Modular forms and special cubic fourfolds
We study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ADE singularities. In this paper, we show that the generating series of the degrees of such divisors is a level three modular form.
The classical Noether-Lefschetz locus for degree hypersurfaces in is the locus in the Hilbert scheme space where the Picard rank is greater than one. For , the Noether-Lefschetz loci are known to be a countable union of proper subvarieties of by Griffiths and Harris . A natural question is to find the degrees of these subvarieties. For quartic surfaces in , Maulik and Pandharipande  showed that the Noether-Lefschetz loci are divisors and the degrees of these divisors are the Fourier coefficients of certain modular forms.
In higher dimensional cases, cubic fourfolds have received a lot of attention since their period map behaves quite nicely. Specifically, the period domain for cubic fourfolds is a bounded symmetric domain of type IV and the global Torelli theorem holds (cf. [25, 26]).
The analogues of Noether-Lefschetz loci for surfaces are the loci of special cubic fourfolds studied by Hassett . A smooth cubic fourfold in is special of discriminant if it contains an algebraic surface , and the discriminant of the saturated lattice spanned by and in is , where . The Zariski closure of the collection of such cubic fourfolds forms an irreducible divisor in the moduli space (cf. ) of cubic fourfolds with at worst isolated ADE singularities and it is nonempty if and only if . We interpret as the set of singular cubics in . The special cubic divisor in the Hilbert scheme of cubic hypersurfaces is the lift of (see 2 for more details). In the present paper, we study the degree of special cubic divisors . Our main result is:
Let be the generating series for the degrees of the special cubic divisors. Then is a modular form of weight and level with expansion:
are level three modular forms of weight 1 and 3 that generate the space of modular forms with respect to the group (see 3.2). Here, denotes the Legendre symbol.
The approach to Theorem 1 is via the result of Borcherds  and Kudla-Milson . The degrees of are the Fourier coefficients of a vector-valued modular form. As in , the Noether-Lefschetz numbers are related to the reduced Gromov-Witten (GW) invariants of K3 surfaces. We hope there is a similar GW-theory interpretation of .
Outline of paper
In section 2, we review some classical result on cubic fourfolds and describe the special cubic divisors
from an arithmetic perspective. Section 3 is the central section of this paper. We recap Borcherds’ work on Heegner divisors to prove the modularity of a
vector-valued generating series of . This vector-valued modular form can be expressed explicitly in terms of some well-known modular forms. The proof of our main theorem is presented in the last section.
After posting our paper on the arXiv server, we learned from Atanas Iliev that, in forthcoming work, Atanas Iliev, Emanuel Scheidegger,
and Ludmil Katzarkov in  have independently proved Theorem 3 using a different basis of the space of vector-valued modular forms.
Acknowledgements. The first author was supported by NSF grant 0901645. The authors are grateful to their advisor Brendan Hassett for introducing this problem, and many useful discussions. We have benefited from discussions with Radu Laza. We would also like to thank the referee for many helpful comments.
2. Special Cubic fourfolds and Heegner divisors
In this section, we review some results on special cubic divisors and the relation with Heegner divisors associated to a signature lattice, which is defined from an arithmetic perspective. Throughout the paper, we denote by the dual of a lattice and its associated orthogonal group.
2.1. Period domain
under the intersection form , with an associated period domain that is a connected component of
Here, and are root lattices corresponding to the root systems of the same names, and is the hyperbolic lattice of rank two. The monodromy group (i.e. the identity connected component of ) is defined by
acts on and the arithmetic quotient is a quasi-projective variety parametrizing the periods of cubics.
Hassett  has defined irreducible divisors as follows:
Let be a rank-two positive definite saturated sublattice of containing of discriminant . There is an associated hyperplane
in . Then is defined as the quotient by of the union of all such hyperplanes .
