Cubic threefolds, hyperkähler manifolds and the complex ball

Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball

Abstract.

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of -type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square ; both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkähler fourfolds of -type with non-symplectic automorphism of order three belonging to different families. Finally, we construct a cyclic Pfaffian cubic fourfold to give an explicit construction of a non-natural automorphism of order three on the Hilbert square of a K3 surface.

1. Introduction

The aim of this paper is to study certain moduli spaces of -dimensional irreducible holomorphic symplectic manifolds endowed with a non-symplectic group action of prime order, constructed in full generality by the authors in [BCS_ball], which are uniformized by the same -dimensional arithmetic ball quotient as the smooth cubic threefolds [ACT], and to explore some deep consequences of this isomorphism.

Arithmetic complex ball quotients have attracted great interest in the last decades since they have been proven to be ubiquitous targets of period maps, allowing to show unexpected isomorphisms between different moduli spaces. For example, an arithmetic quotient of the complement of a hyperplane arrangement in the -dimensional complex ball arising as period domain of smooth cubic surfaces in [ACT_surf] has been studied by Dolgachev–van Geemen–Kondō [DvGK], who show that and degenerations to can also be interpreted as moduli spaces of K3 surfaces endowed with certain non-symplectic automorphisms of order three.

Another famous example, which is the core of the present paper, is given by the quotient of the -dimensional complex ball which contains as a Zariski open set the moduli space of smooth cubic threefolds, as shown by Allcock–Carlson–Toledo [ACT] and by Looijenga–Swierstra [LS]. We know nowadays that the hyperplane arrangements corresponding respectively to nodal and chordal degenerations of cubic threefolds are in fact birational to two other arithmetic complex ball quotients (see [CMJL]), first studied by Kondō [Kondo] and appearing as the moduli spaces of K3 surfaces endowed with two different kinds of non-symplectic automorphisms of order three.

The main result of the first part of this paper is to establish an isomorphism between the moduli space  of smooth cubic threefolds and the moduli space  of fourfolds deformation equivalent to the Hilbert square of a K3 surface endowed with a special non-symplectic automorphism of order three, following [BCS_ball, Section 7.1]. We recall in Section 2 the construction of Fano varieties of cyclic cubic fourfolds and of their automorphism, as done in [BCS_class, Example 6.4]; then in Section 3 we prove our first main result:

Theorem 1.1.

The moduli spaces and are isomorphic.

This result will be proven by using the period map of irreducible holomorphic symplectic manifolds and Allcock–Carlson–Toledo’s period map for cubic threefolds. One of the consequences of Theorem 1.1 is Corollary 5.2: although a priori inside we could have deformations which are non-isomorphic to Fano varieties of cyclic cubic fourfolds, this is not the case and every element in this moduli space is isomorphic to such a Fano variety of lines.

The rest of the paper is devoted to the exploration of more consequences of this isomorphism, in particular we give a geometric description of the degenerations of the non-symplectic automorphism of order three. Indeed, the main part of  [ACT] is devoted to the extension of the period map to singular cubics, either nodal or chordal. In Section 4 we will use this extension to understand geometrically the degenerations of the automorphism along the corresponding hyperplanes in the arithmetic ball quotient. The final outcome of this study are Propositions 4.6 and 4.10, which can be summarized as follows:

Theorem 1.2.

The stable discriminant locus (corresponding to nodal degenerations), respectively the stable chordal locus (corresponding to stable chordal degenerations) are birational to -dimensional moduli spaces of fourfolds which are deformations of a Hilbert square of a K3 surface endowed with a non-symplectic automorphism of order three, having respectively invariant lattice isometric to and to .

Finally, in Section 5, looking at Hassett divisors on the moduli space of cubic fourfolds, we produce a Pfaffian cyclic cubic fourfold, thus we show:

Corollary 1.3.

There exists a smooth complex K3 surface whose Hilbert square admits a non-natural non-symplectic automorphism of order three, with fixed locus isomorphic to the Fano surface of a cubic threefold and invariant lattice isometric to .

We further explicitly describe this automorphism in terms of the Pfaffian geometry of the K3 surface, in analogy with Beauville’s construction of the non-natural non-symplectic involution on the Hilbert scheme of a general quartic surface [B_remarks].

