1.1 Overview

Let X be a (irreducible) holomorphic-symplectic variety. The lattice $H^{2}(X,{\mathbb {Z}})$ is equipped with the integral and nondegenerate Beauville–Bogomolov–Fujiki quadratic form. We will also equip $H^{\ast }(X,{\mathbb {Z}})$ with the usual Poincaré pairing. Both pairings are extended to the ${\mathbb {C}}$ -valued cohomology groups by linearity. Let $\mathsf {Mon}(X)$ be the subgroup of $O(H^{\ast }(X,{\mathbb {Z}}))$ generated by all monodromy operators, and let $\mathsf {Mon}^{2}(X)$ be its image in $O(H^{2}(X,{\mathbb {Z}}))$ .

The goal of this section is to describe the monodromy group in the case that X is of $K3^{[n]}$ -type, and we will assume so from now on. The main references for the sections are Markman’s papers [Reference Markman33, Reference Markman34].

1.2 Monodromy

Let X be of $K3^{[n]}$ -type. By work of Markman [Reference Markman34, Thm.1.3], [Reference Markman35, Lemma 2.1], we have that

(3) $$ \begin{align} \mathsf{Mon}(X) \cong \mathsf{Mon}^{2}(X) = \widetilde{O}^{+}(H^{2}(X,{\mathbb{Z}})) , \end{align} $$

where the first isomorphism is the restriction map and $\widetilde {O}^{+}(H^{2}(X,{\mathbb {Z}}))$ is the subgroup of $O(H^{2}(X,{\mathbb {Z}}))$ of orientation preserving lattice automorphisms which act by $\pm 1$ on the discriminant.Footnote ⁵ The first isomorpism implies that any parallel transport operator $H^{\ast }(X_{1}, {\mathbb {Z}}) \to H^{\ast }(X_{2}, {\mathbb {Z}})$ between two $K3^{[n]}$ -type varieties is uniquely determined by its restriction to $H^{2}(X_{1},{\mathbb {Z}})$ .

If $g \in \mathsf {Mon}^{2}(X)$ , we let $\tau (g) \in \{ \pm 1 \}$ be the sign by which g acts on the discriminant lattice. This defines a character

$$ \begin{align*} \tau : \mathsf{Mon}^{2}(X) \to {\mathbb{Z}}_{2}. \end{align*} $$

1.3 Zariski closure

By [Reference Markman33, Lemma 4.11] if $n \geq 3$ , the Zariski closure of $\mathsf {Mon}(X)$ in $O(H^{\ast }(X,{\mathbb {C}}))$ is $O(H^{2}(X,{\mathbb {C}})) \times {\mathbb {Z}}_{2}$ . The inclusion yields the representation

(4) $$ \begin{align} \rho : O(H^{2}(X,{\mathbb{C}})) \times {\mathbb{Z}}_{2} \to O(H^{\ast}(X,{\mathbb{C}})) \end{align} $$

which acts by degree-preserving orthogonal ring isomorphism. There is a natural embedding

$$ \begin{align*} \widetilde{O}^{+}(H^{2}(X,{\mathbb{Z}})) \to O(H^{2}(X,{\mathbb{C}})) \times {\mathbb{Z}}_{2},\, g \mapsto (g, \tau(g)) \end{align*} $$

under which $\rho $ restricts to the monodromy representation. In case $n \in \{ 1 ,2 \}$ , the Zariski closure is $O(H^{2}(X,{\mathbb {C}}))$ . In this case, we define the representation (4) by letting it act through $O(H^{2}(X,{\mathbb {C}}))$ .

The representation $\rho $ is determined by the following properties.

Property 0. For any $(g, \tau ) \in O(H^{2}(X,{\mathbb {C}})) \times {\mathbb {Z}}_{2}$ , we have

$$ \begin{align*} \rho(g,\tau)|_{H^{2}(X,{\mathbb{C}})} = g. \end{align*} $$

Property 2. We have

$$ \begin{align*} \rho(1, -1) = D \circ \rho(-\mathrm{id}_{H^{2}(X,{\mathbb{C}})}, 1), \end{align*} $$

where D acts on $H^{2i}(X,{\mathbb {C}})$ by multiplication by $(-1)^{i}$ .

Property 3. Assume that $X = \mathsf {Hilb}_{n}(S)$ , and identify $H^{2}(X,{\mathbb {Z}})$ with $H^{2}(S,{\mathbb {Z}}) \oplus {\mathbb {Z}} \delta $ . Then the restriction of $\rho $ to $O(H^{2}(X,{\mathbb {C}}))_{\delta } \times 1$ (identified naturally with $O(H^{2}(S,{\mathbb {C}}))$ ) is the Zariski closure of the induced action on the Hilbert scheme by the monodromy representation of S.

In particular, the action is equivariant with respect to the Nakajima operators: For $g \in O(H^{2}(X,{\mathbb {C}}))_{\delta }$ let $\tilde {g} = g|_{H^{2}(S,{\mathbb {C}}))} \oplus \mathrm {id}_{H^{0}(S,{\mathbb {Z}}) \oplus H^{4}(S,{\mathbb {Z}})}$ . Then

$$ \begin{align*} \rho(g, 1) \left( \prod_{i} {\mathfrak{q}}_{k_{i}}(\alpha_{i}) 1 \right) = \prod_{i} {\mathfrak{q}}_{k_{i}}( \tilde{g} \alpha_{i} ) 1. \end{align*} $$

Property 4. Let $P_{\psi } : H^{\ast }(X_{1}, {\mathbb {Z}}) \to H^{\ast }(X_{2}, {\mathbb {Z}})$ be a parallel transport operator with $\psi = P_{\psi }|_{H^{2}(X_{1},{\mathbb {Z}})}$ . Then

$$ \begin{align*} P_{\psi}^{-1} \circ \rho(g, \tau) \circ P_{\psi} = \rho( \psi^{-1} g \psi, \tau). \end{align*} $$

Property 1 follows by [Reference Markman33, Lemma 4.13]. The other properties also follow from the results of [Reference Markman33]. Properties 1–3 determine the action $\rho $ completely in the Hilbert scheme case. Moreover, by [Reference Oberdieck46], this description is explicit in the Nakajima basis. The last condition extends this presentation then to arbitrary X. The parallel transport operator between different moduli spaces of stable sheaves can also be described more explicitly [Reference Markman33].

1.4 Parallel transport

Let

$$ \begin{align*} \Lambda = E_{8}(-1)^{\oplus 2} \oplus U^{4} \end{align*} $$

be the Mukai lattice. For $n \geq 2$ , any holomorphic-symplectic variety of $K3[n]$ -type is equipped with a canonical choice of a primitive embedding

$$ \begin{align*} \iota_{X} : H^{2}(X,{\mathbb{Z}}) \to \Lambda \end{align*} $$

unique up to composition by an element by $O(\Lambda )$ ; see [Reference Markman34, Cor.9.4].

Theorem 3 [Reference Markman34, Thm.9.8]

An isometry $\psi : H^{2}(X_{1}, {\mathbb {Z}}) \to H^{2}(X_{1}, {\mathbb {Z}})$ is the restriction of a parallel transport operator if and only if it is orientation preserving and there exists an $\eta \in O(\Lambda )$ such that

$$ \begin{align*} \eta \circ \iota_{X_{1}} = \iota_{X_{2}} \circ \psi. \end{align*} $$

Orientation preserving is here defined with respect to the canonical choice of orientation of the positive cone of $X_{i}$ given by the real and imaginary part of the symplectic form and a Kähler class. If $X_{1} = X_{2}$ , the theorem reduces to the second isomorphism in equation (3).

1.5 Curve classes

By Eichler’s criterion [Reference Gritsenko, Hulek and Sankaran19, Lemma 3.5], Theorem 3 yields a complete set of deformation invariants of curve classes in $K3^{[n]}$ -type. To state the result we need the following constriction:

The orthogonal complement

$$ \begin{align*} L = \iota_{X}(H^{2}(X,{\mathbb{Z}}))^{\perp} \subset \Lambda \end{align*} $$

is isomorphic to the lattice ${\mathbb {Z}}$ with intersection form $(2n-2)$ . Let $v \in L$ be a generator, and consider the isomorphism of abelian groups

(5) $$ \begin{align} L^{\vee} / L \xrightarrow{\cong} {\mathbb{Z}} / (2n-2){\mathbb{Z}} \end{align} $$

determined by sending $v/(2n-2)$ to the residue class of $1$ . Since the generators of L are $\pm v$ , equation (5) is canonical up to multiplication by $1$ .

Since $\Lambda $ is unimodular, there exists a natural isomorphism ([Reference Huybrechts22, Sec.14])

$$ \begin{align*} H^{2}(X,{\mathbb{Z}})^{\vee} / H^{2}(X,{\mathbb{Z}}) \xrightarrow{\cong} L^{\vee}/L. \end{align*} $$

If we use Poincaré duality to identify $H_{2}(X,{\mathbb {Z}})$ with $H^{2}(X,{\mathbb {Z}})^{\vee }$ , this yields the residue map

$$ \begin{align*} r_{X}:H_{2}(X,{\mathbb{Z}})/H^{2}(X,{\mathbb{Z}}) \xrightarrow{\cong} L^{\vee}/L \xrightarrow{\cong} {\mathbb{Z}}/(2n-2) {\mathbb{Z}}. \end{align*} $$

The map depends on the choice of the generator v and, hence, is unique up to multiplication by $\pm 1$ .

Definition 1. The residue set of a class $\beta \in H_{2}(X,{\mathbb {Z}})$ is defined by

$$ \begin{align*} \pm [\beta] = \{ \pm r_{X}([\beta]) \} \subset {\mathbb{Z}} / (2n-2) {\mathbb{Z}} \end{align*} $$

if $n \geq 2$ and by $\pm [\beta ]=0$ otherwise.

Note that, since $r_{X}$ is canonical up to sign, the residue set is independent of the choice of map $r_{X}$ .

Remark 2. (i) Since parallel transport operators respect the embedding $i_{X}$ up to composing with an isomorphism of $\Lambda $ , the residue set $[\beta ]$ is preserved under deformation. (This is also reflected in the fact that the monodromy acts by $\pm 1$ on the discriminant.) (ii) In the case of the Hilbert scheme $X = \mathsf {Hilb}_{n}(S)$ of a K3 surface, let $A \in H_{2}(X)$ be the class of an exceptional curve, that is, the class of a fiber of the Hilbert–Chow morphism $\mathsf {Hilb}_{n}(S) \to \mathrm {Sym}^{n}(S)$ over a generic point in the singular locus. We have a natural identification

$$ \begin{align*} H_{2}(X,{\mathbb{Z}}) = H_{2}(S,{\mathbb{Z}}) \oplus {\mathbb{Z}} A. \end{align*} $$

The morphism $r_{X}$ then sends (up to sign) the class $[A]$ to $1 \in {\mathbb {Z}}_{2n-2}$ . (iii) In other words, we could have defined the residue class also by first deforming to the Hilbert scheme and then taking the coefficient of A modulo $2n-2$ . This is usually the practical way to compute the residue class.□

Let $\beta \in H_{2}(X,{\mathbb {Z}})$ be a class. The class $\beta $ has then the following deformation invariants:

1. (i) the divisibility $\mathsf {div}(\beta )$ in $H_{2}(X,{\mathbb {Z}})$ ,

2. (ii) the Beauville–Bogomolov norm $(\beta , \beta ) \in {\mathbb {Q}}$ , and

3. (iii) the residue set $\pm \left [ \frac {\beta }{\mathsf {div}(\beta )} \right ] \in {\mathbb {Z}} / (2n-2) {\mathbb {Z}}$ .

The global Torelli theorem and Eichler’s criterion yields the following.

1.6 Lifts of isometries of $H^{2}$

Let $X_{1}, X_{2}$ be of $K3^{[n]}$ -type, and let

$$ \begin{align*} g : H^{2}(X_{1},{\mathbb{C}}) \to H^{2}(X_{2},{\mathbb{C}}) \end{align*} $$

be an isometry. An operator $\widetilde {g} : H^{\ast }(X_{1},{\mathbb {C}}) \to H^{\ast }(X_{2},{\mathbb {C}})$ is a parallel transport lift of g if it is of the form

$$ \begin{align*} \widetilde{g} = \rho(g \circ \psi^{-1}, 1) \circ P_{\psi} \end{align*} $$

for a parallel transport operator $P_{\psi } : H^{\ast }(X_{1}, {\mathbb {Z}}) \to H^{\ast }(X_{2},{\mathbb {Z}})$ with restriction $\psi = P|_{H^{2}(X_{1},{\mathbb {Z}})}$ . In particular, any parallel transport lift is a degree-preserving orthogonal ring isomorphism.

