Moduli Spaces of Symmetric Cubic Fourfolds and Locally Symmetric Varieties
In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type symmetric domains, and study their compactifications. Our results mainly depend on the well-known works about moduli space of cubic fourfolds, including the global Torelli theorem proved by Voisin ([Voi86]) and the characterization of the image of the period map, proved independently by Looijenga ([Loo09]) and Laza ([Laz09, Laz10]). The key input for our study of compactifications is the functoriality of Looijenga compactifications, which we formulate in the appendix (section A).
Cubic fourfold is an intensively studied object in algebraic geometry. The remarkable work by Voisin in 1986 ([Voi86]) showed the global Torelli theorem for smooth cubic fourfolds. Based on this, Allcock-Carlson-Toledo ([ACT11]) and Looijenga-Swierstra ([LS07]) realized the moduli space of smooth cubic threefolds as an arrangement complement in an arithmetic ball quotient. Recently Laza-Pearlstein-Zhang ([LPZ17]) realized the moduli space of pairs consisting of a cubic threefold and a hyperplane as an arrangement complement in a type arithmetic quotient. In both cases, the authors studied compactifications of the moduli spaces. In this paper we characterize the moduli spaces of cubic fourfolds with specified automorphism groups, and identify the -compactifications with Looijenga compactifications. This generalizes the two results mentioned above.
Let be the normalization of the irreducible subvariety parameterizing smooth cubic fourfolds with specified action by finite group (see section 2 for the setup). Let . Let be a cubic fourfold in . Consider the induced action of on , and let be the character corresponding to . Denote to be the -eigenspace, which admits a natural Hermitian form induced by the topological intersection pairing on (see section 4.1). Then has signature if ; otherwise (see proposition 4.1). The first main theorem of the paper is the following:
Theorem 1.1 (Main Theorem 1).
We have equality .
The Hodge structure on gives an algebraic isomorphism . Here is a complex hyperbolic ball if has signature ; a type symmetric domain otherwise. The group is an arithmetic group acting proper discontinuously on and is a -invariant hyperplane arrangement in .
The period map extends naturally to an algebraic isomorphism , where is a natural partial completion of , adding cubic fourfolds with at worst -singularities, and is a -invariant hyperplane arrangement contained in .
Denote to be the -compactification of , see section 2.2. We characterize via:
Theorem 1.2 (Main Theorem 2).
There is an isomorphism between projective varieties .
Here is the Looijenga compactification of , see section A.5 in appendix.
Notice that in [GAL11], smooth cubic fourfolds with prime-order automorphisms are classified and form 13 irreducible subvarieties in the moduli of cubic fourfolds (see section 6.1). Two of the examples in the list are exactly the cases dealt in [ACT11], [LS07] and [LPZ17].
Structure of the Paper: We briefly introduce the main content of each section.
In section 2 we introduce the notion of symmetry type, and set up the geometric invariant theory of hypersurfaces with specified symmetry type.
In section 4 we define the moduli of T-marked cubic fourfolds, and the local period map for those cubic fourfolds. We show that the local period map is an open embedding and characterize its image. Finally we discuss the global period maps by passing to certain quotients.
In section 5 we investigate the compactifications of both sides of the period map for symmetric cubic fourfolds, and identify them.
In section 6 we give some examples and relate them to the previous works.
In section A, we review Looijenga compactification of an arrangement complement in a complex hyperbolic ball or type domain. We prove functoriality of Looijenga compactifications.
Convention: All algebraic varieties are defined over the field of complex numbers. The adjectives open, closed refer to analytic topology and Zariski-open, Zariski-closed are used for Zariski topology.
: dimension and degree of a hypersurface
: complex vector space of dimension
: degree polynomial in variables
: degree -fold; cubic fourfold when
: a finite subgroup of , containing the center of
: image of in
: character of with specified restriction to
: equivalence class of , called symmetry type of degree -fold
: eigenspace of corresponding to
: centralizer of in
: a reductive group acting on
: subspace of smooth/semi-stable points in
: quotient of by
: moduli space of cubic fourfolds with -markings
: moduli space of cubic fourfolds of type , which admits at worst singularities
: quotient of by , which is a compactification of
: moduli space of smooth cubic fourfolds
: compactification of
: (primitive) middle cohomology lattice of cubic fourfold
: topological intersection pairing
: square of hyperplane class
: local period domain for cubic fourfolds
: monodromy group of the universal family of smooth cubic fourfolds
: -invariant hyperplane arrangement in
: character of , induced by the action of on
: eigenspace of corresponding to the character
: representation of on /
: Hermitian form on
: local period domain for cubic fourfolds of symmetry type
: monodromy group for universal family of smooth cubic fourfolds of symmetry type
: locally Hermitian symmetric variety (used in Appendix)
: -invariant hyperplane arrangements in
: Looijenga compactification of
: local period map
: global period map
2 General Setup: Symmetric Hypersurfaces
2.1 Space of Symmetric Polynomials
Let be a complex vector space of dimension . Denote to be the space of degree polynomials on . We have the natural action of on , namely, for and .
