜ d mod p if and only if there is precisely onesolution to kc ≡ α mod p . (cid:3) Example: Cubic fourfolds.
We consider Fano varieties of lines X ⊂ Gr (2 , H ( X, Z ). Hence their deformation type is governedby the discussion in Section 3.8 for p = 3. The Borcherds modular form forthe generic pencil of Fano varieties is computed in [30].Let U ⊂ P ( H ( P , O (3))) be the open locus corresponding to cubic four-folds with at worst ADE singularieties. There is a period mapping p : U → M H to the corresponding moduli space. The pullback of the divisors C e underthis mapping are the special cubic fourfolds of discriminant d = 2 e , see[20, 30]. (A cubic fourfold Y ⊂ P is special if it contains an algebraicsurface S such that the saturation of [ S ] and h is of discriminant d ).For the 1-parameter family π of Fano varieties of lines of a generic pencilof cubic fourfolds the Noether-Lefschetz numbers of the second type NL πs,d and of first type(16) NL ( D ) π = deg ι ∗ π NL ( D )are then related to the classical geometry of special cubic fourfolds. Forexample, NL − , = NL π ( D = 3) = 192 NL − , = NL π ( D = 7) = 917568are the degrees of the (closure of the) divisors in P ( H ( P , O (3))) parametriz-ing nodal and Pfaffian cubics respectively. The locus of determinantal cubicfourfolds p − C is of codimension ≥
2, see e.g. [22, Rmk 3.23], and hence NL − / , = NL π ( D = 1) = 0 . Thus one gets that NL π − / , = NL π ( D = 4) = 3402which is the degree of the locus p − C of cubics containing a plane. Theequalities of the Noether-Lefschetz numbers of first and second type abovefollow from Proposition 5: in the first three cases since D is square free, andthe last case we use that C = 0. ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 31
Example II: Debarre-Voisin fourfolds.
A Debarre-Voisin fourfold[14] is the holomorphic-symplectic variety X ⊂ Gr (6 , U ∨ , where U ⊂ C ⊗ O is theuniversal subbundle on the Grassmannian. These varieties are of K [2] -typeand the Pl¨ucker polarization is of degree H = 22 and pairs evenly with anyclass in H ( X, Z ). Hence we are in the situation of Section 3.8 for p = 11.The Noether-Lefschetz numbers for a generic pencil of these varieties willbe computed below.3.11. Refined Noether-Lefschetz divisors.
We will need refined Noether-Lefschetz divisors which also depend on the divisibility m ≥ K [ n ] -type. Let s ∈ Q , d = ( d , . . . , d ℓ ) ∈ Z ℓ and r ∈ Z n − be fixed.If ∆( s, d ) = 0 we set NL m,s,d, ± r = X L ⊂ ˜ L ⊂ V µ ( m, s, d, r | L ⊂ ˜ L ⊂ V ) · NL ˜ L where the refined multiplicity µ ( . . . ) is the number of classes β ∈ V ∨ whichare contained in e L ⊗ Q , satisfy β · β = s , β · L i = d i and such that thefollowing new conditions hold: div ( β ) = m, (cid:20) β div ( β ) (cid:21) = ± r ∈ Z / (2 n − Z . Note that we treat the residue different from the non-refined case.If ∆( s, d ) = 0 we define NL m,s,d, ± r := NL s,d, ± m · r if m is the gcd of d , . . . , d r and the unique class β ∈ L ⊗ Q with β · L i = d i /m lies in V ∨ and has residue [ β ] = ± r . Otherwise, we set NL m,s,d, ± r = 0.We then have(17) NL s,d, ± r = X m ≥ X ± r ′ ± m · r ′ = ± r NL m,s,d, ± r ′ . and NL m,s,d, ± r = NL ,s/m ,d/m, ± r . By a simple induction argument as in [25, Lemma 1], these two equationsshow that the data of the unrefined Noether-Lefschetz numbers are equiva-lent to the the data of the refined Noether-Lefschetz numbers/divisors.
