Géométrie algébrique/Algebraic geometry

A Complex Hyperbolic Structure for Moduli of Cubic Surfaces

Daniel Allcock, James A. Carlson, and Domingo Toledo

Department of Mathematics University of Utah Salt Lake City, Utah, USA. E-mail: allcock, carlson and toledo at math.utah.edu

Abstract. We show that the moduli space M of marked cubic surfaces is biholomorphic to (B^{4} − H)/Γ_{0} where B^{4} is complex hyperbolic four-space, where Γ_{0} is a specific group generated by complex reflections, and where H is the union of reflection hyperplanes for Γ_{0}. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant holomorphic sectional curvature.

Une structure hyperbolique complexe pour les modules des surfaces cubiques

Résumé. Nous montrons que l’espace des modules M des surfaces cubiques marquées est biholomorphe à (B^{4} − H)/Γ_{0} oú B^{4} est l’espace complexe hyperbolique de dimension quatre, oú Γ_{0} est un groupe spécifique généré par des réflections complexes, et oú H est l’union de l’ensemble d’hyperplans de réflection de Γ_{0}. Donc M admet une structure hyperbolique complexe, c’est à dire une métrique (incomplète) de courbure holomorphe sectionnelle constante.

Version française abrégée

A une surface cubique (marquée) correspond une variété cubique de dimension trois (marquée), à savoir le revêtement de ℙ^{3} ramifié le long de la surface. L’application des périodes f pour ces variétés de dimension trois est définie sur l’espace des modules M des cubiques marquées, et cette application f prend ses valeurs dans une quotient de la boule unitaire dans ℂ^{4} par l’action du groupe de monodromie projective. Ce groupe Γ_{0} est généré par des réflections complexes dans un ensemble d’hyperplans dont nous notons la réunion par H. Alors nous avons le resultat suivant:

Théorème. L’application des périodes définit une biholomorphisme

f : M ⟶ (B^{4} − H) /Γ_{0}.

De ce théorème on obtient des résultats sur la structure métrique de M et sur son groupe fondamental:

First author partially supported by an NSF postdoctoral fellowship. Second and third authors partially supported by NSF grant DMS 9625463 Corollaires. (1) L’espace M des modules des surfaces cubiques marquées admet une structure hyperbolique complexe: une métrique (incomplète) de courbure holomorphe sectionnelle constante. (2) Le groupe fondamental de M contient un sous-groupe normale qui n’est pas de génération finie. (3) Le groupe fondamental de M n’est pas un réseau dans un groupe de Lie semisimple.

Remarques. (1) Nos méthodes montrent aussi que la complétée métrique de (B^{4}−H)/Γ_{0} est l’orbifolde B^{4}/Γ_{0}, isomorphe à l’espace des modules des surfaces cubiques marquées stables. (2) Récemment E. Looijenga a trouvé une présentation remarkable du groupe fondamental orbifolde de l’espace des modules des cubiques lisses non-marquées.

Afin de préciser la notion de surface cubique lisse marquée, fixons un réseau L, un ℤ-module libre avec une base e_{0} , …, e_{6} qui est muni de la forme quadratique telle que la base soit orthogonale et telle que ( e_{0}, e_{0} ) = 1, ( e_{k}, e_{k} ) = −1 pour k > 0. Soit η = 3e_{0} − (e_{1} + ⋯ + e_{6}). Alors une surface cubique marquée est composée d’une surface cubique lisse S et d’une isométrie ψ : L ⟶ H^{2}(S, ℤ) qui envoie η sur la classe d’un hyperplan. L’ensemble M de classes d’isomorphisme des surfaces cubiques marquées porte la structure d’une variété et de plus est un éspace de modules fines. Une construction de cette éspace a été donnée dans [9], oú on trouve aussi une compactification lisse C de M telle que les points de C − M forment un diviseur à croisements normaux.

Pour définir le groupe Γ_{0}, soit E l’anneau des entiers d’Eisenstein ℤ[ω] oú ω = (−1 + \sqrt{−3})/2 est une troisième racine d’unité, et considérons le produit Cartesien E^{5} muni d’une forme hermitienne h(v, w) = −v_{1}w̄_{1}+v_{2}w̄_{2}+v_{3}w̄_{3}+v_{4}w̄_{4}+v_{5}w̄_{5}. Alors (E^{5}, h) est l’unique réseau autodual sur les entiers d’Eisenstein qui est de signature (4, 1). Donc Aut(E^{5}, h) est un sous-groupe discrèt du groupe unitaire U(h), qui agit sur B^{4} = { ℓ ∈ ℙ^{4} : h|ℓ < 0 }. Notons que E/\sqrt{−3}E ≅ 𝔽_{3} est un corps de trois eléments et notons aussi qu’il y a un homomorphisme naturel Aut(E^{5}, h) ⟶ Aut(𝔽_{3}^{5}, q) oú q est la forme quadratique obtenue par réduction de h modulo \sqrt{−3}. Notons par “P ” la projectivisation, et définissons un groupe Γ_{0} d’automorphismes de B^{4} par la suite exacte

1 ⟶ Γ_{0} ⟶ P Aut(E^{5}, h) ⟶ P Aut(𝔽_{3}^{5}, q) ⟶ 1.