Recall from Section 1.3, Property 2, the operator

$$ \begin{align*} \widetilde{D} = \rho(\mathrm{id}, -1) = D \circ \rho(-\mathrm{id}_{H^{2}(X_{2},{\mathbb{Z}})}, 1). \end{align*} $$

Proof. Consider two parallel transport lifts of g,

$$ \begin{align*} \widetilde{g_{i}} = \gamma(g \circ \psi_{i}^{-1}) \circ P_{\psi_{i}}, \quad i=1,2 \end{align*} $$

for parallel transport operators $P_{\psi _{1}}, P_{\psi _{2}}$ . We will show that

$$ \begin{align*} \widetilde{g}_{1} = \widetilde{g}_{2} \quad \text{or} \quad \widetilde{g}_{1} = \widetilde{D} \circ \widetilde{g}_{2}. \end{align*} $$

Let $\gamma (h) = \rho (h, 1)$ . If $\tau (\psi _{1} \circ \psi _{2}^{-1}) = 1$ , then

$$ \begin{align*} \gamma(g \circ \psi_{1}^{-1}) \circ P_{\psi_{1}} \circ P_{\psi_{2}}^{-1} = \gamma(g \psi_{1}^{-1}) \circ \gamma(\psi_{1} \circ \psi_{2}^{-1}) = \gamma( g \psi_{2}^{-1}). \end{align*} $$

If $\tau (\psi _{1} \circ \psi _{2}^{-1}) = -1$ , then

$$ \begin{align*} \gamma(g \circ \psi_{1}^{-1}) \circ P_{\psi_{1}} \circ P_{\psi_{2}}^{-1} = \gamma(g \psi_{1}^{-1}) \circ D \circ \gamma(-\psi_{1} \circ \psi_{2}^{-1}) = \widetilde{D} \circ \gamma( g \psi_{2}^{-1}) \end{align*} $$

2.1 Overview

Let X be a variety of $K3^{[n]}$ -type, and let $\beta \in H_{2}(X,{\mathbb {Z}})$ be an effective curve class. The moduli space ${\overline M}_{g,N}(X,\beta )$ of N-marked genus g stable maps to X in class $\beta $ carries a reduced virtual fundamental class $[ {\overline M}_{g,N}(X,\beta ) ]^{\text {red}}$ of dimension $(2n-3)(1-g)+N+1$ . Gromov–Witten invariants of X are defined by pairing with this class:

(6) $$ \begin{align} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{n} \big\rangle^{X}_{g,\beta} := \int_{ [ {\overline M}_{g,n}(X,\beta) ]^{\text{red}} } \pi^{\ast}(\alpha) \cup \prod_{i} \operatorname{ev}_{i}^{\ast}(\gamma_{i}), \end{align} $$

where $\operatorname {ev}_{i} : {\overline M}_{g,n}(X,\beta ) \to X$ are the evaluation maps, $\tau : {\overline M}_{g,n}(X,\beta ) \to {\overline M}_{g,n}$ is the forgetful map and $\alpha \in H^{\ast }({\overline M}_{g,n})$ is a tautological class [Reference Faber and Pandharipande16].

In this section, we state a conjecture to express the invariants (6) for $\beta $ an arbitrary class in terms of invariants where $\beta $ is primitive.

2.2 Invariance

Let $\beta \in H_{2}(X,{\mathbb {Z}})$ be an effective curve class, and let

$$ \begin{align*} \widetilde{O}^{+}(H^{2}(X,{\mathbb{Z}}))_{\beta} \subset \widetilde{O}^{+}(H^{2}(X,{\mathbb{Z}})) \end{align*} $$

be the subgroup fixing $\beta $ (either via the monodromy representation or equivalently, via the dual action on $H^{2}(X,{\mathbb {Z}})^{\vee } \cong H_{2}(X,{\mathbb {Z}})$ under the Beauville–Bogomolov form). Applying Remark 3 and the deformation invariance of the reduced Gromov–Witten invariants, we find that

$$ \begin{align*} \Big\langle \alpha\, ;\, \gamma_{1}, \ldots, \gamma_{n} \Big\rangle^{X}_{g,\beta} = \Big\langle \alpha\, ;\, \mu(\varphi) \gamma_{1},\ \ldots,\ \mu(\varphi) \gamma_{n} \Big\rangle^{X}_{g,\beta} \end{align*} $$

for all $\varphi \in \widetilde {O}^{+}(H^{2}(X,{\mathbb {Z}}))_{\beta }$ , where we have used

$$ \begin{align*} \mu : \widetilde{O}^{+}(H^{2}(X,{\mathbb{Z}})) \to O(H^{\ast}(X,{\mathbb{Z}})) \end{align*} $$

to denote the monodromy representation (defined by the isomorphism (3) composed with the inclusion $\mathsf {Mon}(X) \subset O(H^{\ast }(X,{\mathbb {Z}}))$ ).

The image of $\widetilde {O}^{+}(H^{2}(X,{\mathbb {Z}}))_{\beta }$ is Zariski dense in

$$ \begin{align*} G_{\beta} &= (O(H^{2}(X,{\mathbb{C}})) \times {\mathbb{Z}}/2)_{\beta} \\ &\kern-3pt:= \{ g \in O(H^{2}(X,{\mathbb{C}})) \times {\mathbb{Z}}/2 {\mathbb{Z}}\ |\ \rho(g) \beta = \beta \}. \end{align*} $$

It follows that for all $g \in G_{\beta }$ we have

(7) $$ \begin{align} \Big\langle \alpha\, ;\, \gamma_{1}, \ldots, \gamma_{n} \Big\rangle^{X}_{g,\beta} = \Big\langle \alpha\, ;\, \rho(g) \gamma_{1},\ \ldots,\ \rho(g) \gamma_{n} \Big\rangle^{X}_{g,\beta}. \end{align} $$

Equivalently, the pushforward of the reduced virtual class lies in the invariant part of the diagonal $G_{\beta }$ action:

$$ \begin{align*} \operatorname{ev}_{\ast} \left( \tau^{\ast}(\alpha) \cap [{\overline M}_{g,n}(X,\beta)]^{\text{red}} \right) \in H^{\ast}(X^{n}, {\mathbb{Q}})^{G_{\beta}}. \end{align*} $$

The representation $\rho $ restricted to $\{ \mathrm {id} \} \times {\mathbb {Z}}_{2}$ acts trivially on $H^{2}(X,{\mathbb {C}})$ . Hence, for X a K3 surface, we obtain the invariance of the Gromov–Witten class under the group $O(H^{2}(X,{\mathbb {C}}))_{\beta }$ . This matches what was conjectured in [Reference Oberdieck, Pandharipande, Faber, Farkas and van der Geer45, Conj.C1] and then proven along the above lines in [Reference Buelles9].

2.3 Multiple-cover conjecture

The main conjecture is the following: Let $\beta \in H_{2}(X,{\mathbb {Z}})$ be any effective curve class. For every divisor $k | \beta $ , let $X_{k}$ be a variety of $K3^{[n]}$ type and let

$$ \begin{align*} \varphi_{k} : H^{2}(X,{\mathbb{R}}) \to H^{2}(X_{k},{\mathbb{R}}) \end{align*} $$

be a real isometry such that:

• ○ $\varphi _{k}(\beta /k)$ is a primitive curve class

• ○ $\pm \left [ \varphi _{k}(\beta /k) \right ] = \pm [\beta /k]$ .

We extend $\varphi _{k}$ as a parallel transport lift (Section 1.6) to the full cohomology:

$$ \begin{align*} \varphi_{k} : H^{\ast}(X, {\mathbb{R}}) \to H^{\ast}(X_{k}, {\mathbb{R}}). \end{align*} $$

By Section 2.6 below, pairs $(X_{k}, \varphi _{k})$ satisfying these properties can always be found.

Conjecture B. For any effective curve class $\beta \in H_{2}(X,{\mathbb {Z}})$ , we have

$$ \begin{align*} &\Big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \Big\rangle^{X}_{g,\beta} \\ &\qquad= \sum_{k | \beta} k^{3g-3+N - \deg(\alpha)} (-1)^{[\beta] + [\beta/k]} \Big\langle \alpha; \varphi_{k}(\gamma_{1}), \ldots, \varphi_{k}(\gamma_{N}) \Big\rangle^{X_{k}}_{g,\varphi_{k}(\beta/k)}. \end{align*} $$

The invariance property discussed in Section 7 and Property 4 of Section 1.3 imply that the right-hand side of the conjecture is independent of the choice of $(X_{k},\varphi _{k})$ . Using that $\sum _{k|a} \mu (k) = \delta _{a1}$ , Conjecture B is also seen to be equivalent to Conjecture A of the introduction.

The reduced Gromov–Witten invariants of X can only be nonzero if the dimension constraint

$$ \begin{align*} (\dim X - 3)(1-g) + N + 1 = \deg(\alpha) + \sum_{i} \deg(\gamma_{i}) \end{align*} $$

is satisfied. Hence, the conjecture can also be rewritten as

$$ \begin{align*} \Big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \Big\rangle^{X}_{g,\beta} = \sum_{k | \beta} k^{\dim(X)(g-1) - 1 + \sum_{i} \deg(\gamma_{i})} (-1)^{[\beta] + [\beta/k]} \Big\langle \alpha; \varphi_{k}(\gamma_{1}), \ldots, \varphi_{k}(\gamma_{N}) \Big\rangle^{X}_{g,\varphi_{k}(\beta/k)}. \end{align*} $$

Since for K3 surfaces the residue always vanishes, Conjecture B specializes for K3 surfaces to the conjecture made in [Reference Oberdieck, Pandharipande, Faber, Farkas and van der Geer45, Conj C2].

Remark 4. The condition that we ask of the residue, i.e., $\pm \left [ \varphi _{k}(\beta /k) \right ] = \pm [\beta /k]$ , is necessary for the conjecture to hold. For example, consider X of $K3^{[5]}$ -type and two primitive classes $\beta _{1}, \beta _{2}$ with $(\beta _{1} \cdot \beta _{1}) = (\beta _{2} \cdot \beta _{2}) = 16$ but $[\beta _{1}]=0$ and $[\beta _{2}] = 4$ in ${\mathbb {Z}}/8 {\mathbb {Z}}$ . Then by [Reference Oberdieck42] one has

$$ \begin{align*} \operatorname{ev}_{\ast}[{\overline M}_{0,1}(X,\beta_{1})]^{\text{red}} = 1464 \beta_{1}^{\vee}, \quad \operatorname{ev}_{\ast}[{\overline M}_{0,1}(X,\beta_{2})]^{\text{red}} = 480 \beta_{2}^{\vee}, \end{align*} $$

so an isometry taking $\beta _{1}$ to $\beta _{2}$ does not preserve Gromov–Witten invariants.

2.4 Uniruled divisors

An uniruled divisor $D \subset X$ which is swept out by a rational curve in class $\beta $ is a component of the image of $\operatorname {ev}:{\overline M}_{0,1}(X,\beta ) \to X$ . The virtual class of these uniruled divisors is given by the pushforward $\operatorname {ev}_{\ast } [{\overline M}_{0,1}(X,\beta )]^{\text {red}}$ . For $\beta $ primitive and $(X,\beta )$ very generally, the virtual class is closely related to the actual class [Reference Oberdieck, Shen and Yin47].

By monodromy invariance (e.g., [Reference Oberdieck, Shen and Yin47, Section 2.6]), there exists $N_{\beta } \in {\mathbb {Q}}$ such that

(8) $$ \begin{align} \operatorname{ev}_{\ast} [{\overline M}_{0,1}(X,\beta)]^{\text{red}} = N_{\beta} \cdot h , \end{align} $$

where $h = ( \beta , - ) \in H_{2}(X,{\mathbb {Q}})^{\vee } \cong H^{2}(X,{\mathbb {Q}})$ is the dual of $\beta $ with respect to the Beauville–Bogomolov–Fujiki form. Since by deformation invariance $N_{\beta }$ only depends on the divisibility $m = \mathsf {div}(\beta )$ , the square $s = (\beta , \beta )$ and the residue $r = [\beta / \mathsf {div}(\beta )]$ we write

$$ \begin{align*} N_{\beta} = N_{m,s,r}. \end{align*} $$

Conjecture B then says that

(9) $$ \begin{align} N_{m,s,r} = \sum_{k | \beta} \frac{1}{k^{3}} (-1)^{mr + \frac{m}{k} r} N_{1, \frac{s}{k^{2}}, \frac{mr}{k}}. \end{align} $$

The primitive numbers $N_{1,s,r}$ have been determined in [Reference Oberdieck42].

Example 1. Let $X = \mathsf {Hilb}_{2}(S)$ , and let $A \in H_{2}(X,{\mathbb {Z}})$ be the class of the exceptional curve. We have

$$ \begin{align*} \operatorname{ev}_{\ast} [{\overline M}_{0,1}(X,\beta)]^{\text{red}} = \Delta_{\mathsf{Hilb}_{2}(S)} = -2 \delta. \end{align*} $$

Since $A^{\vee } = -\frac {1}{2} \delta $ , we see $N_{1,-\frac {1}{2},1} = 4$ . The multiple cover formula then predicts that for even $\ell \in {\mathbb {Z}}_{\geq 1}$ we have

$$ \begin{align*} N_{\ell A} = -\frac{1}{\ell^{3}} N_{1,-\frac{1}{2},1} + \frac{1}{(\ell/2)^{3}} N_{1,-2,0} = \frac{-4}{\ell^{3}} + \frac{8}{\ell^{3}} = \frac{4}{\ell^{3}}, \end{align*} $$

where we have used $N_{1,-2,0} = 1$ . This matches the degree-scaling property discussed in [Reference Maulik and Oblomkov36, Reference Okounkov and Pandharipande48].

2.5 Fourfolds

We consider genus $0$ Gromov–Witten invariants of a variety X of $K3^{[2]}$ -type. By dimension considerations, all genus $0$ Gromov–Witten invariants are determined by the two-point class:

$$ \begin{align*} \operatorname{ev}_{\ast} [{\overline M}_{0,2}(X,\beta)]^{\text{red}} \in H^{\ast}(X \times X). \end{align*} $$

Following the arguments of [Reference Oberdieck, Shen and Yin47, Sec.2]Footnote ⁶ , for every effective $\beta \in H_{2}(X,{\mathbb {Z}})$ there exist constantsFootnote ⁷ $F_{\beta }, G_{\beta } \in {\mathbb {Q}}$ such that

$$ \begin{align*} \operatorname{ev}_{\ast}[{\overline M}_{0,2}(X,\beta)]^{\text{red}} &= F_{\beta} (h^{2} \otimes h^{2}) + G_{\beta} \big(h \otimes \beta + \beta \otimes h + (h \otimes h) \cdot c_{BB} \big) \\ & \quad + \left( -\frac{1}{30} (h^{2} \otimes c_{2} + c_{2} \otimes h^{2}) + \frac{1}{900} (\beta, \beta) c_{2} \otimes c_{2} \right) (G_{\beta} + (\beta, \beta) F_{\beta} ), \end{align*} $$

where

• ○ $h = ( \beta , - ) \in H^{2}(X,{\mathbb {Q}})$ is the dual of the curve class,

• ○ $c_{2} = c_{2}(X)$ is the second Chern class, and

• ○ $c_{BB} \in H^{2}(X) \otimes H^{2}(X)$ is the inverse of the Beauville–Bogomolov–Fujiki form.