Remark . If the residue of a class is determined by d and s , the inverserelation between refined and unrefined is easy to state. We simply have NL ,s,d = X k | gcd( d ,...,d ℓ ) µ ( k ) · NL s/k ,d/k , parallel to the multiple cover rule we study in this paper.4. Gromov–Witten theory and Noether-Lefschetz theory
Let V be the K [ n ] -lattice and let L ⊂ V be a fixed primitive sublatticewith integral basis L i . We consider a 1-parameter family π : X → C, L , . . . , L ℓ ∈ Pic( X )of L -quasipolarized holomorphic-symplectic varieties of K [ n ] -type.The goal of this section is to relate Gromov-Witten invariants of X infiber classes to the Noether-Lefschetz numbers of the family and the reducedGromov-Witten invariants in K [ n ] -type.4.1. Gromov-Witten invariants of the family.
Let γ i ∈ H ∗ ( X ) be co-homology classes which can be written in terms of polynomials p i in theChern classes of L i , γ i = p i ( c ( L ) , . . . , c ( L ℓ )) . Let M g,N ( X , d ) for d ∈ Z ℓ be the moduli space of N -marked genus g stablemaps f : C → X such that • f maps into the fibers of X , that is π ∗ f ∗ [ C ] = 0, and • f is of degree d i against L i , Z [ C ] f ∗ ( c ( L i )) = d i . We consider the invariants (cid:10) α ; γ , . . . , γ N (cid:11) X g,d = Z [ M g,n ( X ,d )] τ ∗ ( α ) ev ∗ ( γ ) · · · ev ∗ N ( γ N )where α ∈ H ∗ ( M g,n ) is tautological and τ is the forgetful map.4.2. Gromov-Witten invariants of the fiber.
Let X be any holomorphic-symplectic variety of K [ n ] -type and let β ∈ H ( X, Z ) be an effective curveclass. Assume there exists an embedding L ⊗ R ֒ → H ( X, R )which is an isometry onto its image such that β · L i = d i for all i . As usualwe let L i ∈ H ( X, R ) denote the image of L i ∈ L under this map. Let also γ i = p i ( L , . . . , L ℓ ) . By deformation invariance and the invaraince property of Section 2.2 thereduced Gromov–Witten invariant (cid:10) α ; γ , . . . , γ N (cid:11) Xg,β only depends on the
ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 33 degree d = ( d , . . . , d ℓ ), the polynomials p i , s = β · β , and the curve invariants m = div ( β ) and the residue set ± r = ± [ β/ div ( β )]. We write (cid:10) α ; γ , . . . , γ N (cid:11) Xg,β = (cid:10) α ; p , . . . , p N (cid:11) Xg,m,s,d, ± r . The relation.
Consider the refined Noether-Lefschetz numbers of thefamily NL πm,s,d, ± r := Z C ι ∗ π NL m,s,d, ± r ′ where ι π : C → M L is the morphism defined by the family. Proposition 6.
Let γ i = p i ( L , . . . , L ℓ ) ∈ H ∗ ( X ) . Then we have: (cid:10) α ; γ , . . . , γ N (cid:11) X g,d = X m,s, ± r NL πm,s,d, ± r · (cid:10) α · ( − g λ g ; p , . . . , p N (cid:11) Xg,m,s,d, ± r Here λ i are the i -th Chern classes of the Hodge bundle on the modulispace of stable curves. The proposition can be extended to more generalclasses γ i . It is enough to assume that γ i is the product of some polynomialin the L i and a class which restricts to a monodromy invariant class on eachfiber, for example a Chern class. Proof.
Identical to the case of K3 surfaces discussed in [36]. (cid:3)
BPS formulation.