Acknowledgements

The second author was partially supported by FIRB 2012 “Moduli spaces and their applications” and by “Laboratoire Internationale LIA LYSM”. The authors thank Klaus Hulek, Shigeyuki Kondō, Christian Lehn, Gregory Sankaran and Davide Veniani for their helpful comments and suggestions.

2. Fano varieties of lines of cyclic cubic fourfolds

We work over the field of complex numbers.

2.1. Cyclic cubic fourfolds

A general element of the linear system defines a smooth cubic threefold  in . The ramified triple covering branched along  is then a cyclic cubic fourfold in . We denote by a generator of the fundamental group of the covering, it is a biregular automorphism of order three of .

By Hassett [Hassett, Proposition 2.1.2] the lattice endowed with the Poincaré pairing is an odd unimodular lattice isometric to

where denotes the rank one lattice with quadratic form taking the value on the generator. Denote by the class of a hyperplane section. The class in  has square  and the primitive cohomology space is by definition the orthogonal complement of in . Then is an even lattice of signature , isometric to the lattice

where is the hyperbolic plane and are the positive definite root lattices (see also [ACT, proof of Theorem 1.7] or [LS, Section 1] for another point of view). We fix for the rest of this paper a polarization class of square and a primitive embedding such that .

Denote by an equation of . Then has equation

and the automorphism maps to , where is a primitive third root of the unity that we fix for the rest of this paper. By Griffiths residue theorem, the Hodge group is one-dimensional, generated by , where Since the residue map is -equivariant, we compute . Denoting by the eigenspace of associated to the eigenvalue , the line is called, following Allcock–Carlson–Toledo [ACT], the period of the cubic threefold .

2.2. Fano variety of lines on cyclic cubic fourfolds

Consider the Fano variety of lines , which parametrizes the projective lines contained in . By Beauville–Donagi [BD], the variety is irreducible holomorphic symplectic, deformation equivalent to the Hilbert square of a K3 surface. As a consequence, the second cohomology group with integer coefficients , endowed with the Beauville–Bogomolov–Fujiki bilinear form is isometric to the lattice

The automorphism of leaves globally invariant the lines of contained in the cubic , so the fixed locus of the action of induced on is isomorphic to the Fano variety of lines , which is a surface of general type with Hodge numbers , and (see [BCS_class, Example 6.4],[CG, Formula (0.7)]). It follows from the lattice-theoretical classification of non-symplectic automorphisms in [BCS_class, Example 6.4] that the invariant lattice is isometric to , with orthogonal complement in , denoted by , isometric to the lattice .

The apparent coincidence with the results of the preceding section is due to the Abel–Jacobi map. Denote by the universal family and by , resp.  , the projection to , resp.  . By [BD, Proposition 4] the Abel–Jacobi map

is an isomorphism of Hodge structures. Denote by the class of a hyperplane section in the Plücker embedding. It is easy to see geometrically that . Since , we can directly deduce that the invariant lattice has rank one, hence it is generated by . By -equivariance of we observe that the invariant lattice is generated by .

Moreover, by [BD, Proposition 6] we have and

Since , by -equivariance of we deduce from the results of the previous section that the line , which is the period of the irreducible holomorphic symplectic manifold , lives in the eigenspace of associated to the eigenvalue . The transition from the lattice to the opposite lattice is explained by the Abel–Jacobi map being an anti-isometry at the level of the primitive cohomology.

3. Occult period maps and unexpected isomorphisms

3.1. Two useful lemmas

We state two classical results of Nikulin [Nikulin] on lattice theory which will be used several times below. For any integral lattice , we denote by the dual lattice and by its discriminant group, endowed with its quadratic form . The length of is its minimal number of generators.

Lemma 3.1.

[Nikulin, Theorem 1.14.2] Assume that is an even indefinite lattice. If , then the natural map is surjective.

Lemma 3.2.

Let be an integral non-degenerate lattice, a primitive sublattice of  and its orthogonal complement. Let and . The isometry of extends to an isometry of if and only if its action on leaves globally invariant the subgroup . In particular, if acts by on , with , then extends to an isometry of .

Proof.