In $K3^{[2]}$ -type, the residue $r = [\beta ]$ of a curve class is determined by $s = (\beta ,\beta )$ via $r = 2 s \text { mod } 2$ . So we can write $F_{\beta } = F_{m,s}$ and $G_{\beta } = G_{m,s}$ . The multiple cover conjecture for $K3^{[2]}$ -type in genus $0$ is then equivalent to

$$ \begin{gather*} F_{m,s} = \sum_{k|m} \frac{1}{k^{5}} (-1)^{2 (s+s/k^{2})} F_{1,\frac{s}{k^{2}}} \\ G_{m,s} = \sum_{k|m} \frac{1}{k^{3}} (-1)^{2 (s+s/k^{2})} G_{1,\frac{s}{k^{2}}}. \end{gather*} $$

We can also define

$$ \begin{gather*} f_{m,s} = \sum_{k|m} \frac{\mu(k)}{k^{5}} (-1)^{2(s + s/k^{2})} F_{\frac{m}{k},\frac{s}{k^{2}}} \\ g_{m,s} = \sum_{k|m} \frac{\mu(k)}{k^{3}} (-1)^{2(s + s/k^{2})} G_{\frac{m}{k},\frac{s}{k^{2}}}. \end{gather*} $$

and arrive at the following.

Lemma 2. Conjecture B in $K3^{[2]}$ -type and genus $0$ is equivalent to

$$ \begin{align*} \forall m,s: \ f_{m,s} = f_{1,s}, \quad g_{m,s} = g_{1,s}. \end{align*} $$

The first few cases are known.

For later use, we will also need to following expression for one-pointed descendence invariants:

$$ \begin{align*} \operatorname{ev}_{\ast} [{\overline M}_{0,1}(X,\beta) ] & = G_{\beta} \beta^{\vee} \\ \operatorname{ev}_{\ast} \left( \psi_{1} \cdot [{\overline M}_{0,1}(X,\beta) ] \right) & = 2 F_{\beta} h^{2} - \frac{1}{15} \left( G_{\beta} + (\beta, \beta) F_{\beta} \right) c_{2}(X) \\ \operatorname{ev}_{\ast} \left( \psi_{1}^{2} \cdot [{\overline M}_{0,1}(X,\beta) ] \right) & = -12 F_{\beta} \\ \operatorname{ev}_{\ast} \left( \psi_{1}^{3} \cdot [{\overline M}_{0,1}(X,\beta) ] \right) & = 24 F_{\beta}. \end{align*} $$

This follows by monodromy invariance and topological recursions. In particular, to check the multiple cover formula in $K3^{[2]}$ -type and genus $0$ it is enough to consider one-point descendent invariants.

Remark 5. For convenience, we recall the evaluation of $f_{1,m,s}$ and $g_{1,m,s}$ . Let

$$ \begin{gather*} \vartheta(q) = \sum_{n \in {\mathbb{Z}}} q^{n^{2}}, \quad \alpha(q) = \sum_{\substack{\text{odd }n> 0 \\ d|n}} d q^{n}, \\ G_{2}(q) = -\frac{1}{24} + \sum_{n \geq 1} \sum_{d|n} d q^{n}, \quad \Delta(q) = q \prod_{n \geq 1} (1-q^{n})^{24}. \end{gather*} $$

Then by [Reference Oberdieck42] one has

$$ \begin{align*} \sum_{s} f_{1,s} (-q)^{4s} &= \frac{1}{4} \frac{-1}{\vartheta(q) \alpha(q) \Delta(q^{4})} \\ \sum_{s} g_{1,s} (-q)^{4s} &= \frac{1}{12} \frac{\vartheta^{4}(q) + 4 \alpha(q) + 24 G_{2}(q^{4})}{\vartheta(q) \alpha(q) \Delta(q^{4})}. \end{align*} $$

2.6 Hilbert schemes of elliptic K3 surfaces

Let

$$ \begin{align*} \pi : S \to \mathbb{P}^{1} \end{align*} $$

be an elliptic K3 surface with a section. Let $B, F \in H^{2}(S,{\mathbb {Z}})$ be the class of the section and a fiber, respectively, and let

$$ \begin{align*} W = B+F. \end{align*} $$

We consider the Hilbert scheme $X = \mathsf {Hilb}_{n} S$ and the generating series of Gromov–Witten invariants

$$ \begin{align*} F_{g,m}(\alpha; \gamma_{1}, \ldots, \gamma_{N}) = \sum_{d = -m}^{\infty} \sum_{r \in {\mathbb{Z}}} \left\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \right\rangle^{\mathsf{Hilb}_{n} S}_{g, mW+dF+rA} q^{d} (-p)^{r}. \end{align*} $$

We state a characterization of the multiple cover formula:

Consider the basis of $H^{\ast }(S,{\mathbb {R}})$ defined by

$$ \begin{align*} {\mathcal B} = \{ 1, {\mathsf{p}}, W, F, e_{3}, \ldots, e_{22} \}, \end{align*} $$

where ${\mathsf {p}}$ is the class of a point and $e_{3}, \ldots , e_{22}$ is a basis of ${\mathbb {Q}}\langle W,F \rangle ^{\perp }$ in $H^{2}(S,{\mathbb {R}})$ . An element $\gamma \in H^{\ast }(X,{\mathbb {R}})$ is in the Nakajima basis with respect to ${\mathcal B}$ if it is of the form

$$ \begin{align*} \gamma = \prod_{i} {\mathfrak{q}}_{k_{i}}(\alpha_{i})1, \quad \alpha_{i} \in {\mathcal B}. \end{align*} $$

Let $w(\gamma )$ and $f(\gamma )$ be the number of classes $\alpha _{i}$ which are equal to W and F, respectively, and define a modified degree grading $\underline {\deg }$ by:

$$ \begin{align*} \underline{\deg}(\gamma) = \deg(\gamma) + w(\gamma) - f(\gamma). \end{align*} $$

For a series $f = \sum _{d,r} c(d,r) q^{d} p^{r}$ , define the formal Hecke operator by

$$ \begin{align*} T_{m,\ell} f = \sum_{d,r} \left( \sum_{k|(m,d,r)} k^{\ell-1} c\left( \frac{md}{k^{2}}, \frac{r}{k} \right) \right) q^{d} p^{r}. \end{align*} $$

Lemma 3. Conjecture B holds if and only if for all $m>0$ , all $g, N, \alpha $ and all $(\deg , \underline {\deg })$ -bihomogeneous classes $\gamma _{i}$ we have

(10) $$ \begin{align} F_{g,m}(\alpha; \gamma_{1}, \ldots, \gamma_{N}) = m^{\sum_{i} \deg(\gamma_{i}) - \underline{\deg}(\gamma_{i})} \cdot T_{m,\ell} F_{g,1}(\alpha; \gamma_{1}, \ldots, \gamma_{N}) \end{align} $$

where $\ell = 2n(g-1) + \sum _{i} \underline {\deg }(\gamma _{i})$ .

Proof. Given the class $\beta = mW + dF + rA$ and a divisor $k | \beta $ , consider the real isometry $\varphi _{k} : H^{2}(X,{\mathbb {R}}) \to H^{2}(X,{\mathbb {R}})$ defined by

$$ \begin{align*} W & \mapsto \frac{k}{m} W \\ F & \mapsto \frac{m}{k} F \\ \gamma & \mapsto \gamma \text{ for all } \gamma \perp W,F. \end{align*} $$

We extend this map to the full cohomology by $\varphi _{k} = \rho (\phi _{k},0): H^{\ast }(X,{\mathbb {R}}) \to H^{\ast }(X,{\mathbb {R}})$ . We then have

$$ \begin{align*} \varphi_{k}\left( \frac{\beta}{k} \right) = W + \frac{dm}{k^{2}} F + \frac{r}{k} A. \end{align*} $$

The Nakajima operators are equivariant with respect to the action of $\varphi _{k}$ , and the isometry of $H^{\ast }(S,{\mathbb {R}})$ given by

$$ \begin{align*} \widetilde{\varphi}_{k} = \varphi_{k}|_{H^{2}(S,{\mathbb{R}})} \oplus \mathrm{id}_{H^{0}(S,{\mathbb{R}}) \oplus H^{4}(S,{\mathbb{R}})}; \end{align*} $$

see Property 3 of Section 1.3.

Let $\gamma _{i} \in H^{\ast }(X,{\mathbb {Q}})$ be elements in the Nakajima basis with respect to ${\mathcal B}$ . If Conjecture B holds, then its application with respect to $\varphi _{k}$ yields:

(11) $$ \begin{align} F_{g,m}(\alpha; \gamma_{1}, \ldots, \gamma_{N}) &= \sum_{d,r} q^{d} p^{r} \sum_{k|(m,d,r)} (-1)^{r/k} k^{3g-3+N-\deg(\alpha)} \nonumber\\ &\quad \times \left( \frac{m}{k} \right)^{\sum_{i} f(\gamma_{i}) -w(\gamma_{i})} \left\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \right\rangle^{\mathsf{Hilb}_{n}}_{g, W + \frac{md}{k^{2}} F + \frac{r}{k} A}. \end{align} $$

Using the dimension constraint, our modified degree function and the formal Hecke operators this becomes: Define the weight of a cohomology class in the Nakajima basis with respect to ${\mathcal B}$ by

$$ \begin{align*} {\mathsf{wt}}\left( \prod_{i} {\mathfrak{q}}_{a_{i}}(\alpha_{i}) 1 \right) = \sum_{i} {\mathsf{wt}}(\alpha_{i}), \end{align*} $$

where

$$ \begin{align*} {\mathsf{wt}}(\alpha) = \begin{cases} 1 & \text{ if } \alpha \in \{ {\mathsf{p}}, W \} \\ -1 & \text{ if } \alpha \in \{ 1, F \} \\ 0 & \text{ if } \alpha \in \{ 1,F,W,{\mathsf{p}} \}^{\perp}. \end{cases} \end{align*} $$

For a homogeneous $\gamma \in H^{\ast }(\mathsf {Hilb}_{n}(S))$ , we then set $\underline {\deg }(\gamma ) = {\mathsf {wt}}(\gamma ) + n$ .

Since the Nakajima operator ${\mathfrak {q}}_{i}(\alpha )$ has degree $i-1 + \deg (\alpha )$ , we have for $\gamma = \prod _{i=1}^{\ell } {\mathfrak {q}}_{i}(\alpha _{i}) 1 \in H^{\ast }(\mathsf {Hilb}_{n} S )$ that

$$ \begin{align*} \deg( \gamma ) = n - \ell + \sum_{i} \deg(\alpha_{i}). \end{align*} $$

Hence, we can rewrite

$$ \begin{align*} \sum_{i=1}^{n} \deg(\gamma_{i}) + w - f & = n N - \sum_{i} \ell(\gamma_{i}) + \sum_{i,j} \deg(\alpha_{ij}) + w - f \\ & = n M + \sum_{i} {\mathsf{wt}}(\gamma_{i}) \\ & = \sum_{i} \underline{\deg}(\gamma_{i}) \end{align*} $$

(12) $$ \begin{align} F_{g,m}(\alpha; \gamma_{1}, \ldots, \gamma_{N}) &= m^{\sum_{i} \deg(\gamma_{i}) - \underline{\deg}(\gamma_{i})} \nonumber\\ &\quad \times \sum_{d,r} q^{d} p^{r} \sum_{k|(m,d,r)} (-1)^{r/k} k^{2n(g-1)-1 + \sum_{i} \underline{\deg}(\gamma_{i})} \left\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \right\rangle^{\mathsf{Hilb}_{n}}_{g, W + \frac{md}{k^{2}} F + \frac{r}{k} A} \end{align} $$

which is equation (10).

Conversely, equality (10) implies Conjecture B since any pair $(X,\beta )$ is deformation equivalent to some $(\mathsf {Hilb}_{n}(S), \widetilde {\beta } = mW+dF+kA)$ with $m>0$ , and the right-hand side of Conjecture B is independent of choices.

We will reinterpret the lemma in terms of Jacobi forms in [Reference Oberdieck44]. See also [Reference Bae and Buelles1] for a parallel discussion in the case of K3 surfaces.

3.1 Lattice polarized holomorphic-symplectic varieties

Let V be the abstract lattice, and let $L \subset V$ be a primitive nondegenerate sublatticeFootnote ⁸ .

An L-polarization of a holomorphic-symplectic variety X is a primitive embedding

$$ \begin{align*} j : L \hookrightarrow \mathop{\textrm{Pic}}\nolimits(X) \end{align*} $$

such that

• ○ there exists an isometry $\varphi : V \xrightarrow {\cong } H^{2}(X,{\mathbb {Z}})$ with $\varphi |_{L} = j$ , and

• ○ the image $j(L)$ contains an ample class.

We call the isometry $\varphi $ 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 ${\mathcal M}_{L}$ be the moduli space of L-quasipolarized holomorphic-symplectic varieties of a given fixed deformation type.