We rewrite the relation of Proposition 6 in termsof BPS numbers. For simplicity assume that for β ∈ H ( X, Z ) the residue r ([ β ]) is determined by the degrees d i = β · L i . We write r ( d ) for the residue.Proposition 6 then says that (cid:10) α ; γ , . . . , γ N (cid:11) X g,d = X m,s NL πm,s,d · (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) Xg,m,s,d . Define the BPS numbers for the family invariants (cid:10) α ; γ , . . . , γ N (cid:11) X , BPS g,d := X k | d ( − r ( d )+ r ( d/k ) µ ( k ) k g − N − deg( α ) (cid:10) α ; γ , . . . , γ N (cid:11) X , BPS g,d/k and that of the reduced Gromov-Witten invariants: (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) X, BPS g,m,s,d := X k | m ( − r ( d )+ r ( d/k ) k g − N − deg( α ) µ ( k ) (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) Xg,m/k,s/k ,d/k . The following is the result of a short calculation:
Lemma 6.
We have (cid:10) α ; γ , . . . , γ N (cid:11) X , BPS g,d = X m,s NL m,s,d · (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) X, BPS g,m,s,d
In particular, if the multiple cover holds, the BPS invariants of X do notdepend on the divisibility m and so with (17) we obtain:(18) (cid:10) α ; γ , . . . , γ N (cid:11) X , BPS g,d = X s (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) X, BPS g, ,s,d X m,s NL m,s,d = X s NL s,d · (cid:10) α ( − g λ g ; p , . . . , p N (cid:11) X, BPS g, ,s,d . Mirror symmetry
Overview.
In this section we review how to use mirror symmetryformulas to compute the genus 0 Gromov-Witten invariants for the totalspace X of generic pencils of Fano varieties of lines of cubic fourfolds and ofDebarre-Voisin varieties.Mirror symmetry here means an application of the following results: Given-tal’s description of the I -function for complete intersections in toric varieties[18], the proof of the abelian/non-abelian correspondence by Webb that re-lates the I -function of a GIT quotient with that of its abelian quotient[50], and the genus 0 wallcrossing formula between quasi-maps and Gromov-Witten invariants for GIT quotients by Ciocan-Fontanine and Kim [11].We first determine the small I -function for the cases we are interested in,then we shortly recall how to relate the I and J functions. We assume basicfamiliarity with the language of [11, 50] throughout.5.2. I -functions. We work in the following setup: Let V be a vector spaceover C , and let G be a connected reductive group acting faithfully on V on the left. We also fix a character of G for which we assume that thesemistable and stable locus, denoted by V s ( G ), agrees. For simplicity wealso assume that the G -action on the stable locus is free. We consider theGIT quotient Y = V //G = V s ( G ) /G. Let T ⊂ G be a maximal torus and consider also the abelian quotient V s ( T ) /T . We have then the following diagram relating the abelian andnon-abelian quotients: V s ( G ) /T V s ( T ) /TV s ( G ) /G. jξ The Weyl group W of G acts naturally on the cohomology of V s ( G ) /T andone has the isomorphism: ξ ∗ : H ∗ ( V s ( G ) /G, Q ) ∼ = −→ H ∗ ( V s ( G ) /T, Q ) W . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 35