The center of is the group consisting of -th roots of unity. Let be a finite subgroup of containing and denote the image of in . Then is a representation of .
Notice that for any and , we have . Let be a character of such that sends to . Let be the -eigenspace of . We write for short. Geometrically, an element in determines a degree hypersurface (not necessarily smooth) in , whose automorphism group contains .
Two pairs and are called equivalent if and only if there exists such that and . We call an equivalence class a symmetry type, denoted by . There is a poset structure on the space of symmetry types, namely, if are represented by respectively, such that and . Notice that the space depends on the representative of .
For , we denote to be the hypersurface determined by in . For , we denote to be the group of elements in preserving , and to be the preimage of in . From [MM64] (theorem 1 and theorem 2) we have:
Theorem 2.1 (Matsumura-Monsky).
When is smooth, , ,
the group is finite,
if , the group contains all biregular automorphisms of .
Apparently, the group embeds into , for any . We propose the following conditions on the symmetry type :
The linear space contains a point determining smooth hypersurface.
The linear space contains a point with the determined hypersurface smooth and .
If satisfies condition 2.3, then a generic element in determines a smooth hypersurface with .
Suppose with smooth, and . Then any small deformation of in determines a smooth hypersurface . By theorem 2.5 in [Zhe17], when is sufficiently close to , there exists such that . Since , we have , hence . ∎
2.2 Geometric Invariant Theory for Symmetric Hypersurfaces
Now we assume that , . Given a symmetry type satisfying condition 2.2, let
be two reductive subgroups of . For reductivity, see [LR79], lemma 1.1.
There is a natural action of on , under which the points in defining smooth hypersurfaces are stable.
For any and , we need to show . For any , we have:
which implies by definition of . Therefore, there is a natural action of on .
Now take with smooth. Then is finite by theorem 2.1. Since the stabilizer group of under action of is a subgroup of , hence also finite. Moreover, is closed in , and the latter is closed in since is smooth. Thus is closed in , hence also closed in . We conclude that is stable under the action of . ∎
Denote smooth, the set of semi-stable elements in under the action of , and , their projectivizations. By lemma 2.5, we can take to be the quotient, with compactification . Different representatives of the symmetry type induce canonically isomorphic -quotients. Define to be the moduli space of smooth degree hypersurfaces in , with compactification . We have the following proposition:
There is a natural morphism sending to for any . This morphism is finite. When satisfies condition 2.3, the morphism is a normalization of its image.
Here we use a projective version of the main theorem in [Lun75]. See the argument of proposition in [Res10]. Since is a finite group, there exists certain symmetric power on which the -action is trivial. Consider the -action on the coordinate ring of . Notice that is of finite index in the normalizer of in . By the main theorem in [Lun75], we have a finite morphism
sending semi-stable points to semi-stable points, and preserving the cone structures. Thus does not contract any line, so descends to a finite morphism . The morphism sends to for any .
We claim that when satisfies condition 2.3, the morphism is generically injective. Take generically and assume in . Then there exists with . By the calculation
we have that is an automorphism of . By the genericity of , we have , which implies that . Then by equation (1) and , we have . This implies that , hence in . Thus is generically injective.
Moreover, since is normal and projective, is a normalization of its image. ∎
Let be a symmetry type satisfying condition 2.2. Consider the automorphism groups for all . There exists such that is minimal. Fix this polynomial , and denote . We have a symmetry type , where for all . We have , and satisfies condition 2.3. For , there are the corresponding and . We have the following proposition:
There exists a natural finite morphism .
By proposition 2.6, we have two finite morphisms and , and the latter one is a normalization of its image. We show that and have the same image. Firstly, we have that since . By lemma 2.5 in [Zhe17], when is sufficiently close to , there exists , such that . By minimality of , we have . This implies that , hence . We then have that . By irreducibilities of the two images, they are the same.