Ce groupe est le groupe discrèt du théorème principal. Les hyperplans de H sont définis par les équations h(x, v) = 0 pour des vecteurs v dans E^{5} avec h(v) = 1. Notons aussi que P Aut(𝔽_{3}^{5}, q) est isomorphe au groupe de Weyl du réseau E_{6}.

1. Main results To a (marked) cubic surface corresponds a (marked) cubic threefold defined as the

triple cover of ℙ^{3} ramified along the surface. The period map f for these threefolds is defined on the moduli space M of marked cubic surfaces and takes its values in the quotient of the unit ball in ℂ^{4} by the action of the projective monodromy group. This group Γ_{0} is generated by complex reflections in a set of hyperplanes whose union we denote by H. Then we have the following result:

Theorem. The period map defines a biholomorphism

f : M ⟶ (B^{4} − H) /Γ_{0}.

From this identification we obtain results on the metric structure and the fundamental group:

Corollaries. (1) The moduli space of marked cubic surfaces carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature. (2) The fundamental group of the space of marked cubic surfaces contains a normal subgroup which is not finitely generated. (3) The fundamental group of the space of marked cubic surfaces is not a lattice in a semisimple Lie group.

Remarks. (1) Our methods also show that the metric completion of (B^{4} − H)/Γ_{0} is the complex hyperbolic orbifold B^{4}/Γ_{0}, which is isomorphic to the moduli space of marked stable cubic surfaces. (2) Recently E. Looijenga found a remarkable presentation of the orbifold fundamental group of the moduli space of smooth unmarked cubic surfaces.

To make precise the notion of smooth marked cubic surface, fix the lattice L to be the free ℤ-module with basis e_{0} , …, e_{6} endowed with the quadratic form for which the given basis is orthogonal and such that ( e_{0}, e_{0} ) = 1, ( e_{k}, e_{k} ) = −1 for k > 0. Let η = 3e_{0} − (e_{1} + ⋯ + e_{6}). Then a marked cubic surface consists of a smooth cubic surface S and an isometry ψ : L ⟶ H^{2}(S, ℤ) which carries η to the hyperplane class. The set M of isomorphism classes of marked cubic surfaces has the structure of a variety and is a fine moduli space. A construction of it is described in [9], and a smooth compactification C is given for which the points of C − M constitute a normal crossing divisor.

To define the group Γ_{0}, let E denote the ring of Eisenstein integers ℤ[ω] where ω = (−1 + \sqrt{−3})/2 is a cube root of unity, and consider the Cartesian product E^{5} endowed with the hermitian inner product h(v, w) = −v_{1}w̄_{1} + v_{2}w̄_{2} + v_{3}w̄_{3} + v_{4}w̄_{4} + v_{5}w̄_{5}. Then (E^{5}, h) is the unique self-dual lattice over the Eisenstein integers with signature (4, 1). Thus Aut(E^{5}, h) is a discrete subgroup of the unitary group U(h), which acts on B^{4} = \{ ℓ ∈ ℙ^{4} : h|ℓ < 0 \}. Observe that E/\sqrt{−3}E ≅ 𝔽_{3} is a field of three elements and that there is a natural homomorphism Aut(E^{5}, h) ⟶ Aut(𝔽_{3}^{5}, q) where q is the quadratic form obtained by reduction of h modulo \sqrt{−3}. Let “P ” denote projectivization, and define a group Γ_{0} of automorphisms of B^{4} by

1 ⟶ Γ_{0} ⟶ P Aut(E^{5}, h) ⟶ P Aut(𝔽_{3}^{5}, q) ⟶ 1.

This is the discrete group of the main theorem. The hyperplanes of H are defined by the equations h(x, v) = 0 for vectors v in E^{5} with h(v) = 1. Note that P Aut(𝔽_{3}^{5}, q) is isomorphic to the Weyl group of the E_{6} lattice.