Let E be a G -representations and consider a smooth zero locus of theassociated homogeneous bundle E on Y , X ⊂ Y. The small I -function of X in Y is a formal series I X = I Y,E = 1 + X β =0 q β I β ( z )where β ∈ H ( Y, Z ) runs over all curve classes, q β is a formal variable and I β ( z ) is a formal series in z ± with coefficients in H ∗ ( Y, Q ). It can then bedetermined in the following steps: Abelian/Non-Abelian correspondence ([50]). ξ ∗ I Y,Eβ = j ∗ X ˜ β β Y α Q ˜ β · c ( L α ) k = −∞ ( c ( L α ) + kz ) Q k = −∞ ( c ( L α ) + kz ) I V//T,E ˜ β where α runs over the roots of G and L α is the associated line bundle on theabelian quotient, ˜ β ∈ H ( V //T, Z ) = Hom( χ ( T ) , Z ) runs over the charactersof T that restrict to the given character β ∈ H ( Y, Z ) = Hom( χ ( G ) , Z ) underthe map induced by χ ( G ) → χ ( T ). When it is clear from context, we willoften omit the pullbacks ξ ∗ and j ∗ from the notation. Twisting ([18]). When restricting the G -representation E to T , it decom-poses into a direct sum of 1-dimensional representations M i . We write M i also for the associated line bundles on V //T . Then I V//T,Eβ = rk( E ) Y i =1 c ( M i ) · β Y k =1 ( c ( M i ) + kz ) · I Xβ . Toric varieties ([18]). Let D i , i = 1 , . . . , n be the torus invariant divisorson the toric variety V //T . I V//Tβ = n Y i =1 Q k = −∞ ( D i + kz ) Q D i · βk = −∞ ( D i + kz ) . Example 2. (Projective space P n − ) We have I P n − d = ( Q dk =1 ( H + kz ) n ) − . Example 3. (Grassmannian) Let M k × n be the space of k × n -matrices actedon by GL( k ) by the left. Taking the determinant character, the stable locusis the locus of matrices of full rank and the associated GIT quotient is theGrassmannian Gr ( k, n ) = M k × n // det GL( k ) . The stable locus for the maximal torus T ⊂ GL( k ) of diagonal matrices isgiven by matrices where each row is non-zero. The abelian quotient is M k × n //T = P n − × . . . × P n − | {z } k times The roots of GL( k ) are e ∗ i − e ∗ j and correspond to O ( H i − H j ) where H i isthe hyperplane class pulled back from the i -th factor.The universal subbundle U → C n ⊗O Gr on the Grassmannian correspondsto the inclusion of G -representations M k × n × C k → M k × n × C n where a column vector w ∈ C k is acting on by g · w := ( g t ) − w , and C n carries the trivial representation. The Pl¨ucker polarization on Gr ( k, n ) thuscorresponds to the line bundle O ( H + . . . + H k ) on ( P n − ) k . Hence if weconsider degree d curves on the Grassmannian, in the abelian/non-abeliancorrespondence we have to sum over ( d , . . . , d k ) adding up to d .Calculating the I -function is then easy. For example, for k = 2 (anddropping the pullbacks ξ ∗ , j ∗ from notation), one obtains I Gr (2 ,n ) d = X d = d + d ( − d H − H + ( d − d ) zH − H Q d k =1 ( H + kz ) n Q d k =1 ( H + kz ) n where the division by H − H is to take place formally. Example 4. (Fano variety of a cubic fourfold) The Fano variety of a cubicfourfold X ⊂ Gr (2 ,
6) is a zero locus of a section of Sym ( U ∨ ). On theabelian quotient P × P this vector bundle corresponds to O (3 H ) ⊕ O (2 H + H ) ⊕ O ( H + 2 H ) ⊕ O (3 H ) . We find the I -function I X ⊂ Gr (2 , = I Gr (2 , d · Y i + i i ,i ≥ i d + i d Y k =1 ( i H + i H + kz ) . Example 5. (A pencil of cubic fourfolds) We consider a generic pencil ofcubic fourfolds
X ⊂ Gr (2 , × P . Since X is the zero locus of a genericsection of the globally generated bundle Sym ( U ∨ ) ⊗ O P (1), it is smoothby a Bertini type argument. The abelian quotient is P × P × P . Let h bethe hyperplane class on P . Then the I -function for the fiber part reads: I X ( d, = ( − d X d = d + d H − H + ( d − d ) zH − H · Q d k =1 ( H + kz ) Q d k =1 ( H + kz ) Y i + i i ,i ≥ i d + i d Y k =1 ( i H + i H + h + kz ) ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 37