By universal property of normalization, the morphism factors through . Therefore, we have naturally a finite morphism . ∎
The fiber of the finite morphism over is bijective to the orbit of in the set of subdatas of under the action of .
2.3 Universal Deformation
We fix a type satisfying condition 2.2, and assume and . Next we use Luna’s étale slice theorem to describe the local structure of , and construct the universal family of smooth degree -folds of type . We follow the argument in [Zhe17] (section 2). For Luna’s étale slice theorem and its proof, one can refer to [Lun73] or [PV94].
Denote to be the centralizer of in , which acts on the affine variety . For any , we denote to be the orbit of and to be the stabilizer of . By lemma 2.5, is closed in the affine variety and is finite. By Luna’s étale slice theorem, there exists a smooth, locally closed, -invariant subvariety containing , such that:
The image of , denoted by , is Zariski-open and -invariant,
The morphism is étale,
The morphism is étale,
The above two morphisms induce an isomorphism
We can shrink in the analytic category such that:
is -invariant, contractible and contains , with a -invariant open subset of ,
the morphism between analytic spaces: is an isomorphism.
From (2), we have an isomorphism between analytic spaces:
by which we have a principal -bundle . In particular, covers .
For any symmetry type , we define a category as follows. The objects are families of degree -folds of type with a specified central fiber, and the morphisms are holomorphic maps between families, sending central fiber to central fiber and compatible with the action of .
The family of degree -folds of type over has the following universal property. For any subfamily of degree -folds of type containing a central fiber with isomorphism commuting with , we have a unique morphism in the category :
such that the restriction of to is .
Moreover, for any two fibers of with an isomorphism commuting with , we can extend uniquely to a morphism in
Proof of proposition 2.10.
The base lies in and is covered by . Thus we have a unique lifting , sending to . In other words, we have uniquely , which restricts to on .
Now suppose , are two fibers of with isomorphism . Denote , the corresponding base points in . Then have the same image in . Since is a principal -bundle, the two pairs and are -equivalent, hence . The corollary follows. ∎
We have the following lemma, which will be used in the proof of proposition 4.8. Since it holds for general degree -folds, we state and prove it here.
be a family of smooth degree -folds, with the base contractible. Suppose there is a group , such that for all , the fiber admits a biregular action of , with induced actions on compatible with respect to the local trivialization. Then there is an action of on the whole family inducing on each fiber the existed action.
For , , and a smooth degree -fold , the induced action of on is faithful.
Proof of lemma 2.11.
Take any . By theorem 2.5 in [Zhe17], there is a universal hypersurface family of , such that any isomorphism between two fibers (may coincide) of comes from an automorphism of the central fiber . There exists an open neighbourhood of in , with a unique morphism . Then for any , the action of on is induced by a subgroup of . By lemma 2.12, and compatibility of induced action of on and , we have that as subgroups of . Therefore, the actions of on fibers of glue to an action of on the whole family. ∎
3 Review: Period Map for Smooth Cubic Fourfolds
Take . Then we have the moduli of smooth cubic fourfolds, as a Zariski-open subset of its compactification . Let be a smooth cubic fourfold. We denote to be the intersection pairing on . Then is an odd unimodular lattice of signature . Denote to be square of the hyperplane class of . Then is an even sublattice of discriminant . Now we define to be an abstract data isomorphic to , this does not depend on the choice of the cubic fourfold .
A marking of the cubic fourfold is an isomorphism sending to .
Two marked cubic fourfolds and are called equivalent if there exists a linear isomorphism such that . Let be the set of equivalence classes of marked cubic fourfolds. From [Zhe17], section 3, we have:
The set is a complex manifold in a natural way.
Next we define the period domain and period map for cubic fourfolds. Let
this is an analytically open subset of a quadric hypersurface in , and has two connected components. We have naturally a holomorphic map
sending to . It is called the local period map for cubic fourfolds.
Let be one connected component of and the index subgroup of which respects the component . Then is an arithmetic group acting on and descends to
which is called the (global) period map for cubic fourfolds.
The subgroup consists of elements in with spinor norm 1. Since there exist vectors in with self intersection , the group is of index in .
Theorem 3.4 (Voisin).
The period map is an open embedding.
In fact, the period map is algebraic, see discussion in [Has00b] (proposition 2.2.3).
We give a lemma which will be constantly used. See [Zhe17] (theorem 1.1).
Take a smooth cubic fourfold, then .
We have a refined version of theorem 3.4:
Proposition 3.7 (Voisin, Hassett, Looijenga, Laza).