The inclusions produce a subgroup . For any , one has for some . Consider . If this group is -stable in , then , so and extends to an isometry of . ∎

3.2. Moduli space of smooth cubic threefolds

We denote by the GIT moduli space of -stable points in the linear system . Inside this quasi-projective variety, the locus whose points parametrize projective equivalence classes of smooth cubic -folds in is the open set determined by the nonvanishing of the discriminant (see [Mukai, Chapter 5]).

By Allock–Carlson–Toledo [ACT], the moduli space is isomorphic to the quotient by an arithmetic group of the complementary of a hyperplane arrangement in a -dimensional complex ball. We briefly recall the main ingredients for later use. Given (we choose a representative of the projective equivalence class) we denote by the ramified cyclic triple covering branched along and by  the generator of the covering group which acts by multiplication by on the period . Since has no fixed points for the action of , it inherits a natural structure of free module over the ring of Eisenstein integers , which does not depend on the cubic . Hence any marking (i.e. isometry of -lattices) induces a representation

whose isomorphism class does not depend on by the local topological triviality of the family of cubic threefolds. We call framing a marking compatible with the representation, in the sense that . Said differently, a framing is an isometry of -lattices and an isomorphism of -modules.

We denote by the moduli space of framed smooth cubic threefolds: it parametrizes pairs , where and is a framing considered up to composition on the target with the group of th roots of the unity, which is also the group of units of . We denote by the group of isometries  of  such that . The group is equivalently the group of isometries of the -valued Hermitian form on naturally associated to  (see Section 3.4). It acts on  on the left by for all , and the units act trivially so we can consider the action of the projective group . We thus have a natural isomorphism .

Any framing induces an isomorphism , where  is the eigenspace of for the eigenvalue of the isometry . The Poincaré pairing on extends to a hermitian form on . Using the Hodge–Riemann relations [GH, p.123] we get that has signature and that the line is negative-definite. Denoting by the hermitian form on induced by the integral bilinear form on , we see that the line defines a point in the -dimensional complex ball

The period map , commutes with the action of and defines a quotient period map

By [ACT, Theorem 1.9] the period map is an isomorphism onto its image, which is the complementary of the quotient of a hyperplane arrangement (see Section 4).

3.3. Relation with the period map for cubic fourfolds

As initial data, we choose a cubic threefold , its associated cyclic fourfold and a framing . The period of is the line inside . The period domain is isomorphic to the Grassmannian variety of negative definite -planes in , and has two connected components interchanged by complex conjugation. We denote by the connected component which contains . By deformation of the complex structure we get a holomorphic period map from the moduli space of marked cubic fourfolds . By results of Ebeling and Beauville (see [B_monodromy, Theorem 2]), the monodromy group of a cubic fourfold is isometric to the group of isometries of which act trivially on the discriminant group and which respect the orientation of a negative-definite -plane in (see Laza [Laza_period, §2.2]). Thus acts on and by a theorem of Voisin [VoisinCubic], the period map is an open embedding. Note that is not a subgroup of : since , the elements of act by on (see Lemma 3.1). Observe that : the isometry  comes from the covering automorphism of so it fixes the polarization  and it acts trivially on the discriminant group since the lattice is unimodular. Thus the -orbits in are contained in the corresponding -orbits and we have a commutative diagram:

It is well-known that the covering morphism mapping to is generically injective (see Beauville [B_modcub, proof of Theorem 4.6] and references therein for a similar argument).

3.4. The Hermitian module

The ring of Eisenstein integers is euclidean with invertible group . Its irreducible elements are the prime numbers such that and the elements whose norm is a prime integer. In particular is irreducible in  and . Recall that acts on by for all . Following [ACT, Chapter 1], we define a Hermitian form on by

It is an easy exercise to check that is a Hermitian form and that it takes values in the prime ideal of . One has if and only if : this characterizes the roots of . For such a root , the image of the -linear morphism is an ideal generated by an element such that . It follows easily that either ( is called nodal) or ( is called chordal).