Example 6. (A pencil of Debarre-Voisin fourfolds) We consider a pencil
X ⊂ Gr (6 , × P of DV fourfolds which is cut out by ∧ U ∨ ⊗ O (1). Theabelian quotient is ( P ) × P . We let h be the hyperplane class of P , and H i be the hyperplane class pulled back from the i -th copy of P . Then the I -function in the fiber class is: I X ( d, = ( − d X d = d + ... + d Y ≤ i Cubic fourfolds. We consider a generic pencil of Fano varieties ofcubic fourfolds X ⊂ Gr (2 , × P . This defines a 1-parameter family π : X → P polarized by the Pl¨uckerembeddings. The family has precisely 192 singular fibers X t , which are sin-gular along a smooth K3 surface with locally A -singularities in the normaldirection [10]. The blowup Bl S X t along the singular locus is isomorphic tothe Hilbert scheme Hilb ( S ), and the blowdown map contracts a P -bundleover S along its fibers which are ( − ǫ : C → P which is ramified along the 192 base points of nodal fibers. The family ǫ ∗ X → C then has double point singularities along the surfaces S which can be resolvedby a small resolution ˜ π : ˜ X → C. The family ˜ π is a well-defined 1-family of quasi-polarized K [2] -type vari-eties, polarized by the pullback of the Pl¨ucker polarization.The Noether-Lefschetz numbers of the family π in terms of ˜ π are then: NL πs,d = 12 NL ˜ πs,d . We also have the following comparision of Gromov-Witten invariants ofthe total spaces of X and ˜ X in fiber classes. We consider 1-pointed invariantsto simplify the notation. Lemma 7. For any i, α , h α ; H i i X , BPS g,d = 12 h α ; H i i e X , BPS g,d . Proof. We need to prove that h α ; H i i X g,d = 12 h α ; H i i e X g,d . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 39 This follows from the same argument as in [36, Lem. 4]: The conifoldtransition is taken relative to the K3 surface S . The extra components whichappear in the degeneration argument is a bundle (with fiber P ( O P ( − ⊕O P ( − ⊕ O P ) or a quadric in P ) over the K3 surface S . Because of theexistence of the symplectic form, it follows that the curve classes which maycontribute non-trivially have to be fiber. The argument of [29] then goesthrough without change. (cid:3) By the lemma the Gromov-Witten/Noether-Lefschetz relation of Propsi-tion 6 extends to the family π . Specializing to genus 0 we obtain that(19) (cid:10) α ; H i (cid:11) X , BPS0 ,d = X m,s (cid:10) α ; H i (cid:11) X, BPS0 ,m,s,d X m,s NL πm,s,d . The left hand side can be computed using the mirror symmetry formulasof Section 5. The primitive invariants appearing on the right hand side aregiven by Remark 4. The Noether-Lefschetz numbers NL s,d and hence theirrefinements NL m,s,d are determined by [30]and the formulas in Section 3.11.By using a computer (see the author’s website for the code) one finds that fordegree 6, 9, and 15 this equation uniquely determines the invariants f β , g β for β = mα where m = 2 , , α · α = 3 / 2. Moreover, one checks thenchecks that for these degrees we have(20) (cid:10) α ; H i (cid:11) X , BPS0 ,d = X s NL s,d · (cid:10) α ; H i (cid:11) X, BPS0 , ,s,d . which implies that Conjecture A holds in these cases. Together with Propo-sition 2 this proves Proposition 