The period domain associated to $L \subset V$ is

$$ \begin{align*} {\mathcal D}_{L} = \{ x \in \mathbb{P}(L^{\perp} \otimes {\mathbb{C}}) \, | \, \langle x, x \rangle = 0, \langle x, \bar{x} \rangle> 0 \} \end{align*} $$

and has two connected components. Let ${\mathcal D}_{L}^{+}$ be one of these components. Consider the subgroup

$$ \begin{align*} \mathsf{Mon}(V) \subset O(V) \end{align*} $$

which, for some choice of marking $\varphi : V \to H^{2}(X,{\mathbb {Z}})$ for some $(X,j)$ defining a point in ${\mathcal M}_{L}$ , can be identified with the monodromy group $\mathsf {Mon}^{2}(X)$ of X.Footnote ⁹ Let $\mathsf {Mon}(V)_{L}$ be the subgroup of $\mathsf {Mon}(V)$ that acts trivially on L. Then the global Torelli theorem says that the period mapping

$$ \begin{align*} \mathrm{Per} : {\mathcal M}_{L} \to {\mathcal D}_{L}^{+} / \mathsf{Mon}(V)_{L} \end{align*} $$

is surjective, restricts to an open embedding on the open locus of L-polarized holomorphic symplectic varieties and any fiber consists of birational holomorphic symplectic varieties; see [Reference Markman34] for a survey and references.

3.2 One-parameter families

Let $L_{i}, i = 1, \ldots , \ell $ be an integral basis of L.

Given a compact complex manifold ${\mathcal X}$ of dimension $2n+1$ , line bundles

$$ \begin{align*} {\mathcal L}_{1}, \ldots, {\mathcal L}_{\ell} \in \mathop{\textrm{Pic}}\nolimits({\mathcal X}) \end{align*} $$

and a morphism $\pi : {\mathcal X} \to C$ to a smooth proper curve, following [Reference Klemm, Maulik, Pandharipande and Scheidegger26, 0.2.2], we call the tuple $({\mathcal X}, {\mathcal L}_{1}, \ldots , {\mathcal L}_{\ell }, \pi )$ a one-parameter family of L-quasipolarized holomorphic-symplectic varieties if the following holds:

1. (i) For every $t \in C$ , the fiber $(X_{t}, L_{i} \mapsto {\mathcal L}_{i}|_{X_{t}})$ is a L-quasipolarized holomorphic-symplectic variety.

2. (ii) There exists an vector $h \in L$ which yields a quasi-polarization on all fibers of $\pi $ simultaneously.

Any one-parameter family as above defines a morphism $\iota _{\pi } : C \to {\mathcal M}_{L}$ into the moduli space of L-quasipolarized holomorphic-symplectic varieties (of the deformation type specified by a fiber).

3.3 Noether–Lefschetz cycles

Given primitive sublattices $L \subset \widetilde {L} \subset V$ , consider the open substack

$$ \begin{align*} {\mathcal M}_{\widetilde{L}}^{\prime} \subset {\mathcal M}_{\widetilde{L}} \end{align*} $$

parametrizing pairs $(X, j : \widetilde {L} \hookrightarrow \mathop {\text {Pic}}\nolimits (X))$ such that $j(L)$ contains a quasi-polarization. There exists a natural proper morphism $\iota : {\mathcal M}_{\widetilde {L}}^{\prime } \to {\mathcal M}_{L}$ defined by restricting j to L. The Noether–Lefschetz cycle associated to $\widetilde {L}$ is the class of the reduced image of this map:

$$ \begin{align*} \mathsf{NL}_{\widetilde{L}} = [ \iota( {\mathcal M}_{\widetilde{L}}^{\prime} ) ] \in A^{c}({\mathcal M}_{L}). \end{align*} $$

The codimension c of the cycle is given by $\mathrm {rank}(\widetilde {L}) - \mathrm {rank}(L)$ . For $c=1$ , we call $\mathsf {NL}_{\widetilde {L}}$ 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 for intersection numbers with Noether–Lefschetz divisors.

Consider the lattice

$$ \begin{align*} M = L^{\perp} \subset V \end{align*} $$

and the subgroup

$$ \begin{align*} \Gamma_{M} = \{ g \in O^{+}(M) | g \text{ acts trivially on } M^{\vee}/M \}, \end{align*} $$

where $O^{+}(M)$ stands for those automorphisms which preserve the orientation, or equivalently, the component ${\mathcal D}_{L}^{+}$ . We consider the quotient

$$ \begin{align*} {\mathcal D}_{L}^{+} / \Gamma_{M}. \end{align*} $$

For every $n \in {\mathbb {Q}}_{<0}$ and $\gamma \in M^{\vee }/M$ , the associated Heegner divisor is:

$$ \begin{align*} y_{n,\gamma} = \left( \sum_{\substack{ v \in M^{\vee} \\ \frac{1}{2} v \cdot v = n,\, [v] = \gamma } } v^{\perp} \right) / \Gamma_{M}. \end{align*} $$

For $n=0$ , we define $y_{n,\gamma }$ by the descent ${\mathcal K}$ of the tautological line bundle ${\mathcal O}(-1)$ on ${\mathcal D}_{L}$ equipped with the natural $\Gamma _{M}$ -action. Concretely, we set

$$ \begin{align*} y_{0,\gamma} = \begin{cases} c_{1}({\mathcal K}^{\ast}) & \text{ if } \gamma = 0 \\ 0 & \text{ otherwise.} \end{cases} \end{align*} $$

In case $n>0$ , we set $y_{n,\gamma } = 0$ in all cases.

Define the formal power series of Heegner divisors

$$ \begin{align*} \Phi(q) = \sum_{n \in {\mathbb{Q}}_{\leq 0}} \sum_{\gamma \in M^{\vee}/M} y_{n,\gamma} q^{-n} e_{\gamma} \end{align*} $$

which is an element of $\mathop {\text {Pic}}\nolimits ({\mathcal D}_{L}^{+} / \Gamma _{M})[[q^{1/N}]] \otimes {\mathbb {C}}[M^{\vee }/M]$ , where $e_{\gamma }$ are the elements of the group ring ${\mathbb {C}}[M^{\vee }/M]$ indexed by $\gamma $ and N is the smallest integer for which $M^{\vee }(N)$ is an even lattice.

We recall the modularity result of Borcherds in the formulation of [Reference Maulik and Pandharipande37].

Theorem 4. ([Reference Borcherds5, Reference McGraw39]) The generating series $\Phi (q)$ is the Fourier-expansion of a modular form of weight $\mathrm {rank}(M)/2$ for the dual of the Weil representation $\rho _{M}^{\vee }$ of the metaplectic group $\mathrm {Mp}_{2}({\mathbb {Z}})$ :

$$ \begin{align*} \Phi(q) \in \mathop{\textrm{Pic}}\nolimits({\mathcal D}_{L}^{+} / \Gamma_{M}) \otimes \mathsf{Mod}( \mathrm{Mp}_{2}({\mathbb{Z}}) , \mathrm{rank}(M)/2, \rho_{M}^{\vee} ). \end{align*} $$

The modular forms for the dual of the Weil representations can be computed easily by a Sage program of Brandon Williams [Reference Williams53].

3.5 Noether–Lefschetz divisors of the second type

The precise relationship between Noether–Lefschetz and Heegner divisors for arbitrary holomorphic-symplectic varieties is somewhat painful to state. For once the monodromy group $\mathsf {Mon}^{2}(X)$ is not known in general, and even if it is known it usually does not contain $\Gamma _{M}$ or is contained in $\Gamma _{M}$ . To simplify the situation, we from now on restrict to the case of $K3[n]$ -type for $n \geq 2$ . Hence, we let

$$ \begin{align*} V = E_{8}(-1)^{\oplus 2} \oplus U^{3} \oplus (2-2n), \end{align*} $$

and we fix an identification $V^{\vee }/V = {\mathbb {Z}}/(2n-2){\mathbb {Z}}$ .

We define the Noether–Lefschetz divisors of second type:

$$ \begin{align*} \mathsf{NL}_{s,d, \pm r} \in A^{1}({\mathcal M}_{L}), \end{align*} $$

where $d = (d_{1}, \ldots , d_{\ell }) \in {\mathbb {Z}}^{\ell }$ , $s \in {\mathbb {Q}}$ and $r \in {\mathbb {Z}}/(2n-2){\mathbb {Z}}$ are given. Consider the intersection matrix of the basis $L_{i}$ :

$$ \begin{align*} \mathsf{a} = \big( a_{ij} \big)_{i,j=1}^{\ell} = \big( L_{i} \cdot L_{j} \big)_{i,j=1}^{\ell}. \end{align*} $$

We set

$$ \begin{align*} \Delta(s, d) = \det \begin{pmatrix} a & d^{t} \\ d & s \end{pmatrix} = \det \begin{pmatrix} a_{11} & \cdots & a_{1 \ell} & d_{1} \\ \vdots & & \vdots & \vdots \\ a_{\ell 1} & \cdots & a_{\ell \ell} & d_{\ell} \\ d_{1} & \cdots & d_{\ell} & s \end{pmatrix}. \end{align*} $$

Case: $\Delta (s, d) \neq 0$ . We define

$$ \begin{align*} \mathsf{NL}_{s,d, \pm r} = \sum_{L\subset \tilde{L} \subset V} \mu(s,d, r | L\subset \tilde{L} \subset V ) \cdot \mathsf{NL}_{\tilde{L}}, \end{align*} $$

where the sum runs over all isomorphism classes of primitive embeddings $L \subset \tilde {L} \subset V$ with $\mathrm {rank}(\tilde {L}) = \ell +1$ . The multiplicityFootnote ¹⁰ $\mu (s,d, r | L\subset \tilde {L} \subset V )$ is the number of elements $\beta \in V^{\vee }$ which are contained in $\widetilde {L} \otimes {\mathbb {Q}}$ and satisfy:

$$ \begin{align*} \beta \cdot L_{i} = d_{i}, \quad \beta \cdot \beta = s, \quad \pm [\beta] = \pm r \text{ in } {\mathbb{Z}}/(2n-2). \end{align*} $$

Here we have used the canonical embeddings $V^{\vee } \subset V \otimes {\mathbb {Q}}$ and $\widetilde {L} \otimes {\mathbb {Q}} \subset V \otimes {\mathbb {Q}}$ .

Case: $\Delta (s, d) = 0$ . In this case, any curve class with these invariants has to lie in $L \otimes {\mathbb {Q}}$ and is uniquely determined by the degree d. Hence, we let $\beta \in L \otimes {\mathbb {Q}}$ be the unique class so that $\beta \cdot L_{i} = d_{i}$ for all i.Footnote ¹¹ If $\beta $ lies in $V^{\vee }$ and has residue $[\beta ] = \pm r$ , we define

$$ \begin{align*} \mathsf{NL}_{s,d, \pm r} = c_{1}({\mathcal K}^{\vee}), \end{align*} $$

and we define $\mathsf {NL}_{s,d, \pm r} = 0$ otherwise.

Remark 6. Often the residue set $\pm [\beta ]$ of a class $\beta \in H_{2}(X,{\mathbb {Z}})$ is determined by the degrees $d_{i} = \beta \cdot L_{i}$ . For example, if L contains a class $\ell $ such that

$$ \begin{align*} \langle \ell, H^{2}(X,{\mathbb{Z}}) \rangle = (2n-2) {\mathbb{Z}}, \quad \langle \ell, H_{2}(X,{\mathbb{Z}}) \rangle = {\mathbb{Z}} \end{align*} $$

we may define a natural isomorphism by cupping with $\ell $ :

$$ \begin{align*} H_{2}(X,{\mathbb{Z}}) / H^{2}(X,{\mathbb{Z}}) \xrightarrow{\cong} {\mathbb{Z}} / (2n-2){\mathbb{Z}}, \gamma \mapsto \gamma \cdot \ell. \end{align*} $$

(Not every polarization is of that form, for example the case of double covers of EPW sextics.) In other cases, the residue set is determined by the norm $\beta \cdot \beta $ , for example in $K3^{[2]}$ -type. When the residue is determined by s and d, we will drop it from the notation of Noether–Lefschetz divisors.□

3.6 Heegner and Noether–Lefschetz divisors

By the result of Markman, $\mathsf {Mon}(V) \subset O(V)$ is the subgroup of orientation-preserving isometries which act by $\pm \mathrm {id}$ on the discriminant. Hence, we have the inclusion $\Gamma _{M} \subset \mathsf {Mon}(V)_{L}$ of index $1$ or $2$ . This yields the diagram

where $\pi $ is either an isomorphism or of degree $2$ .

Let $C \subset {\mathcal M}_{L}$ be a complete curve, and define the modular form

$$ \begin{align*} \Phi_{C}(q) = \langle \Phi(q), \pi^{\ast} [\mathrm{Per}(C)] \rangle. \end{align*} $$

We write $\Phi _{C}[n,\gamma ]$ for the coefficient of $q^{n} e^{\gamma }$ in the Fourier-expansion of $\Phi _{C}$ .

We will need also

$$ \begin{align*} \widetilde{\Delta}(s, d) := -\frac{1}{2} \cdot \frac{1}{\det(a)} \det \begin{pmatrix} a & d^{t} \\ d & s \end{pmatrix}. \end{align*} $$

The following gives the main connection between the Noether–Lefschetz divisors of the second type and the Heegner divisors.

In $K3^{[2]}$ -type, we have $V^{\vee }/V = {\mathbb {Z}}_{2}$ so that $\pi $ is an isomorphism. Moreover, the residue r of any $\beta \in V^{\vee }$ is determined by its norm $s= \beta \cdot \beta $ . Hence, omitting r from the notation we find the following.