3.5. A moduli space of non-symplectic automorphisms

We start again from a ramified cyclic covering branched along the cubic , with covering automorphism  acting by multiplication by . Any marking induces through the Abel–Jacobi map an isometry , which extends to a marking as follows. By [BCS_class, Theorem 3.7] the lattice admits a unique primitive embedding in , up to an isometry of , and its orthogonal complement is isometric to the lattice . We implicitly fix this embedding. This determines, up to an isometry of , a primitive embedding that we fix for the rest of this paper. We extend  to an isometry which extends using Lemma 3.2 to an isometry . The action of the automorphism defines as above a representation

whose isomorphism class does not depend on the cubic , and whose invariant subrepresentation is . The restriction of to is nothing else than viewed as an isometry of instead of . Following Boissière–Camere–Sarti [BCS_ball], the -tuple is the initial data of the construction of a moduli space of -polarized irreducible holomorphic symplectic manifolds. We briefly recall the main ingredients for later use.

Denote by the moduli space of equivalence classes of pairs where is an irreducible holomorphic symplectic manifold deformation equivalent to the Hilbert square of a K3 surface and is a marking (i.e. an isometry of -lattices), where two pairs and are considered equivalent if there exists a biregular isomorphism such that . Following [Markman_Torelli, Section 2] and  [BCS_ball] we fix once and for all a connected component of and we define a -polarisation of an irreducible holomorphic symplectic manifold  deformation equivalent to the Hilbert square of a K3 surface as the data of:

  • a marking ,

  • a primitive embedding such that ,

  • an automorphism such that  is a framing for (i.e. ) and .

If such an automorphism exists, then it is uniquely determined by this property (see [Mongardi_CRAS, Lemme 1.2]). We denote by (the embedding is implicit in this notation) the set of equivalence classes of such tuples , where two tuples and are equivalent if there exists a biregular isomorphism such that , and . We similarly define the space parametrizing -polarized triples up to equivalence. We have then the inclusions .

Remark 3.3.

The polarisation contains always an ample class: the variety  is projective since it admits a non-symplectic automorphism (see [B_remarks, §4]), so if is an ample class, the invariant ample class is necessarily a multiple of the generator of the rank one invariant lattice. Denote the primitive ample invariant class; since the invariant lattice has rank one, this class is uniquely defined by the automorphism . It follows that the map embeds in the moduli space of irreducible holomorphic symplectic manifolds polarized by an ample primitive class of square studied by Gritsenko–Hulek–Sankaran [GHS_moduli].

Starting from the initial data as above, we consider the monodromy group and we define

It is easy to check that this group does not depend on the representative of the equivalence class in and that it is conjugated to the monodromy group of any variety deformation equivalent to . Following [BCS_ball, Section 6], we define the group as the image in of those elements which act trivially on and commute with . The group acts on by and we define .

Using the properties of the Beauville–Bogomolov–Fujiki quadratic form, we see that the line defines a point in the -dimensional complex ball

The period map , commutes with the action of and defines a quotient period map

By [BCS_ball, Theorem 4.5, Theorem 5.6, Section 7.1], the moduli space is connected, the period map is an open embedding and its image is the complementary of the quotient of a hyperplane arrangement. Note that there is a tiny difference between this definition of the moduli space and those of  [BCS_ball] since we include and in the data. Since they are uniquely determined, this is of course equivalent and our period map is, strictly speaking, the composition of a bijective forgetful map and of the period map of [BCS_ball]. We define the structure of complex manifold of as the one inherited by the one of its period domain.

Theorem 3.4.

The moduli spaces and are isomorphic.

Proof.

The period domains and are clearly identical, we identify them with the complex ball . By [ACT, Theorem 6.1] the image of the period map is the complementary of the quotient of the union of the hyperplanes for all roots of , i.e. all elements of norm (in [ACT] these have norm for the naturally associated -valued Hermitian form on , see Section 3.4). By [BCS_ball, Theorem 4.5, Theorem 5.6, Section 7.1], the image of the period map is the complementary of the quotient of the union of the hyperplanes for all which are monodromy birationally minimal (MBM), see [AV]. By [BHT, Mongardi] those are exactly the classes either of norm or of norm and divisibility (i.e. ). The latter cannot happen since has discriminant . This shows that both hyperplane arrangements are identical.