1. (As mentioned in the introduction, wehave checked (20) up to degree 38, which provides plenty of evidence forConjecture A.)6.2. Debarre-Voisin fourfolds. We consider a generic pencil of Debarre-Voisin fourfolds X ⊂ Gr (6 , × P , π : X → P . The case is very similar to the case of cubic fourfolds. We have the samedescription of singular fibers as in the Fano case [14, 3]. In particular,we may use the same double cover construction and conclude the Gromov-Witten/Noether-Lefschetz relation (19) for π .We want to determine the generating series of Noether-Lefschetz numbers ϕ ( q ) = X D ≥ q D/ NL π ( D )where D runs over squares modulo 11, and we used the notation of Sec-tion 3.8. Recall that we have NL πs,d = NL π ( D ) , where D = − 14 det (cid:18) dd s (cid:19) . We first prove the following basic invariants: Lemma 8. NL π (0) = − and NL π (11) = 640 .Proof. We have NL π (0) = Z P ι ∗ π c ( K ∗ )where K → D L / Γ M is the descent of the tautological bundle O ( − ι ∗ π K corresponds to the Hodge bundle π ∗ Ω π . Hence ι ∗ π K ∗ isisomorphic to L = R π ∗ O X which has fiber H ( X t , O X t ) over t ∈ P . In K -theory we have Rπ ∗ O = O P + L + L ⊗ By a Riemann-Roch calculation (using the software package [28]) we findthat 3 c ( L ) = Z ch ( Rπ ∗ O ) = Z π ∗ (td X / td P ) = − NL π (11) = NL − , is the number of singular fibers. To com-pute these, we recall that the singular locus of every singular fiber is asmooth K3 surface and the blowup along the singular locus has exceptionaldivisor a P -bundle over the K3 surface. Hence the topological Euler char-acteristic of a singular fiber is 300. By a standard computation (using [28])the topological Euler number of the total family is e ( X ) = − δ is the number of singular fibers we get − e ( X ) = 324(2 − δ ) + δ · , hence δ = 640 . The last part also follows from [14, Proof of Prop.3.1]. (cid:3) To further constrain the Noether-Lefschetz numbers we argue as fol-lows. By a computer check (see again the author’s webpage) the Gromov-Witten/Noether-Lefschetz relation(21) (cid:10) H (cid:11) X , BPS0 ,d = X m,s (cid:10) H (cid:11) X, BPS0 ,m,s,d NL πm,s,d . involves for d ≤ 13 only terms for which the multiple cover conjecture isknown by Proposition 2. Hence for d ≤ 13 we may rewrite it (cid:10) H (cid:11) X , BPS0 ,d = X s NL s,d · (cid:10) H (cid:11) X, BPS0 , ,s,d . The left hand side can be computed using the mirror symmetry formalism.The primitive invariants on the right are given in Remark 4. For 1 ≤ d ≤ ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 41 one obtains:(22) 0 = 264 NL (3)130680 = 3960 NL (1) + 132 NL (12)0 = 792 NL (5)3020160 = 264 NL (15) + 7920 NL (4)0 = 1320 NL (9)Using equations (22) and NL (0) = − 10, and employing William’s program[51] we find that:(23) NL (1) = NL (3) = NL (4) = NL (5) = NL (9) = 0 . This in turn determines the modular form Φ π uniquely. (One independentlychecks that indeed NL (11) = 640 matches the second result of Lemma 8.)Theorem 2 follows from this, from Proposition 4 and straightforward lin-ear algebra (by [4, Sec.12] we have that E , ∆ , E generate the ring ofmodular forms for character χ : M k ≥ Mod k (Γ (11) , χ k ) = C [ E , ∆ , E ] / (relations) ) . The proof of Corollary 1 follows from (23) and Proposition 5. Appendix A. A multiple cover rule for abelian surfaces In this appendix we state a conjectural rule that expressed reduced