Corollary 3. In $K3^{[2]}$ type, there exists a canonical class $\gamma = \gamma (d,s)$ with

$$ \begin{align*} C \cdot \mathsf{NL}_{s,d} = \Phi_{C}[ \widetilde{\Delta}(s,d), \gamma ]. \end{align*} $$

Remark 7. In fact, in $K3^{[2]}$ type, the proof below will imply the equality of divisors

$$ \begin{align*} \mathsf{NL}_{s,d} = \Phi[ \widetilde{\Delta}(s,d), \gamma ] = y_{-\tilde{\Delta}(s,d), \gamma} \end{align*} $$

on ${\mathcal M}_{L}$ , where we have omitted the pullback by the period map $\mathrm {Per}$ on the right-hand side.

For the proof of Proposition 3, we will repeatedly use the following basic linear algebra fact whose proof we skip.

Lemma 4. Consider an ${\mathbb {R}}$ -vector space $\Lambda $ with inner product $\langle -, - \rangle $ and an orthogonal decomposition $L \oplus M = \Lambda $ . Let $L_{i}$ be a basis of L with intersection matrix $a_{ij} = L_{i} \cdot L_{j}$ . For $\beta \in \Lambda $ with $d_{i} = \beta \cdot L_{i}$ , let $v = \beta - \sum _{i,j} d_{i} a^{ij} L_{j}$ be the projection of $\beta $ onto M, where $a^{ij}$ are the entries of $a^{-1}$ . Then we have

$$ \begin{align*} \langle v,v \rangle = \frac{1}{\det(a)} \det \begin{pmatrix} a & d^{t} \\ d & \langle \beta, \beta \rangle \end{pmatrix}, \end{align*} $$

where $d = (d_{1}, \ldots , d_{\ell })$ .

The main step in the proof of the proposition is given by the following lemma: For fixed $d = (d_{1}, \ldots , d_{\ell }), s$ and $r \in {\mathbb {Z}}_{2n-2}$ consider the divisor on ${\mathcal D}_{L} / \Gamma _{M}$ given by

$$ \begin{align*} \mathsf{NL}_{s, d,r} = \left( \sum_{\beta} \beta^{\perp} \right)/\Gamma_{M}, \end{align*} $$

where the sum is over all classes $\beta \in V^{\vee }$ such that

(13) $$ \begin{align} \beta \cdot \beta = s, \quad \beta \cdot L_{i} = d_{i}, i=1,\ldots, \ell, \quad\text{and} \quad [\beta] = r \in V^{\vee}/V. \end{align} $$

Moreover, $\beta ^{\perp }$ stands for the hyperplane in $\mathbb {P}(V)$ orthogonal to $\beta $ intersected with the period domain $D_{L}$ .

Lemma 5. There exists a canonically defined class $\gamma = \gamma (s,d, r) \in M^{\vee }/M$ such that

$$ \begin{align*} \mathsf{NL}_{s, d,r} = y_{n,\gamma} \ \in A^{1}({\mathcal D}_{L}/\Gamma_{M}), \end{align*} $$

where $n = \frac {1}{2} \frac {1}{\det (a)} \det \binom {a\ d}{d\ s}$ .

Proof. In view of the definition of both sides of the claimed equation, it is enough to establish a bijection between

1. (a) the set of classes $\beta \in V^{\vee }$ satisfying (13), and

2. (b) the set of classes $v \in M^{\vee }$ satisfying $v^{2} = \frac {1}{\det (a)} \det \binom {a\ d}{d\ s}$ and $[v]=\gamma $ for an appropriately defined $\gamma $ .

Consider a primitive embedding $V \subset \Lambda $ into the Mukai lattice

$$ \begin{align*} \Lambda = E_{8}(-1)^{\oplus 2} \oplus U^{\oplus 4}. \end{align*} $$

Let $e,f$ be a symplectic basis of one summand of U. We choose the embedding such that $V^{\perp } = {\mathbb {Z}} L_{0}$ , where $L_{0} = e+(n-1)f$ . Since $\Lambda $ is unimodular, there exists a canonical isomorphism

(14) $$ \begin{align} V^{\vee}/V \cong ({\mathbb{Z}} L_{0})^{\vee} / {\mathbb{Z}} L_{0} ,\end{align} $$

and we may assume that under this isomorphism the class $L_{0}/(2n-2)$ mod ${\mathbb {Z}} L_{0}$ corresponds to $1 \in {\mathbb {Z}}/(2n-2) {\mathbb {Z}}$ .

Step 1. Let $d_{0} \in {\mathbb {Z}}$ be any integer such that $d_{0} \equiv r$ modulo $2n-2$ , and let $\widetilde {s} \in 2 {\mathbb {Z}}$ such that

$$ \begin{align*} s = \frac{1}{2n-2} \det \begin{pmatrix} 2n-2 & d_{0} \\ d_{0} & \widetilde{s} \end{pmatrix} \ \Longleftrightarrow \ \widetilde{s} = s + \frac{d_{0}^{2}}{2n-2}. \end{align*} $$

(We may assume such $\widetilde {s}$ exists: Otherwise, the set in (a) is empty and by the argument below also the set in (b)). Then we claim that there exists a bijection between the set in (a) and

1. (c) the set of classes $\widetilde {\beta } \in \Lambda $ such that $\widetilde {\beta } \cdot L_{i} = d_{i}$ for all $i=0,\ldots , \ell $ and $\widetilde {\beta } \cdot \widetilde {\beta } = \widetilde {s}$ .

Proof of Step 1

Given $\widetilde {\beta }$ satisfying the conditions in (c), then

$$ \begin{align*} \beta = \widetilde{\beta} - \frac{d_{0}}{2n-2} L_{0} \end{align*} $$

lies in $V^{\vee }$ . Moreover, $[\beta ]$ is the class in $V^{\vee }/V$ corresponding to $d_{0}/(2n-2)$ in $({\mathbb {Z}} L_{0})^{\vee } / {\mathbb {Z}} L_{0}$ under (14); hence, $[\beta ] = r$ . Also, $\beta \cdot L_{i}=d_{i}$ for $i=1,\ldots , \ell $ . The equality $\beta \cdot \beta = s$ is by definition of $\widetilde {s}$ and Lemma 4.

Conversely, let $\delta = -e + (n-1) f$ , and observe that $L_{0} \cdot \delta = 0$ and $L_{0} / (2n-2) + \delta /(2n-2) = f$ . Hence, if $\beta $ satisfies (a) then $\beta $ is an element of $d_{0} \cdot \frac {\delta }{2n-2} + V$ , and hence,

$$ \begin{align*} \beta \mapsto \beta + \frac{d_{0}}{2n-2} L_{0} \in \Lambda \end{align*} $$

defines the required inverse.

We consider now the embedding $M \subset \Lambda $ and the orthogonal complement

$$ \begin{align*} \widehat{L} = M^{\perp}. \end{align*} $$

Since $\Lambda $ is unimodular, we have an isomorphism

$$ \begin{align*} \widehat{L}^{\vee} / \widehat{L} \cong M^{\vee} / M. \end{align*} $$

We specify the class $\gamma $ via this isomorphism. Concretely, we set

$$ \begin{align*} \gamma := \left[ \sum_{i,j = 0}^{\ell} d_{i} a^{ij} L_{j} \right] \in \widehat{L}^{\vee} / \widehat{L}, \end{align*} $$

where we let $a^{ij}$ denote the entries of the inverse of the extended intersection matrix $\hat {a} = ( L_{i} \cdot L_{j} )_{i,j=0, \ldots , \ell }$ .

If we replace $d_{0}$ by $d_{0} + (2n-2)$ , then since $a^{00} = 1/(2n-2)$ and $a^{0j} = 0$ for $j \neq 0$ , the expression $\sum d_{i} a^{ij} L_{j}$ gets replaced by the same expression plus $L_{0}$ . Hence, the class $\gamma $ only depends on $s,(d_{1}, \ldots , d_{\ell }),r$ .

Step 2. There exists a bijection between the classes in (c) and (b).

Proof of Step 2

We have the bijection

$$ \begin{align*} \widetilde{\beta} \mapsto \widetilde{\beta} - \sum_{i,j=0}^{r} d_{i} a^{ij} L_{j} \in M^{\vee}. \end{align*} $$

Combining Step 1 and 2 finished the proof of the lemma.

Proof of Proposition 3

By definition we have:

$$ \begin{align*} \mathsf{NL}_{s,d_{1}, \ldots, d_{r}, \pm r} = \mathrm{Per}^{\ast}\left[ \left( \sum_{\beta} \beta^{\perp} \right) / \mathsf{Mon}(V)_{L} \right], \end{align*} $$

where the sum is over all $\beta \in V^{\vee }$ such that

$$ \begin{align*} \beta \cdot \beta = s, \quad \beta \cdot L_{i} = d_{i}, \quad [\beta ] = \pm r. \end{align*} $$

Hence, if $\pi $ is an isomorphism, the result follows from this by Lemma 5.

Hence, assume now $\pi $ is of degree $2$ , and let $\widetilde {C} = \mathrm {Per}(C)$ .

If $-r = r$ , then the morphism $\mathsf {NL}_{s,d, r} \to \mathsf {NL}_{s,d, \pm r}$ given by restriction of $\pi $ is of degree $2$ . Therefore,

$$ \begin{align*} C \cdot \mathsf{NL}_{s,d, \pm r} = \frac{1}{2} \widetilde{C} \cdot \pi_{\ast} \mathsf{NL}_{s,d, r} = \frac{1}{2} \pi^{\ast}[\widetilde{C}] \cdot \mathsf{NL}_{s,d, r} \end{align*} $$

which then implies the claim by Lemma 5. If $r \neq -r$ , then we have $\pi _{\ast } \mathsf {NL}_{s,d, r} = \mathsf {NL}_{s,d, \pm r}$ from which the result follows.

3.7 Noether–Lefschetz numbers

Let $({\mathcal X}, {\mathcal L}_{1}, \ldots , {\mathcal L}_{\ell }, \pi )$ be a one-parameter family of L-quasipolarized holomorphic-symplectic varieties of $K3^{[n]}$ -type. We have the associated classifying morphism

$$ \begin{align*} \iota_{\pi} : C \to {\mathcal M}_{L}. \end{align*} $$

We define the Noether–Lefschetz numbers of the family by

$$ \begin{align*} \mathsf{NL}^{\pi}_{s,d, \pm r} = \int_{C} \iota_{\pi}^{\ast}\mathsf{NL}^{\pi}_{s,d, \pm r}. \end{align*} $$

Intuitively, the Noether–Lefschetz numbers are the number of fibers of $\pi $ for which there exists a Hodge class $\beta $ with prescribed norm $\beta \cdot \beta = s$ , degree $\beta \cdot L_{i} = d_{i}$ and residue $[\beta ] = \pm r$ .

In $K3^{[2]}$ -type we will also often write $\Phi ^{\pi }(q) = \iota _{\pi }^{\ast } \Phi (q)$ .

The families of holomorphic-symplectic varieties we will encounter in geometric constructions often come with mildly singular fibers. The definition of Noether–Lefschetz numbers can be extended to these families as follows. Let $\pi : {\mathcal X} \to C$ be a projective flat morphism to a smooth curve, and let ${\mathcal L}_{1}, \ldots , {\mathcal L}_{\ell } \in \mathop {\text {Pic}}\nolimits ({\mathcal X})$ . We assume that over a nonempty open subset of C this defines a one-parameter family of L-quasipolarized holomorphic-symplectic varieties of $K3^{[n]}$ type. We also assume that around every singular point the monodromy is finite. Then there exists a cover

$$ \begin{align*} f:\widetilde{C} \to C \end{align*} $$

such that the pullback family $f^{\ast } {\mathcal X} \to \widetilde {C}$ is bimeromorphic to a one-parameter family of L-quasipolarized holomorphic-symplectic varieties of $K3^{[n]}$ -type,

$$ \begin{align*} \widetilde{\pi} : \tilde{{\mathcal X}} \to \widetilde{C}. \end{align*} $$

See, for example, [Reference Kollár, Laza, Saccà and Voisin28]. Concretely, around each basepoint of a singular fiber, after a cover that trivializes the monodromy, the rational map $C \dashrightarrow {\mathcal M}_{L}$ can be extended. (In the examples we will consider, we can construct the cover $\tilde {C} \to C$ and the birational model $\tilde {{\mathcal X}}$ explicitly). We define the Noether–Lefschetz numbers of $\pi $ by

$$ \begin{align*} \mathsf{NL}^{\pi}_{s,d,\pm r} := \frac{1}{k} \mathsf{NL}^{\widetilde{\pi}}_{s,d,\pm}, \end{align*} $$

where k is the degree of the cover $\widetilde {C} \to C$ . Since the Noether–Lefschetz divisors are pulled back from the separated period domain, the definition is independent of the choice of cover.

3.8 Example: Prime discriminant in $K3^{[2]}$ -type

Let

$$ \begin{align*} V = E_{8}(-1)^{\oplus 2} \oplus U^{\oplus 3} \oplus {\mathbb{Z}} \delta, \quad \delta^{2} = -2 \end{align*} $$

be the $K3^{[2]}$ -lattice and consider a primitive vector $H \in V$ satisfying

• ○ $H \cdot H = 2p$ for a prime p with $p \equiv 3$ mod $4$ ,

• ○ $\langle H, V \rangle = 2 {\mathbb {Z}}$ .