On the other hand, as constructed by Laza  via geometric invariant theory (GIT), the moduli space of cubic fourfolds with at worst isolated ADE singularities is a Zariski open subset of , where and denotes the GIT stable points of under the action of . Together with Voisin’s Global Torelli theorem (cf. [25, 26]), Laza  and also Looijenga  have shown that there is an extended period map
which is an open immersion. The complement of the image of in is corresponding to degenerations of determinant cubic hypersurfaces (cf.  4.4). Moreover, is exactly the pullback of via .
Let be the natural quotient map; then the special cubic divisor is the Zariski closure of the pullback of via .
2.2. Heegner divisors
In general, let be an even lattice of signature with an associated domain as a connected component of
there is an arithmetic group
acting on . For and , the Heegner divisor  on is defined by
where is a hyperplane of . When , we take to be the -Cartier divisor coming from on . Similarly to , it is clear from definition that is actually the Hodge bundle on , where is the universal family and is the relative sheaf of holomorphic 1-forms.
One can see that is the arithmetic quotient of a Hermitian symmetric subdomain of . They are called special cycle on in Kudla’s program .
Taking , we have and . Note that , one can choose representatives with for
The Heegner divisors on , where
The redundancy is because of the symmetry . Let be a rank 2 negative sublattice containing and of discriminant . Assume that is generated by and . Then there is a bijection between the two sets of hyperplanes as follows:
since one can verify that
As an application, we show the following result:
The Picard group has rank two and is spanned by Heegner divisors and .
Let be the subgroup of generated by the Heegner divisors. By applying the general formula from Bruinier  5.2 to the lattice , we get
On the other hand, the moduli space of cubic fourfolds with at worst isolated ADE singularities is an open subset of via the extended period map, and the complement of is the irreducible Heegner divisor . If the Picard number of is at most one, then is at most two and has to be the same as by dimension considerations.
We prove that the dimension of is at most one. Observe that is constructed via the GIT quotient , where is the open subset of the Hilbert scheme parameterizing all cubic hypersurfaces with at worst ADE singularities. Then has rank one since the boundary of in has codimension at least two.
Let be the set of -linearized line bundles on . There is an injection
by [12, Proposition 4.2.]. Our assertion follows from the fact the forgetful map is an injection.
The moduli space of quasi-polarized K3 surface is a 19-dimensional locally Hermitian symmetric variety associated to . It is conjectured by Maulik and Pandharipande that the Picard group of is rationally spanned by Noether-Lefschetz divisors. This conjecture has been verified for low degree surfaces (cf. , , ). We also refer the readers to  and  for some recent results in this subject.
2.3. The degree of special cubic divisors
Analogous to Noether-Lefschetz numbers of K3 surfaces , the degree of can be computed via intersection with a test curve. Let be a Lefschetz pencil of cubic hypersurfaces in . It yields a natural morphism
which factors through the rational map . It is not difficult to see that is the same as the intersection number .
Let be the composition of and . If we set
then we have for , and since there are no determinantal cubic fourfolds in a Lefschetz pencil . The generating series can be rewritten as
One can see that there is a natural enumerate geometry interpretation of , and some of them can be computed using geometric methods when is small.
The degree of counts the number of singular fibers in . The first jet bundle  of parametrizes all nodal cubic hypersurfaces in . Then equals to the top Chern class of , which is .
The degree of counts the number of planes contained in the fibers of . Let be the Grassmannian parametrizing all planes in . The planes contained in a cubic hypersurface of are parametrized by certain vector bundle on . Via standard Schubert calculus, one can show equals , which is the top chern class of .
The degree of counts the number of Pfaffian cubic fourfolds (cf. ) in , which equals to according to our main theorem.
3. Modular forms associated to signature lattices
In this section, we introduce the vector-valued modular form associated to an even lattice and prove the modularity of the generating series of special cubic divisors from Borcherds’ work .
3.1. Vector valued modular forms
The metaplectic double cover of consists of pairs , where
It is well-known that is generated by
Let be the complex upper half-plane. Suppose is a representation of on a finite dimensional complex vector space , such that factors through a finite quotient. For any , a vector-valued modular form of weight and type on is a holomorphic function on , such that
When , this recovers the definition of scalar-valued modular forms with a character.