By a result of Markman [Markman_constraints, Theorem 1.2],[Markman_Torelli, §9] is the subgroup of of isometries leaving globally invariant the positive cone of . By Remark 3.3, isometries acting trivially on the polarization do automatically fix an ample class, so they leave globally invariant the positive cone. The elements of are thus the restrictions to of those isometries of acting trivially on and commuting with the representation . By Lemma 3.2, any isometry of commuting with extends to an isometry of acting by on and commuting with . So we have an isomorphism and the ball quotients are equal. ∎

Concretely, we have a natural map which maps the equivalence class of a framed cubic to the equivalence class of the -polarized irreducible holomorphic manifold constructed above, and the theorem says that this map is an isomorphism. We thus have a commutative diagram

where and is the hyperplane arrangement discussed in the proof of Theorem 3.4, where we denote

the unions of hyperplanes orthogonal respectively to nodal and chordal roots.

Corollary 3.5.

An irreducible holomorphic symplectic manifold deformation equivalent to the Hilbert square of a K3 surface, polarized by an ample class of square , admits a non-symplectic automorphism of order three with invariant lattice if and only if is isomorphic to the Fano variety of lines of the ramified cyclic triple covering of  branched along a smooth cubic threefold. In this case, the automorphism is induced by the covering automorphism.

Remark 3.6.

Take . A class is algebraic if and only if it is orthogonal to , or equivalently if the period is outside the hyperplane orthogonal to in the period domain . Since such hyperplanes form a countable union, their complementary is dense in the period domain, so generically the Néron–Severi group of is generated by . The moduli space is thus a -dimensional moduli space of irreducible holomorphic symplectic manifolds deformation equivalent to the Hilbert square of a K3 surface, admitting a non symplectic automorphism of order three, and whose general Picard number is one. This answers a question asked to the authors by Olivier Debarre [DM].

3.6. Relation with the moduli space of genus five principally polarized abelian varieties

The term “occult” in the section title is a reference to a paper of Kudla–Rapoport [KudlaRapoport]. As suggested to us by Michael Rapoport and thanks to enlightening discussions with Gregory Sankaran, we can draw direct paths between all the involved period maps, producing in particular a period map from  to a moduli space of principally polarized abelian varieties. The classical period map for cubic threefolds of Clemens–Griffiths [CG] maps a cubic to its intermediate Jacobian , which gives a point in the -dimensional moduli space of principally polarized abelian variety of genus .

Corollary 3.7.

There exists an embedding mapping a -polarized tuple to the Albanese variety of the fixed locus of the automorphism .

Proof.

By [CG, (0.8)] the intermediate Jacobian of is isomorphic to the Albanese variety of the Fano variety of lines on , i.e. . As observed in Section 2.2, the variety is the fixed locus in of the automorphism of the ramified covering map branched along . Since the morphism mapping to (with the appropriate markings and polarizations as above) is an isomorphism, we get the result. ∎

4. Degenerations

Following an idea used by Dolgachev–Kondō [DK, §11] to study the degeneration of K3 surfaces with a non-symplectic automorphism of prime order, we study the degenerations of those irreducible holomorphic symplectic manifolds parametrized by , inside the moduli space of -polarized marked triples. Denote by the connected component of which contains the ball , it is easy to deduce from the theorem of surjectivity of the period map of Huybrechts [Huybrechts_Invent, Theorem 8.1] that the period map is surjective. Note that, comparing with the notation in Section 3.3, we have but we use both notations depending on the context. We thus have a commutative diagram

Let and . By [Markman_Torelli, Theorem 1.2], any two points in this fiber of correspond to birational manifolds.

We first observe that the isometry is not represented by any automorphism of . As explained in the proof of Theorem 3.4, this is a special case of [BCS_class, Theorem 4.0.8]: if , then is orthogonal to some class such that is a MBM class of . If was represented by an automorphism of , this one would be automatically non-symplectic since  is not in the invariant sublattice of , so as explained in Remark 3.3 the line  contains an ample class . But since , the divisor  is orthogonal to . As explained in  [BCS_class, Remark 4.0.7 (2)], this is not possible since MBM classes have nonzero intersection with the Kähler cone of . In particular, in this situation the line  contains no ample class any more.