Equivalently, $H/2$ defines a primitive vector in $V^{\vee }$ and has norm $p/2$ . By Eichler’s criterion [Reference Gritsenko, Hulek and Sankaran19, Lemma 3.5], there exists a unique $O(V)$ orbit of vectors H satisfying these condition. To be concrete, we choose

$$ \begin{align*} H = 2\left( e^{\prime} + \frac{p+1}{4} f^{\prime} \right) + \delta, \end{align*} $$

where $e^{\prime },f^{\prime }$ is a basis of one of the summands U. In V one then has

$$ \begin{align*} M = H^{\perp} \cong E_{8}(-1)^{\oplus 2} \oplus U^{\oplus 2} \oplus \begin{pmatrix} -2 & -1 \\ -1 & -\frac{p+1}{2} \end{pmatrix}, \end{align*} $$

a lattice of discriminant group ${\mathbb {Z}}/p {\mathbb {Z}}$ .

We consider H-quasipolarized holomorphic-symplectic varieties X of $K3^{[2]}$ -type. Examples are the Fano varieties of lines ( $p=3$ ) or the Debarre–Voisin fourfolds ( $p=11$ ); see below. For these varieties, the Borcherds modular forms and the relationship between between Noether–Lefschetz divisors of first and second type can be described very explicitly.

3.8.1 The Borcherds modular forms

Consider the series of Noether–Lefschetz numbers of second type

$$ \begin{align*} \Phi^{\pi}(q) = \sum_{\gamma} \Phi^{\pi}_{\gamma}(q) e_{\gamma} \end{align*} $$

for a one-parameter family $\pi $ of holomorphic-symplectic varieties of this polarization type. This is a modular form of weight $11$ for the Weil representation on $M^{\vee }/M$ . The space of such forms is easily computed through [Reference Williams53], and the first values are given in the following table.

Table 1

The dimension of the space of modular forms of weight $11$ for the Weil representation associated to M.

If we write $y_{1}, y_{2}$ for the standard basis of the lattice $\begin {pmatrix} -2 & -1 \\ -1 & -\frac {p+1}{2} \end {pmatrix}$ , then the discrimimant of M is generated by

$$ \begin{align*} y^{\prime} = \frac{1}{p} (2 y_{2}-y_{1}) \end{align*} $$

which has norm $y^{\prime } \cdot y^{\prime } = -2/p$ . Hence, for any element v of $M^{\vee }$ , written as

$$ \begin{align*} v = w + k y^{\prime} \in M^{\vee}, \quad w \in M, k \in {\mathbb{Z}}, \end{align*} $$

we have $-\frac {1}{2} p v \cdot v = k^{2}$ modulo p. In particular, this determines $[v] \in {\mathbb {Z}}/p{\mathbb {Z}}$ up to multiplication by $\pm 1$ . Thus, for any $v \in M^{\vee }$ , we see that

1. (i) $D := -\frac {p}{2} v \cdot v$ is a square modulo p, and

2. (ii) $r=[v]$ is determined from D via $r^{2} \equiv D$ mod p, up to multiplication by $\pm 1$ .

By the redundancy of Heegner divisors $y_{n,\gamma } = y_{n,-\gamma }$ , the coefficient $q^{n} e_{\gamma }$ of $\Phi (q)$ is thus determined by n alone. It is, hence, enough to consider

(15) $$ \begin{align} \varphi^{\pi}(q) = \frac{1}{2} \Phi^{\pi}_{0}(q) + \frac{1}{2} \sum_{\gamma \in M^{\vee}/M} \Phi^{\pi}_{\gamma}(q). \end{align} $$

Let $\chi _{p}$ be the Dirichlet character given by the Legendre symbol $\left ( \frac { \cdot }{p} \right )$ .

Proposition 4. The series $\Phi ^{\pi }_{0}(q)$ and $\sum _{\gamma \in M^{\vee }/M} \Phi ^{\pi }_{\gamma }(q^{p})$ are modular forms of weight $11$ and character $\chi _{p}$ for the congruence subgroup $\Gamma _{0}(p)$ .

Proof. The modularity of the first series is well-known [Reference Borcherds4]. The second is one direction of the Bruinier–Bundschuh isomorphism [Reference Bruinier and Bundschuh6].

The generators of the ring of modular forms for the character $\chi _{p}$ is easily computable (see, e.g., [Reference Borcherds4, Sec.12]) which yields explicit formulas for $\varphi ^{\pi }$ . One example for Fano varieties can be found in [Reference Li and Zhang31]. We will consider the case of Debarre–Voisin fourfolds below.

Finally, the Noether–Lefschetz numbers of the family are given by

$$ \begin{align*} \mathsf{NL}^{\pi}_{s,d} = \varphi^{\pi}\left[ -\frac{1}{4p} \det \begin{pmatrix} 2p & d \\ d & s \end{pmatrix} \right]. \end{align*} $$

3.8.2 Noether–Lefschetz divisors the first type

The relationship between Noether–Lefschetz divisors of the first and second type is not so easy to state in general. However, here the situation simplifies. For any $w \in H^{\perp } \subset V$ , we consider the intersection of $w^{\perp }$ with the period domain ${\mathcal D}_{H}$ ,

$$ \begin{align*} {\mathcal D}_{w^{\perp}} = \{ x \in {\mathcal D}_{H} | \langle x, w \rangle = 0 \}. \end{align*} $$

The image of this divisor under the quotient map ${\mathcal D}_{H} \to {\mathcal D}_{H}/\Gamma _{M}$ defines an irreducible divisor that by a result of Debarre and Macrì [Reference Debarre and Macrì14] only depends on the discriminant

$$ \begin{align*} -2e := \mathrm{disc}( w^{\perp} \subset M ). \end{align*} $$

Moreover, e is a square modulo p. We write ${\mathcal C}_{2e}$ for this divisor.

The relationship between Noether–Lefschetz divisors of first and second type is given as follows.

Proposition 5. Let $D \geq 1$ be a square modulo p, and let $\alpha \in {\mathbb {Z}}/p {\mathbb {Z}}$ such that $\alpha ^{2} \equiv D$ mod p. The associated Heegner divisor $y_{-D/p,\alpha }$ , denoted also by $\mathsf {NL}(D)$ , is given by

$$ \begin{align*} \mathsf{NL}(D) = \sum_{\substack{a_{0} \geq 0, k \in \{ 0, \ldots, \lfloor \frac{p}{2} \rfloor \} \\ e = p a_{0} + k^{2} \geq 1}} \left| \left\{ c \in {\mathbb{Z}}\, \middle|\, c^{2} = \frac{D}{e},\ kc \equiv \alpha \text{mod } p \right\} \right| {\mathcal C}_{2e}. \end{align*} $$

In particular, we have

$$ \begin{align*} \mathsf{NL}(D) = \begin{cases} {\mathcal C}_{2D} + \ldots & \text{ if } D \neq 0 \text{ mod } 11 \\ 2 {\mathcal C}_{2D} + \ldots & \text{ if } D = 0 \text{ mod } 11, \end{cases} \end{align*} $$

where $\ldots $ stands for terms ${\mathcal C}_{2e}$ with $e<D$ . This shows that the Noether–Lefschetz divisors of the first type are related to the Heegner divisors by an invertible upper triangular matrx. If D is square free, then $\mathsf {NL}(D)$ and ${\mathcal C}_{2D}$ agree up to a constant.

Proof. For any positive $e = p a_{0} + k^{2}$ with $k \in \{ 0, 1 \ldots , \lfloor \frac {p}{2} \rfloor \}$ and $a \geq 1$ , we choose a lattice $K_{e} \subset V$ containing H and such that $\mathrm {disc}(K_{e}^{\perp }) = -2e$ . The lattice is unique up to an automorphism of V that fixes H [Reference Debarre and Macrì14]. Fix $s \in \frac {1}{2} {\mathbb {Z}}$ with $2s \equiv 3 (4)$ and $d \geq 1$ such that $D = -\frac {1}{4} \det \binom {2p\ d}{d\ s} = \frac {1}{4}( d^{2} - 2ps )$ . Then by Proposition 3, Remark 7 and the definition we have

$$ \begin{align*} \mathsf{NL}(D) = \mathsf{NL}_{s,d} = \sum_{e} \mu(K_{e}, s, d) {\mathcal C}_{2e}, \end{align*} $$

where the multiplicity is given by

(16) $$ \begin{align} \mu(K_{e}, s, d) = | \{ \beta \in K_{e} \otimes {\mathbb{Q}} | \beta \in V^{\vee}, \beta \cdot \beta = s, \beta \cdot H = d \} |. \end{align} $$

It remains to calculate the multiplicity. We first embed V into the Mukai lattice $\Lambda $ as the orthogonal of $e+f$ such that $\delta = -e+f$ . Here, $e,f$ is a symplectic basis of a not previously used copy of U. One finds that

$$ \begin{align*} \widehat{L} = (M^{\perp} \subset \Lambda) \cong \begin{pmatrix} 2 & 1 \\ 1 & \frac{p+1}{2} \end{pmatrix} \end{align*} $$

which has the integral basis

$$ \begin{align*} x_{1} = e+f, \quad x_{2} = e^{\prime} + \frac{p+1}{4} f^{\prime} + f. \end{align*} $$

Let us next choose

$$ \begin{align*} K_{e} = {\mathbb{Z}} H \oplus {\mathbb{Z}} (k f^{\prime} + e^{\prime\prime} - a_{0} f^{\prime\prime}), \end{align*} $$

where $e^{\prime \prime }, f^{\prime \prime }$ is a symplectic basis of a third copy of U. The saturation of $K_{e} \oplus {\mathbb {Z}} (e+f)$ inside $\Lambda $ is then given by

$$ \begin{align*} \widetilde{K}_{e} \cong \begin{pmatrix} 2 & 1 & 0 \\ 1 & \frac{p+1}{2} & k \\ 0 & k & -2 a_{0} \end{pmatrix}, \end{align*} $$

where the lattice is generated by $x_{1}, x_{2}$ and $x_{3} = k f^{\prime } + e^{\prime \prime } - a_{0} f^{\prime \prime }$ .

We follow the recipe of the proof of Lemma 5, that is we compare the multiplicity (16) with a simpler multiplicity for $\tilde {K}_{e}$ . If $s \in 2 {\mathbb {Z}}$ , then for D to be an integer, we must have d even. Then as in Lemma 5, one gets

$$ \begin{align*} \mu(K_{e}, s, d) = \left| \left\{ \beta \in \widetilde{K}_{e} \middle| \beta \cdot x_{1} = 0,\ \beta \cdot x_{2} = \frac{d}{2}, \ \beta \cdot \beta = s \right\} \right|. \end{align*} $$

If $s+\frac {1}{2} \in {\mathbb {Z}}$ , then d is odd, and

$$ \begin{align*} \mu(K_{e}, s, d) = \left| \left\{ \beta \in \widetilde{K}_{e} \middle| \beta \cdot x_{1} = 1,\ \beta \cdot x_{2} = \frac{d+1}{2}, \ \beta \cdot \beta = s+\frac{1}{2} \right\} \right|. \end{align*} $$

The result follows from this by a direct calculation. For exposition, we evaluate the multiplicity in the first case. Using that $\beta \cdot x_{1} = 0$ , any element $\beta \in \tilde {K}_{e}$ as on the right-hand side is given by

$$ \begin{align*} \beta = a (-x_{1} + 2 x_{2}) + c x_{e}. \end{align*} $$

Let $\tilde {d} = d/2$ . The condition $\beta \cdot x_{2} = \tilde {d}$ yields $a p + kc = \tilde {d}$ which can be solved if and only if $kc \equiv \tilde {d}$ mod p, in which case $a = (\tilde {d}-kc)/p$ . Inserting this expression into $\beta \cdot \beta $ yields

$$ \begin{align*} c^{2} e = \tilde{d}^{2} - \frac{p}{2} s = -\frac{1}{4}(2ps-d^{2}) = D. \end{align*} $$

Finally, $D = \tilde {d}^{2}$ mod p, and hence, if $\alpha ^{2} = D$ mod p, then $\alpha = \pm \tilde {d}$ . If $\alpha \equiv 0$ mod p, then the result follows. In the other case, among $c \in \{ \pm \sqrt {D/e} \}$ , there is precisely one solution to $kc \equiv \tilde {d}$ mod p if and only if there is precisely one solution to $kc \equiv \alpha $ mod p.

3.9 Example: Cubic fourfolds

We consider Fano varieties of lines

$$ \begin{align*} X \subset \operatorname{Gr}(2,6) \end{align*} $$

of a cubic fourfold. By [Reference Beauville and Donagi2], the Plücker polarization is of square $6$ and pairs evenly with any class in $H^{2}(X,{\mathbb {Z}})$ . Hence, their deformation type is governed by the discussion in Section 3.8 for $p=3$ . The Borcherds modular form for the generic pencil of Fano varieties is computed in [Reference Li and Zhang31].

Let $U \subset \mathbb {P}( H^{0}(\mathbb {P}^{5}, {\mathcal O}(3)))$ be the open locus corresponding to cubic fourfolds with at worst ADE singularities. There is a period mapping

$$ \begin{align*} p : U \to {\mathcal M}_{H} \end{align*} $$

to the corresponding moduli space. The pullback of the divisors ${\mathcal C}_{2e}$ under this mapping are the special cubic fourfolds of discriminant $d=2e$ ; see [Reference Hassett21, Reference Li and Zhang31]. (A cubic fourfold $Y \subset \mathbb {P}^{5}$ is special if it contains an algebraic surface S such that the saturation of $[S]$ and $h^{2}$ is of discriminant d).