Given a lattice M of signature with a bilinear form , there is a Weil representation of on the group ring defined by the action of the generators as follows:
where is the standard basis of for . We denote by the space of modular forms of weight and type .
Now we take and denote by the standard basis of corresponding to element as in 2.2. Our first result is:
Let be the vector-valued generating series of (2.4) defined by
Then is an element of .
is an element in .
In our situation , we have by Lemma 1 and thus the generating series
is a vector-valued modular form of weight and type with coefficients in .
Next, let be a linear function defined by
Then as shown in 2.2, we have and
which is the degree of the Hodge bundle . It follows that is an element in . ∎
In Borcherds’ setting, the lattice has signature and the generating series of Heegner divisors are vector-valued modular forms of type (dual of ). For with signature , one can get (3.2) by transferring the lattice to , which has signature and .
3.2. Construction of modular forms
Here we introduce some modular forms which will be used later.
3.2.1. Scalar-valued Eisenstein series
The classical Eisenstein series
is a modular form of weight for for , where is the Bernoulli number.
Let us denote by (resp. ) the arithmetic subgroups in defined by
and let be the nontrivial Dirichlet character modulo 3 on , i.e.
where is the nontrivial Dirichlet character modulo 3.
Assume that is an odd integer. Then the Eisenstein series
is a modular form of weight with character for congruence group . Moreover, and .
Respectively, is a modular form of weight with character for congruence group .
3.2.2. Vector-valued Eisenstein series
Let and an even lattice. The vector-valued Eisenstein series on constructed via Petersson slash operator  are vector-valued modular forms of weight and type . The equivalence of Weil representations and implies
Let be the vector-valued Eisenstein series associated to of weight . For , it is given by  that
Here, denotes the Dirichlet L-series with character and is the local Euler product defined as following:
where is the Bernoulli polynomial. Thus one obtains that
3.2.3. Rankin-Cohen bracket
Given any two level scalar-valued modular forms on the upper half plane of weight and . The n-th Rankin-Cohen bracket is defined as follows:
where denotes the r-th differential of with respect to .
For a vector-valued modular form
one can extend the Rankin-Cohen bracket to and as follows,
According to [19, Lemma 5], we have the following result:
The vector-valued functions
are vector-valued modular forms of weight and type .
3.3. Expression of the generating series
Now we are ready to give an explicit expression of . From the dimension formula of Bruinier in , we know that
It follows that
Let be the vector-valued modular forms constructed in Lemma 2. Then is a basis of and
By checking the coefficients of the term , we know that and are linearly independent. Thus they form a basis of by dimension considerations. To obtain the expression (3.11), it suffices to use the following two constraints:
The degree of the Hodge bundle is , which gives the coefficient of . By Grothendieck-Riemann-Roch, we have the following Chern character computation:
where is the relative tangent bundle. Since is trivial by the Lefschetz hyperplane theorem, we get .
The coefficient of is as shown in 2.3.
4. Proof of Theorem 1
To prove our main theorem, we first start with a Lemma on the modularity on the components of a vector-valued modular form:
Let be an element in . Then the following are true:
is a scalar-valued modular form for of weight with character .
is a scalar-valued modular form for of weight with character , where
The sum is a scalar-valued modular form for of weight with character .
Statement (i) and (ii) follows from  2. For (iii), since the generators of the congruence subgroup are
then statement (iii) follow from a direct computation by checking the modularity on these generators. ∎
Proof of Theorem 1. Let us write . Then it is not difficult to see that
Next, let (resp. ) denote the space of scalar-valued modular forms with character for (resp. ). It is known by  12 that is a polynomial ring generated by two modular forms of weight and weight .
The coefficients computation shows that
Similarly, is a polynomial ring generated by and . Then the modular form has weight and can be expressed as
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