Following [Markman_Torelli, Section 1], we denote by the subgroup of consisting of those monodromies whose -linear extension preserves the Hodge decomposition. For any nonisotropic vector , we denote by the reflection which leaves invariant. We consider the normal subgroup of generated by the reflections by prime exceptional divisors of (i.e. reduced and irreducible effective divisors of negative Beauville–Bogolomov–Fujiki degree).

Proposition 4.1.

Let and . There exists a unique birational map and a unique element such that:

Proof.

Since , the isometry is a monodromy operator of . Since , we have so . By [Markman_Torelli, Theorem 1.6], there exists a unique element and a birational map such that . The map is unique in this case since the natural map is injective in this deformation class (see [Mongardi_CRAS, Lemma 1.2]). ∎

This results means that the isometry , after composition by a suitable product of reflections, is realized by a birational transformation of . By a deeper analysis of the degeneracy situations, we will show below that the product of reflections is not trivial and that itself is in fact realized by another automorphism and not only by a birational map. This means that, when the period point goes to a point of , the non-symplectic automorphism of the family does not degenerate to a birational non-biregular transformation. What happens instead is that the automorphism degenerates by jumping to another one with a bigger invariant sublattice on a different family. We will show this by describing explicitly the new isometry to be considered and by proving that, for a suitable choice of in the fiber, the birational transformation extends to an automorphism of .

This phenomenon is very common in the context of degenerations of automorphisms of K3 surfaces. Take for instance an ample -polarized K3 surface . It is well-known that carries a non-symplectic involution with invariant lattice isometric to and such that . This exhibits as a ramified double cover of branched along a smooth sextic curve. If the sextic acquires ADE singularities then the double cover is singular and its minimal resolution is a K3 surface containing several -curves coming from the desingularisation of the singular points. In this case the K3 surface carries again a non-symplectic involution but its invariant lattice has higher rank. A concrete example is when the sextic has a node: generically, the surface  has a Néron–Severi lattice isometric to , which is also the invariant lattice of the non-symplectic involution (see [AN]). What happens is that as soon as the sextic admits singular points, the K3 surface contains rational curves which do not meet the class of square (the pullback of the class of a line of  is not an ample polarization any more, it is only big and nef). Then admits a non-symplectic involution which is of “different kind” than , since their invariant lattices are different, so heuristically the automorphism survives by jumping to a different family.

4.1. Degeneracy lattices

For any and , the degeneracy lattice of is the sublattice of generated by MBM classes of which are orthogonal to (see the proof of Theorem 3.4). Here this lattice is simply the lattice generated by the roots of , with , such that . This lattice is clearly globally -invariant and orthogonal to . Generically,  belongs to only one hyperplane so the general degeneracy lattice is .

It is easy to compute that is isometric to the lattice and that  acts trivially on the discriminant group . The lattice is primitive in . Indeed, using the -lattice structure (see Section 3.4) of we have an isomorphism of -modules . Since is a root, with , it is easy to check that the rank one -submodule is primitive in , so is primitive in . Using a result of Nikulin [Nikulin, Proposition 1.15.1] we find that two degeneration situations appear, depending on the two possible isometry classes of the orthogonal complement of in :

Recall that following [ACT] a root is called nodal if and chordal if . The relation with the above dichotomy is explained by the following observation.

Lemma 4.2.

A root is chordal if and only if is unimodular.

Proof.

It is easy to check that and . If is unimodular, then so and is chordal. Otherwise is generated by a class with and we get so and is nodal. ∎

4.2. Degeneration of the representation

Let be a root (). Recall that we have choosen primitive embeddings and such that . This induces an embedding in  of , not necessarily primitive. We thus consider the saturation of in , which is the minimal primitive sublattice of  containing . Observe that the orthogonal complement of in is nothing else than the orthogonal complement of in embedded in .

Lemma 4.3.

Let be a root.

  1. If is nodal, then is primitive in and .

  2. If is chordal, then is not primitive in and .

Proof.

Since has discriminant , its only possible saturation is by adding a -divisible class, so either (primitive case), or (nonprimitive case). Recall that if is a nondegenerate lattice and a primitive sublattice, denoting by the natural map, we have the following formula (see [GHS_handbook, p.47]):

Applying this formula to and either or , we get that if  is nodal, th