For the one-parameter family $\pi $ of Fano varieties of lines of a generic pencil of cubic fourfolds, the Noether–Lefschetz numbers of the second type $\mathsf {NL}_{s,d}^{\pi }$ and of first type

(17) $$ \begin{align} \mathsf{NL}^{\pi}(D) = \deg \iota_{\pi}^{\ast} \mathsf{NL}(D) \end{align} $$

are then related to the classical geometry of special cubic fourfolds. For example,

$$ \begin{gather*} \mathsf{NL}^{\pi}_{-2,0} = \mathsf{NL}^{\pi}(D = 3) = 192 \\ \mathsf{NL}^{\pi}_{-2,4} = \mathsf{NL}^{\pi}(D = 7) = 917,568 \end{gather*} $$

are the degrees of the (closure of the) divisors in $\mathbb {P}( H^{0}(\mathbb {P}^{5}, {\mathcal O}(3)))$ parametrizing nodal and Pfaffian cubics, respectively. The locus of determinantal cubic fourfolds $p^{-1} {\mathcal C}_{2}$ is of codimension $\geq 2$ ; see, e.g., [Reference Huybrechts23, Rmk 3.23], and hence,

$$ \begin{align*} \mathsf{NL}^{\pi}_{-1/2,1} = \mathsf{NL}^{\pi}(D=1) = 0. \end{align*} $$

Thus, one gets that

$$ \begin{align*} \mathsf{NL}^{\pi}_{-5/2,1} = \mathsf{NL}^{\pi}(D = 4) = 3402 \end{align*} $$

which is the degree of the locus $p^{-1} {\mathcal C}_{8}$ of cubics containing a plane. The equalities of the Noether–Lefschetz numbers of first and second type above follow from Proposition 5: In the first three cases, since D is square free, and in the last case we use that ${\mathcal C}_{2}$ does not meet the curve defined by $\pi $ .

3.10 Example II: Debarre–Voisin fourfolds

A Debarre–Voisin fourfold [Reference Debarre and Voisin15] is the holomorphic-symplectic variety

$$ \begin{align*} X \subset \operatorname{Gr}(6,10) \end{align*} $$

given as the vanishing locus of a section of $\Lambda ^{3} {\mathcal U}^{\vee }$ , where ${\mathcal U} \subset {\mathbb {C}}^{10} \otimes {\mathcal O}$ is the universal subbundle on the Grassmannian. These varieties are of $K3^{[2]}$ -type, and the Plücker polarization is of degree $H^{2} = 22$ and pairs evenly with any class in $H^{2}(X,{\mathbb {Z}})$ . Hence, we are in the situation of Section 3.8 for $p=11$ . The Noether–Lefschetz numbers for a generic pencil of these varieties will be computed below.

3.11 Refined Noether–Lefschetz divisors

We will need refined Noether–Lefschetz divisors which also depend on the divisibility $m \geq 1$ of the curve class. Refined Noether–Lefschetz numbers are then defined as usually by intersection with Noether–Lefschetz divisors. As before, we assume that we are in $K3^{[n]}$ -type. Let $s \in {\mathbb {Q}}$ , $d = (d_{1}, \ldots , d_{\ell }) \in {\mathbb {Z}}^{\ell }$ and $r \in {\mathbb {Z}}_{2n-2}$ be fixed.

If $\Delta (s, d) \neq 0$ , we set

$$ \begin{align*} \mathsf{NL}_{m, s,d, \pm r} = \sum_{L\subset \tilde{L} \subset V} \mu(m, s,d, r | L\subset \tilde{L} \subset V ) \cdot \mathsf{NL}_{\tilde{L}}, \end{align*} $$

where the refined multiplicity $\mu ( \ldots )$ is the number of classes $\beta \in V^{\vee }$ which are contained in $\widetilde {L} \otimes {\mathbb {Q}}$ , satisfy $\beta \cdot \beta = s$ , $\beta \cdot L_{i} = d_{i}$ and such that the following new conditions hold:

$$ \begin{align*} \mathsf{div}(\beta) = m, \quad \left[ \frac{\beta}{\mathsf{div}(\beta)} \right] = \pm r \in {\mathbb{Z}}/(2n-2){\mathbb{Z}}. \end{align*} $$

Note that we treat the residue different from the nonrefined case.

If $\Delta (s, d) = 0$ , we define

$$ \begin{align*} \mathsf{NL}_{m, s,d, \pm r} := \mathsf{NL}_{s, d, \pm m \cdot r} \end{align*} $$

if m is the gcd of $d_{1}, \ldots , d_{r}$ and the unique class $\beta \in L \otimes {\mathbb {Q}}$ with $\beta \cdot L_{i} = d_{i}/m$ lies in $V^{\vee }$ and has residue $[\beta ] = \pm r$ . Otherwise, we set $\mathsf {NL}_{m, s,d, \pm r} = 0$ .

We then have

(18) $$ \begin{align} \mathsf{NL}_{s,d, \pm r} = \sum_{m \geq 1} \sum_{\substack{\pm r^{\prime}\\ \pm m \cdot r^{\prime} = \pm r}} \mathsf{NL}_{m, s,d, \pm r^{\prime}} \end{align} $$

and

$$ \begin{align*} \mathsf{NL}_{m,s,d, \pm r} = \mathsf{NL}_{1, s/m^{2}, d/m, \pm r}. \end{align*} $$

By a simple induction argument as in [Reference Klemm, Maulik, Pandharipande and Scheidegger26, Lemma 1], these two equations show that the data of the unrefined Noether–Lefschetz numbers are equivalent to the the data of the refined Noether–Lefschetz numbers/divisors.

Remark 8. If the residue of a class is determined by d and s, the inverse relation between refined and unrefined is easy to state. We simply have

$$ \begin{align*} \mathsf{NL}_{1,s,d} = \sum_{k|\mathrm{gcd}(d_{1}, \ldots, d_{\ell})} \mu(k) \cdot \mathsf{NL}_{s/k^{2}, d/k}, \end{align*} $$

parallel to the multiple cover rule we study in this paper.

4.1 Gromov–Witten invariants of the family

Let $\gamma _{i} \in H^{\ast }({\mathcal X})$ be cohomology classes which can be written in terms of polynomials $p_{i}$ in the Chern classes of ${\mathcal L}_{i}$ ,

$$ \begin{align*} \gamma_{i} = p_{i}(c_{1}({\mathcal L}_{1}), \ldots , c_{1}({\mathcal L}_{\ell}) ). \end{align*} $$

Let ${\overline M}_{g,N}({\mathcal X}, d)$ for $d \in {\mathbb {Z}}^{\ell }$ be the moduli space of N-marked genus g stable maps $f : C \to {\mathcal X}$ such that

• ○ f maps into the fibers of ${\mathcal X}$ , that is $\pi _{\ast } f_{\ast }[C] = 0$ , and

• ○ f is of degree $d_{i}$ against $L_{i}$ ,

  $$ \begin{align*} \int_{[C]} f^{\ast}(c_{1}({\mathcal L}_{i})) = d_{i}. \end{align*} $$

We consider the invariants

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}}_{g,d} = \int_{[{\overline M}_{g,n}({\mathcal X}, d)]} \tau^{\ast}(\alpha) \operatorname{ev}_{1}^{\ast}(\gamma_{1}) \cdots \operatorname{ev}_{N}^{\ast}(\gamma_{N}), \end{align*} $$

where $\alpha \in H^{\ast }({\overline M}_{g,n})$ is tautological and $\tau $ is the forgetful map.

4.2 Gromov–Witten invariants of the fiber

Let X be any holomorphic-symplectic variety of $K3^{[n]}$ -type, and let $\beta \in H_{2}(X,{\mathbb {Z}})$ be an effective curve class. Assume there exists an embedding

$$ \begin{align*} L \otimes {\mathbb{R}} \hookrightarrow H^{2}(X,{\mathbb{R}}) \end{align*} $$

which is an isometry onto its image such that $\beta \cdot L_{i} = d_{i}$ for all i. As usual, we let $L_{i} \in H^{2}(X,{\mathbb {R}})$ denote the image of $L_{i} \in L$ under this map. Let also

$$ \begin{align*} \gamma_{i} = p_{i}( L_{1}, \ldots, L_{\ell} ). \end{align*} $$

By deformation invariance and the invariance property of Section 2.2, the reduced Gromov–Witten invariant $\big \langle \alpha; \gamma _{1}, \ldots , \gamma _{N} \big \rangle ^{X}_{g,\beta }$ only depends on the degree $d = (d_{1}, \ldots , d_{\ell })$ , the polynomials $p_{i}$ , $s = \beta \cdot \beta $ and the curve invariants $m = \mathsf {div}(\beta )$ and the residue set $\pm r = \pm [ \beta / \mathsf {div}(\beta ) ]$ . We write

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{X}_{g,\beta} = \big\langle \alpha; p_{1}, \ldots, p_{N} \big\rangle^{X}_{g,m,s,d,\pm r}. \end{align*} $$

4.3 The relation

Consider the refined Noether–Lefschetz numbers of $\pi $ ,

$$ \begin{align*} \mathsf{NL}^{\pi}_{m, s,d, \pm r} := \int_{C} \iota_{\pi}^{\ast} \mathsf{NL}_{m, s,d, \pm r^{\prime}}, \end{align*} $$

where $\iota _{\pi }: C \to {\mathcal M}_{L}$ is the morphism defined by the family.

Proposition 6. Let $\gamma _{i} = p_{i}({\mathcal L}_{1}, \ldots , {\mathcal L}_{\ell }) \in H^{\ast }({\mathcal X})$ . Then we have

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}}_{g,d} = \sum_{m,s, \pm r} \mathsf{NL}^{\pi}_{m,s, d, \pm r} \cdot \big\langle \alpha \cdot (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X}_{g,m,s,d,\pm r}. \end{align*} $$

Here, $\lambda _{i}$ are the i-th Chern classes of the Hodge bundle on the moduli space of stable curves. The proposition can be extended to more general classes $\gamma _{i}$ . It is enough to assume that $\gamma _{i}$ is the product of some polynomial in the ${\mathcal L}_{i}$ and a class which restricts to a monodromy invariant class on each fiber, for example, a Chern class.

Proof. The proof follows by the identical argument as for the K3 surfaces, as discussed in [Reference Maulik and Pandharipande37, Section 3.2]. The above equality in the K3 case is [Reference Maulik and Pandharipande37, Eqn. (17)]. As in [Reference Maulik and Pandharipande37], for each $\xi \in C$ , we want to group together all curve classes in $H_{2}({\mathcal X}_{\xi },{\mathbb {Z}})$ of degree d which have the same Gromov–Witten invariants. By Corollary 2, we, hence, may group together classes of the same square, the same divisibility and the same residue. Thus, we replace the set $B_{\xi }(m,h,d)$ of [Reference Maulik and Pandharipande37, Sec.3.2] by

$$ \begin{align*} B_{\xi}(m,s,d, \pm r) = \left\{ \beta \in H_{2}({\mathcal X}_{\xi},{\mathbb{Z}}) \middle| \begin{array}{c} (\beta,\beta)=s,\ \mathsf{div}(\beta) = m, \\ {[} \beta/\mathsf{div}(\beta)] = \pm r,\ \beta \cdot L_{i} = d_{i} \\ \beta \perp H^{2,0}({\mathcal X}_{\xi},{\mathbb{C}}) \end{array} \right\}. \end{align*} $$

The rest of the argument of [Reference Maulik and Pandharipande37, Sec.3] goes through without change.

4.4 Reformulation

We can rewrite Proposition 6 in terms of invariants where we have formally subtracted multiple cover contrubtions. For simplicity, assume that for $\beta \in H_{2}(X,{\mathbb {Z}})$ the residue $r([\beta ])$ is determined by the degrees $d_{i} = \beta \cdot L_{i}$ . Write $r(d)$ for the residue. Proposition 6 then says that

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}}_{g,d} = \sum_{m,s} \mathsf{NL}^{\pi}_{m,s, d} \cdot \big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X}_{g,m,s,d}. \end{align*} $$

Let us subtract formally the multiple cover contributions from the invariants of ${\mathcal X}$ ,

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}, \text{mc}}_{g,d} := \sum_{k|d} (-1)^{r(d) + r(d/k)} \mu(k) k^{2g-3+N-\deg(\alpha)} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}}_{g,d/k} \end{align*} $$

as well as from the the reduced Gromov–Witten invariants,

$$ \begin{align*} &\kern-6pt\big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X,\text{mc}}_{g,m,s,d} \\ &\quad := \sum_{k|m} (-1)^{r(d) + r(d/k)} k^{2g-3+N-\deg(\alpha)} \mu(k) \big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X}_{g,m/k,s/k^{2},d/k}. \end{align*} $$

The following is the result of a short calculation.

Lemma 6. We have

$$ \begin{align*} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}, \text{mc}}_{g,d} = \sum_{m,s} \mathsf{NL}_{m,s,d} \cdot \big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X,\text{mc}}_{g,m,s,d}. \end{align*} $$

In particular, if the multiple cover conjecture (Conjecture B) holds, after subtracting the multiple cover contributions, the invariants of X do not depend on the divisibility m, and so with equation (18), we obtain

(19) $$ \begin{align} \begin{aligned} \big\langle \alpha; \gamma_{1}, \ldots, \gamma_{N} \big\rangle^{{\mathcal X}, \text{mc}}_{g,d} & = \sum_{s} \big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X,\text{mc}}_{g,1,s,d} \sum_{m,s} \mathsf{NL}_{m,s,d} \\ & = \sum_{s} \mathsf{NL}_{s,d} \cdot \big\langle \alpha (-1)^{g} \lambda_{g}; p_{1}, \ldots, p_{N} \big\rangle^{X,\text{mc}}_{g,1,s,d}. \end{aligned} \end{align} $$

5.1 Overview

In this section, we review how to use mirror symmetry formulas to compute the genus $0$ Gromov–Witten invariants for the total space ${\mathcal X}$ of generic pencils of Fano varieties of lines of cubic fourfolds and of Debarre–Voisin varieties.