The local period map is an open embedding, with image being the complement of a hyperplane arrangement invariant under the action of on .
4 Period Maps for Symmetric Cubic Fourfolds
4.1 Local Period Map for Symmetric Cubic Fourfolds
In this section we are going to discuss the local and global period maps for symmetric cubic fourfolds. Let , and fix a symmetry type satisfying condition 2.2. We first introduce the local period domains with action of arithmetic groups. Let for a generic point . Recall that the action of on induces an action of on . This action is a character with trivial restriction on . We denote
Define a Hermitian form by . Denote to be the action of on . Let be a action of on , making isomorphic to . Denote to be the -eigenspace of the action of on .
The Hermitian form has signature if (this is also equivalent to ); it has signature otherwise. Here is a non-negative integer independent of the choice of .
Notice that the lattice has signature , with negative part . If is not contained in , we have . Since lies in -eigenspace, the signature of is .
For the case , both and are contained in , hence has signature . ∎
An isomorphisms is called a T-marking of . We consider pairs consisting of a smooth cubic fourfold and its T-marking. Two such pairs and are equivalent if there exists such that . Let be the set of equivalence classes of such pairs, we have:
The set is naturally a complex manifold.
First we describe the local charts on . Take a point , and take a universal deformation of as in proposition 2.10. Since is contractible, the local system is trivializable over and the T-marking of naturally extends to T-marking of every fiber of . Thus we have a map
We first show that is injective. Suppose , are two fibers of , with the induced T-markings by respectively, such that and represent the same point in . Then there exists with . By proposition 2.10 we have and , hence . By lemma 3.6 we have . Thus is injective.
By definition, is covered by countably many such , and they form a basis of a topology. To show is a complex manifold, we need to prove that the topology is Hausdorff. Suppose not, then we have two non-separated points . Then and are isomorphic (because is separated). Without loss of generality, we just assume . Take the universal family as in proposition 2.10, and
induced by and . Now since and are non-separated, we have . Thus there exists with corresponding cubic fourfold , such that the two pairs and represent the same point in . Then there is an automorphism of , such that . Proposition 2.10 implies that is also an automorphism of and satisfies the above relation. Thus in , contradiction. We showed the Hausdorff property, hence conclude that is naturally a complex manifold. ∎
Proposition 4.2 can be generalized to degree -folds (, ) with specified automorphism group. The argument is the same.
When has signature , we define , which is a hyperbolic complex ball of dimension ; when has signature , define to be a component of , which is a type symmetric domain of dimension .
We define local period map for symmetric cubic fourfolds of type as the map from to , sending to , still denoted by . We make the choice of such that has nonempty image in . Write if there is no confusion.
4.2 Properties of Local Period Maps for Symmetric Cubic Fourfolds
Theorem 4.4 ([Laz09]).
A cubic fourfold with at worst -singularities is stable.
Laza proved that the period map extends to the moduli space of cubic fourfolds with at worst singularities, and characterized its image. The results are gathered in the following theorem:
Theorem 4.5 ([Laz10]).
The period map has image , and extends holomorphically to
with image . Here are two -invariant hyperplane arrangements in , with the quotients and irreducible.
This characterization of the image was conjectured by Hassett in [Has00b]. Hassett defined the special cubic fourfolds, some of which correspond to polarized surfaces. The hyperplane arrangements and are two particular ones, parameterizing nodal cubic fourfolds and secent lines of determinantal cubic fourfold, and corresponding to surfaces of degree 6 and 2 respectively. See [Has00b], section 4.2 and 4.4.
We have also the following marked-version of theorem 4.5:
The local period map has image .
For a specified type , we have a natural embedding . Denote and . The local period map has image contained in .
The local period map is an open embedding, with image either or . In particular, .
We have a closed embedding . There is a natural map . Suppose represent the same point in , then there exists a linear isomorphism such that
Since are compatible with the action of on , , so is . Lemma 3.6 implies that is compatible with the actions of on . Thus represent the same point in . We showed the injectivity of .
Combining with the following commutative diagram:
we obtain the injectivity of . In particular, .
Since the differential of is injective everywhere, so is the differential of .
Take . Let and be any point in the component of containing . Since both and are connected, there exists a path
with and . The path has a unique lifting in . By proposition 3.7, we can choose a family of cubic fourfolds, with marking of every fiber, such that and , for all . Since , the Hodge structure on has action of induced by . By lemma 3.6, there exists an action of on for any , inducing compatible action on