Gromov-Witten invariants of an abelian surfaces for any curve class β in terms ofinvariants for which β is primitive. The conjectural formula extends a pro-posal of [7] for the abelian surface analogue of the Katz-Klemm-Vafa for-mula. As in the hyperk¨ahler case the conjecture can be reinterpreted assaying that after subtracting multiple covers, the Gromov-Witten invariantsare independent of the divisibility.A.1. Monodromy. Recall that the cohomology of an abelian surface isdescribed by H i ( A, Q ) = i ^ H ( A, Q ) . The class of a point p ∈ H ( A, Z ) thus defines a canonical element p ∈ ^ H ( A, Q ) . An isomorphism of abelian groups ϕ : H ( A, Z ) → H ( A ′ , Z ) is a paralleltransport operator (i.e. the parallel transport along a deformation from A to A ′ through complex tori) if and only if ϕ preserves the canonical element [8, Sec.1.10]. The Zariski closure of the space of parallel transport operatorsis the set of C -vector spaces homomorphisms: M A,A ′ = { ϕ : H ( A, C ) → H ( A ′ , C ) | ϕ ( p ) = p ′ } . Any ϕ ∈ M A,A ′ extends naturally to a morphism of the full cohomology bysetting ϕ | H i ( A, C ) = ∧ i ϕ . It follows that the restriction to degree 2 H ( A, C ) → H ( A ′ , C )preserves the canonical inner product. If A ′ = A the above just says that themonodromy group is SL( H ( A, Z )) and its Zariski closure SL( H ( A, C )).A.2. Multiple cover rule. Let β ∈ H ( A, Z ) be an effective curve class.For any divisor k | β choose an abelian variety A k and a morphism ϕ k : H ( A, R ) → H ( A k , R ) preserving the canonical element such that the in-duced morphism ϕ k = M i ∧ i ϕ k : H ∗ ( A, R ) → H ∗ ( A k , R )takes β/k to a primitive effective curve class.Let also α ∈ H ∗ ( M g,n ) be a tautological class and γ i ∈ H ∗ ( A, R ) bearbitrary insertions. Conjecture C. For any effective curve class β ∈ H ( A, Z ) . D α ; γ , . . . , γ n E Ag,β = X k | β k g − n − deg( α ) D α ; ϕ k ( γ ) , . . . , ϕ k ( γ n ) E A k g,ϕ k ( β/k ) . A.3. Example. We apply the conjectural multiple cover formula to theanalogue of the Katz-Klemm-Vafa formula for abelian surfaces which is theintegral N FLS g,β = Z [ M g,n ( A,β ) FLS ] red ( − g − λ g − where M g,n ( A, β ) FLS is the substack of M g,n ( A, β ) that maps with image ina fixed linear system (FLS), see [7].To apply the multiple cover rule we specialize to A = E × E ′ . Considersymplectic bases α , β ∈ H ( E, Z ) , α , β ∈ H ( E ′ , Z )which give a basis of H ( A, Z ) (we omit the pullback), and let ω = α β , ω = α β ∈ H ( A, Z ) . We take β = ( d , d ) := d ω + d ω . For every k | gcd( d , d ) define ϕ k ∈ SL( H ( A, Q )) by α α , β kd β , α α , β d k β . ROMOV-WITTEN THEORY AND NOETHER-LEFSCHETZ THEORY 43 The extension to the full cohomology satisfies ϕ k ( β/k ) = ω + d d k ω = (1 , d d /k ) . Recall the result of Bryan ([7, Sec.3.2]) that: N FLS g,β = D ( − g − λ g − ; Y i =1 ξ i E Ag,β where we can take( ξ , ξ , ξ , ξ ) = ( ω α , ω β , α ω , β ω ) . Conjecture C then implies: N FLS g, ( d,d ′ ) = X k | β k g − (cid:10) ( − g − λ g − ; Y i =1 ϕ k ( ξ i ) (cid:11) g, (1 ,dd ′ /k ) = X k | β k g +3 N FLS g, (1 ,dd ′ /k ) which matches precisely Conjecture A in [7]. References [1] Y. Bae, T.-H. Buelles, Curves on K3 surfaces in divisibility two , , Forum of mathe-matics, Sigma, 9, e9 (2021), doi:10.1017/fms.2021.6[2] A. Beauville, R. 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