Mirror symmetry here means an application of the following results: Givental’s description of the I-function for complete intersections in toric varieties [Reference Givental18], the proof of the abelian/nonabelian correspondence by Webb that relates the I-function of a GIT quotient with that of its abelian quotient [Reference Webb52] and the genus $0$ wallcrossing formula between quasi-maps and Gromov–Witten invariants for GIT quotients by Ciocan-Fontanine and Kim [Reference Ciocan-Fontanine and Kim11].

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 basic familiarity with the language of [Reference Ciocan-Fontanine and Kim11, Reference Webb52] throughout.

5.2 I-functions

We work in the following setup: Let V be a vector space over ${\mathbb {C}}$ , 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 the semistable and stable locus, denoted by $V^{s}(G)$ , agrees. For simplicity, we also assume that the G-action on the stable locus is free. We consider the GIT quotient

$$ \begin{align*} Y = V // G = V^{s}(G) / G. \end{align*} $$

Let $T \subset G$ be a maximal torus, and consider also the abelian quotient $V^{s}(T)/T$ . We have then the following diagram relating the abelian and nonabelian quotients:

The Weyl group W of G acts naturally on the cohomology of $V^{s}(G)/T$ , and one has the isomorphism

$$ \begin{align*} \xi^{\ast} : H^{\ast}(V^{s}(G)/G , {\mathbb{Q}}) \xrightarrow{\cong} H^{\ast}( V^{s}(G)/T, {\mathbb{Q}} )^{W}. \end{align*} $$

Let E be a G-representation, and consider a smooth zero locus of the associated homogeneous bundle E on Y,

$$ \begin{align*} X \subset Y. \end{align*} $$

The small I-function of X in Y is a formal series

$$ \begin{align*} I^{X} = I^{Y,E} = 1 + \sum_{\beta \neq 0} q^{\beta} I_{\beta}(z), \end{align*} $$

where $\beta \in H_{2}(Y,{\mathbb {Z}})$ runs over all curve classes, $q^{\beta }$ is a formal variable and $I_{\beta }(z)$ is a formal series in $z^{\pm 1}$ with coefficients in $H^{\ast }(Y,{\mathbb {Q}})$ . It can then be determined in the following steps.

Abelian/Nonabelian correspondence ([Reference Webb52]).

$$ \begin{align*} \xi^{\ast} I_{\beta}^{Y,E} = j^{\ast} \sum_{\tilde{\beta} \mapsto \beta} \left( \prod_{\alpha} \frac{\prod_{k=-\infty}^{\tilde{\beta} \cdot c_{1}(L_{\alpha})} (c_{1}(L_{\alpha}) + kz)}{\prod_{k=-\infty}^{0} (c_{1}(L_{\alpha}) + kz)} \right) I_{\tilde{\beta}}^{V//T,E}, \end{align*} $$

where $\alpha $ runs over the roots of G and $L_{\alpha }$ is the associated line bundle on the abelian quotient, $\tilde {\beta } \in H_{2}(V//T,{\mathbb {Z}}) = \operatorname {Hom}(\chi (T),{\mathbb {Z}})$ runs over the characters of T that restrict to the given character $\beta \in H_{2}(Y,{\mathbb {Z}}) = \operatorname {Hom}(\chi (G),{\mathbb {Z}})$ under the map induced by $\chi (G) \to \chi (T)$ . When it is clear from context, we will often omit the pullbacks $\xi ^{\ast }$ and $j^{\ast }$ from the notation.

Twisting ([Reference Givental18]). When restricting the G-representation E to T, it decomposes into a direct sum of one-dimensional representations $M_{i}$ . We write $M_{i}$ also for the associated line bundles on $V//T$ . Then

$$ \begin{align*} I^{V//T,E}_{\beta} = \left( \prod_{i=1}^{\mathrm{rk}(E)} \prod_{k=1}^{c_{1}(M_{i}) \cdot \beta} (c_{1}(M_{i}) + kz) \right) \cdot I^{X}_{\beta}. \end{align*} $$

Toric varieties ([Reference Givental18]). Let $D_{i}, i=1, \ldots , n$ be the torus invariant divisors on the toric variety $V//T$ .

$$ \begin{align*} I_{\beta}^{V//T} = \prod_{i=1}^{n} \frac{ \prod_{k=-\infty}^{0} (D_{i} + kz) }{ \prod_{k=-\infty}^{D_{i} \cdot \beta} (D_{i} + kz) }. \end{align*} $$

Example 3. (Grassmannian) Let $M_{k \times n}$ be the space of $k \times n$ -matrices acted on by $\mathrm {GL}(k)$ on the left. Taking the determinant character, the stable locus is the locus of matrices of full rank and the associated GIT quotient is the Grassmannian

$$ \begin{align*} \operatorname{Gr}(k,n) = M_{k \times n} //_{\det} \mathrm{GL}(k). \end{align*} $$

The stable locus for the maximal torus $T \subset \mathrm {GL}(k)$ of diagonal matrices is given by matrices where each row is nonzero. The abelian quotient is

$$ \begin{align*} M_{k \times n} // T = \underbrace{\mathbb{P}^{n-1} \times \ldots \times \mathbb{P}^{n-1}}_{k \text{ times }}. \end{align*} $$

The roots of $\mathrm {GL}(k)$ are $e_{i}^{\ast } - e_{j}^{\ast }$ and correspond to ${\mathcal O}(H_{i} - H_{j})$ , where $H_{i}$ is the hyperplane class pulled back from the i-th factor.

The universal subbundle ${\mathcal U} \to {\mathbb {C}}^{n} \otimes {\mathcal O}_{\operatorname {Gr}}$ on the Grassmannian corresponds to the inclusion of G-representations

$$ \begin{align*} M_{k \times n} \times {\mathbb{C}}^{k} \to M_{k \times n} \times {\mathbb{C}}^{n}, \end{align*} $$

where a column vector $w \in {\mathbb {C}}^{k}$ is acting on by $g \cdot w := (g^{t})^{-1} w$ , and ${\mathbb {C}}^{n}$ carries the trivial representation. The Plücker polarization on $\operatorname {Gr}(k,n)$ thus corresponds to the line bundle ${\mathcal O}(H_{1} + \ldots + H_{k})$ on $(\mathbb {P}^{n-1})^{k}$ . Hence, if we consider degree d curves on the Grassmannian, in the abelian/nonabelian correspondence we have to sum over $(d_{1}, \ldots , d_{k})$ adding up to d.

Calculating the I-function is then easy. For example, for $k=2$ (and dropping the pullbacks $\xi ^{\ast }$ , $j^{\ast }$ from notation), one obtains

$$ \begin{align*} I^{\operatorname{Gr}(2,n)}_{d} = \sum_{d=d_{1} + d_{2}} (-1)^{d} \frac{ H_{1} - H_{2} + (d_{1} - d_{2})z }{H_{1} - H_{2}} \frac{1}{ \prod_{k=1}^{d_{1}} (H_{1} + kz)^{n} \prod_{k=1}^{d_{2}} (H_{2} + kz)^{n} }, \end{align*} $$

where the division by $H_{1} - H_{2}$ is to take place formally.

Example 4. (Fano variety of a cubic fourfold) The Fano variety of a cubic fourfold $X \subset \operatorname {Gr}(2,6)$ is a zero locus of a section of $\mathrm {Sym}^{3}(U^{\vee })$ . On the abelian quotient $\mathbb {P}^{5} \times \mathbb {P}^{5}$ , this vector bundle corresponds to

$$ \begin{align*} {\mathcal O}(3 H_{1}) \oplus {\mathcal O}(2 H_{1} + H_{2}) \oplus {\mathcal O}(H_{1} + 2 H_{2}) \oplus {\mathcal O}(3 H_{2}). \end{align*} $$

We find the I-function

$$ \begin{align*} I^{X \subset \operatorname{Gr}(2,6)} = I^{\operatorname{Gr}(2,6)}_{d} \cdot \prod_{\substack{3 = i_{1} + i_{2} \\ i_{1}, i_{2} \geq 0}} \prod_{k=1}^{i_{1} d_{1} + i_{2} d_{2}} (i_{1} H_{1} + i_{2} H_{2} + kz). \end{align*} $$

Example 5. (A pencil of cubic fourfolds) We consider a generic pencil of cubic fourfolds ${\mathcal X} \subset \operatorname {Gr}(2,6) \times \mathbb {P}^{1}$ . Since ${\mathcal X}$ is the zero locus of a generic section of the globally generated bundle $\mathrm {Sym}^{3}(U^{\vee }) \otimes {\mathcal O}_{\mathbb {P}^{1}}(1)$ , it is smooth by a Bertini type argument. The abelian quotient is $\mathbb {P}^{5} \times \mathbb {P}^{5} \times \mathbb {P}^{1}$ . Let h be the hyperplane class on $\mathbb {P}^{1}$ . Then the I-function for the fiber part reads:

$$ \begin{align*} &\quad I^{{\mathcal X}}_{(d,0)} = (-1)^{d} \sum_{d=d_{1} + d_{2}} \frac{ H_{1} - H_{2} + (d_{1} - d_{2})z }{H_{1} - H_{2}} \cdot \\ &\frac{1}{ \prod_{k=1}^{d_{1}} (H_{1} + kz)^{6} \prod_{k=1}^{d_{2}} (H_{2} + kz)^{6} } \prod_{\substack{3 = i_{1} + i_{2} \\ i_{1}, i_{2} \geq 0}} \prod_{k=1}^{i_{1} d_{1} + i_{2} d_{2}} (i_{1} H_{1} + i_{2} H_{2} + h + kz). \end{align*} $$

Example 6. (A pencil of Debarre–Voisin fourfolds) We consider a pencil ${\mathcal X} \subset \operatorname {Gr}(6,10) \times \mathbb {P}^{1}$ of DV fourfolds which is cut out by $\wedge ^{3} {\mathcal U}^{\vee } \otimes {\mathcal O}(1)$ . The abelian quotient is $(\mathbb {P}^{9})^{6} \times \mathbb {P}^{1}$ . We let h be the hyperplane class of $\mathbb {P}^{1}$ and $H_{i}$ be the hyperplane class pulled back from the i-th copy of $\mathbb {P}^{9}$ . Then the I-function in the fiber class is

$$ \begin{align*} I^{{\mathcal X}}_{(d,0)} &= (-1)^{d} \sum_{d=d_{1} + \ldots + d_{6}} \left( \prod_{1 \leq i < j \leq 6} \frac{ H_{i} - H_{j} + (d_{i} - d_{j})z }{H_{i} - H_{j}} \right) \\ &\quad \times \prod_{i=1}^{6} \prod_{k=1}^{d_{i}} \frac{1}{ (H_{i} + kz)^{10} } \\ & \quad\times \prod_{1 \leq i_{1} < i_{2} < i_{3} \leq 6} \prod_{k=1}^{d_{i_{1}} + d_{i_{2}} + d_{i_{3}}} (H_{i_{1}} + H_{i_{2}} + H_{i_{3}} + h + kz). \end{align*} $$

5.3 I and J functions

Given a GIT quotient X as before, let $t \in H^{2}(X,{\mathbb {C}})$ be any element (or a formal variable). The big J-function (at $\epsilon = \infty $ ) with insertion t is

$$ \begin{align*} J^{\infty}(q,t,z) = e^{\frac{t}{z}} \left[ 1 + \sum_{\beta \neq 0} e^{\beta \cdot t} q^{\beta} \operatorname{ev}_{\ast}\left( \frac{ [ {\overline M}_{0,1}( X, \beta) ]^{\text{vir}} }{z (z-\psi)} \right) \right]. \end{align*} $$

Expand the small I-function according to degree:

$$ \begin{align*} I = I_{0}(q) + \frac{I_{1}(q)}{z} + \frac{I_{2}(q)}{z^{2}} + \ldots \,. \end{align*} $$

Then the mirror theorem [Reference Ciocan-Fontanine and Kim11] states that

$$ \begin{align*} J^{\infty}\left( q, \frac{I_{1}(q)}{I_{0}(q)} ,z \right) = \frac{I(q,z)}{I_{0}(q)}. \end{align*} $$

By inverting this relation, we can compute the descendent one-point invariants of X. We sketch the details for the case of the Lefschetz pencil ${\mathcal X} \subset \operatorname {Gr}(2,6) \times \mathbb {P}^{1}$ of Fano’s. In this case (with H the Plücker polarization on $\operatorname {Gr}(2,6)$ ), let us write

$$ \begin{align*} I_{0}(q) = f_{0}(q), \quad I_{1}(q) = f_{1}(q) H + f_{2}(q) h. \end{align*} $$

Then with $t = I_{1}(q)/I_{0}(q)$ and since $\beta $ is fiber we get

$$ \begin{align*} e^{\beta \cdot t} q^{d} = \left( q e^{ \frac{f_{1}(q)}{f_{0}(q)} } \right)^{d} = Q^{d}, \end{align*} $$

where we have identified $q^{\beta } = q^{d}$ and used the variable

$$ \begin{align*} Q = q \exp\left( \frac{f_{1}(q)}{f_{0}(q)} \right). \end{align*} $$

Then we obtain the relation

$$ \begin{align*} \exp\left( -\frac{I_{1}(q)}{I_{0}(q)} \frac{1}{z} \right) \frac{I^{\text{fib}}(q,z)}{I_{0}(q)} = 1 + \sum_{\beta \neq 0} Q^{\beta} \operatorname{ev}_{\ast}\left( \frac{ [ {\overline M}_{0,1}( X, \beta) ]^{\text{vir}} }{z (z-\psi)} \right), \end{align*} $$

where $I^{\text {fib}}(q,z)$ stands for the I-functions involving only fiber classes $\beta $ .