2. Construction of a period mapping To construct the period mapping, we examine in detail the Hodge structures for the

cubic threefolds. The underlying lattice H^{3}(T, ℤ) is ten-dimensional, carries a unimodular symplectic form Ω, and admits a Hodge decomposition of the form H^{3}(T, ℂ) = H^{2,1}⊕H^{1,2}. Choose a generator σ for the group of automorphisms of T over ℙ^{3}, and note that it operates without fixed points on H^{3}(T, ℤ). This action gives H^{3}(T, ℤ) the structure of a five-dimensional module over the Eisenstein integers. It carries a hermitian form

h(x, y) = \frac{1}{2}(Ω((σ − σ^{−1})x, y) + (ω − ω^{−1})Ω(x, y))

which is unimodular and of signature (4, 1).

Now consider the quotient module H^{3}(T, ℤ)/(1 − ω)H^{3}(T, ℤ) and observe that it can be identified isometrically with (𝔽_{3}^{5}, q). We define a marking of T to be choice of such an isometry, and we claim that a marking of a cubic surface determines a marking of the corresponding threefold. Indeed, if γ is a primitive two-dimensional homology class on S then it is the boundary of a three-chain Γ on T. Since Γ and σΓ have the same boundary, the three-chain c(γ) = (1 − σ)Γ is a cycle. However, it is well-defined only up to addition of elements (1 − σ)Δ where Δ is a three-cycle on T. Thus a homomorphism

c : H_{2}^{prim}(S, ℤ) ⟶ H_{3}(T, ℤ)/(1 − σ)

is defined. Since a marking of S can be viewed as a basis of H_{2}^{prim}(S, ℤ), application of c to the basis elements defines a basis of H_{3}(T, ℤ)/(1 − σ), and this gives the required marking of the threefold.

The action of σ decomposes H^{3}(T, ℂ) into eigenspaces H_{λ}^{3} where λ varies over the primitive cube roots of unity. Because σ is holomorphic, the decomposition is compatible with the Hodge decomposition and one has

H_{ω}^{3} = H_{ω}^{2,1} ⊕ H_{ω}^{1,2} H3 ; ω̄ = H2,1 ; ω̄ ⊕ H1,2 ; ω̄.

The dimensions of the Hodge components can be found with the help of Griffiths’ Poincaré residue calculus [5]. Details for this case are found in [3], section 5. One finds that

dim H_{ω}^{2,1} = 4, dim H_{ω}^{1,2} = 1, dim H2,1 ; ω̄ = 1, dim H1,2 ; ω̄ = 4,

and from the Hodge-Riemann bilinear relations one finds that h has signature (4, 1).

Now let ϕ be a generator of the one-dimensional space H2,1 ; ω̄ and let γ_{1} , …, γ_{5} be a standard basis of H^{3}(T, ℤ) considered as an E-module. By this we mean that the γ_{k} are orthogonal and that h(γ_{1}, γ_{1}) = −1 and h(γ_{k}, γ_{k}) = 1 for k > 1. Let v(ϕ, γ) be the vector in ℂ^{5} with components

v_{k} = ∫_{γ_{k}} ϕ.

One verifies that h(v, v) < 0 where now h is the hermitian form −|v_{1}|^{2} +|v_{2}|^{2} +⋯+|v_{5}|^{2}. Thus the line generated by v(ϕ, γ) defines a point in B^{4} ⊂ ℙ^{4}, and one checks that v(ϕ, γ) ∉ H. By well-known constructions (the work of Griffiths), the period vector defines a holomorphic map from the universal cover of M to the ball which transforms according to the projectivized monodromy representation for marked cyclic cubic threefolds. The proof that Γ_{0} is the projective monodromy group relies on the work of Libgober [6] and the first author [1]. Thus our construction yields a period map f : M ⟶ (B^{4} − H)/Γ_{0}.

3. Properties of the period mapping We must now show that f is bijective. For injectivity, consider once again the period

vector v(ϕ, γ). The vectors γ_{k} can be decomposed into eigenvectors γ′_{k} and γ′′_{k} for σ, with eigenvalues ω and ω̄, respectively. Let γ̂′_{k} and γ̂′′_{k} denote elements of the corresponding dual basis. Because ϕ is an eigenvector with eigenvalue ω̄, its integral over γ′_{k} vanishes, so that

ϕ = ∑_{k} γ̂′′_{k} ∫_{γ′′_{k}} ϕ = ∑_{k} γ̂′′_{k} ∫_{γ_{k}} ϕ.

Thus the components of v(ϕ, γ) determine ϕ as an element of H3 ; ω̄. Consequently the line ℂv(ϕ, γ) determines the complex Hodge structure H3 ; ω̄. Viewing the Hodge components of H3 ; ω̄ as subspaces of H^{3}(T, ℂ), we may take their conjugates to determine the complex Hodge structure H_{ω}^{3}. These two complex Hodge structures determine the Hodge structure on H^{3}(T, ℤ). Thus, by the Torelli theorem of Clemens-Griffiths [4], the period vector v(ϕ, γ) determines the cubic threefold T up to isomorphism. It remains to show that T, which perforce is a cyclic cubic threefold, determines its ramification locus uniquely. This follows from the fact that the locus in question is a planar component of the Hessian surface.

To prove surjectivity we first consider a smooth compactification C of M by a normal crossing divisor D, e.g., the one given by Naruki [9], as well as the Satake compactification \overline{B^{4}/Γ_{0}}, obtained by adding fourty points, the “cusps,” each corresponding to a null point of P (𝔽_{3}^{5}, q). By well-known results [2] in complex variable theory, the period map has a holomorphic extension to a map f̄ from C to the Satake compactification. Since C is compact, f̄ is open, and \overline{B^{4}/Γ_{0}} is connected, we conclude that f̄ is surjective.

4. Boundary components To pass from surjectivity of f̄ to surjectivity of f, we must show that f̄ maps the

compactifying divisor D to the complement of (B^{4} − H)/Γ_{0} in the Satake compactification. To this end write D as a sum of irreducible components, D = ⋃ D′_{i} ∪ ⋃ D′′_{j}, where D′_{i} parametrizes nodal cubic surfaces via the map to the geometric invariant theory compactification of the moduli space of smooth cubics, and where in the same way the D′′_{j} parametrize cubics with an A_{2} singularity.

Now consider a one-parameter family of cubic surfaces with smooth total space acquiring a node. Its local equation near the node has the form x^{2} + y^{2} + z^{2} = t and the corresponding family of cyclic cubic threefolds has the form x^{2} + y^{2} + z^{2} + w^{3} = t. The local monodromy of the latter has order six, its eigenvalues are primitive sixth roots of unity, and the space of vanishing cycles is two-dimensional. (These facts are well-known and the relevant literature and arguments are summarized in [3], section 6). From [7] we conclude that coefficients of the period vector on vanishing cycles are of the form A(t)t^{1/6} + B(t)t^{5/6} where A and B are holomorphic. Now the space of vanishing cycles is invariant under the action of σ and so constitutes a rank one E-submodule. One can choose a generator δ for it so that h(δ, δ) = 1, and then one has

lim_{t→0} ∫_{δ} ϕ = 0.

Thus the limiting value of the period vector lies in the orthogonal complement of δ. In other words, f̄(D′_{i}) lies in H/Γ_{0}, as required.

Consider next a one-parameter family of cubic surfaces with smooth total space whose central fiber acquires an A_{2} singularity. Its local equation is x^{2} + y^{2} + z^{3} = t and the corresponding family of cyclic cubic threefolds has local equation x^{2} + y^{2} + z^{3} + w^{3} = t. In this case the local monodromy is of infinite order. After replacing t by t^{3} one finds an expansion of the form ϕ(t) = A(t)(log t) γ̂ + (terms bounded in t), where A(0) ≠ 0 and where γ̂ is an integer cohomology class which is isotropic for h. Consequently the line ℂϕ(t) converges to the isotropic line ℂγ̂ as t converges to zero, hence converges to a cusp in the Satake compactification.

6. The corollaries Finally, we comment on the collaries. Part (a) is immediate. For part (b) let K

denote the kernel of the map π_{1}(M) ⟶ Γ_{0}. Then K is isomorphic to the fundamental group of B^{4} − H and it is easy to see that its abelianization is not finitely generated. We remark that K is not free: there are many sets of commuting elements corresponding to normal crossings of H. For (c) we note first that for lattices in semisimple Lie group of real rank greater than one, the results of Margulis [8] imply finite generation of all normal subgroups. The rank one case can be treated separately, as was shown to us by Michael Kapovich.

References

[1] D. Allcock, New complex and quaternion-hyperbolic reflection groups, submitted, http://www.math.utah.edu/˜ allcock

[2] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays, J. Differential Geometry 6 (1972), 543–560.

[3] J. Carlson and D. Toledo, Discriminant complements and kernels of monodromy representations, 22 pp, submitted, http://www.math.utah.edu/˜ carlson

[4] C. H. Clemens and P. A. Griffiths, the intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972), 281–356.

[5] P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. 90 (1969), 460-541.

[6] A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. 162 (1978), 63–67.

[8] B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. Ecole Norm. Sup., ser. 4, tome 7 (1974), 405–430.

[8] G. A. Margulis, Quotient groups of discrete subgroups and measure theory, Funct. Anal. Appl. 12 (1978), 295–305.

[9] I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982), 1-30.