Hirzebruch-Mumford Proportionality and

Locally Symmetric Varieties of Orthogonal

Type

V. Gritsenko, K. Hulek and G. K. Sankaran

Received: September 28, 2006 Revised: February 17, 2008

Communicated by Thomas Peternell

Abstract. For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2, 8m + 2) is of general type if m ≥ 5.

2000 Mathematics Subject Classification: 14J15, 11F55 Keywords and Phrases: Locally symmetric variety; modular form; Hirzebruch-Mumford proportionality

1 Modular varieties of orthogonal type

Let L be an integral indefinite lattice of signature (2, n) and ( , ) the associated bilinear form. By D_{L} we denote a connected component of the homogeneous type IV complex domain of dimension n

D_{L} = \{[w] ∈ ℙ(L ⊗ ℂ) | (w, w) = 0, (w, \overline{w}) > 0\}^{+}.

O^{+}(L) is the index 2 subgroup of the integral orthogonal group O(L) that leaves D_{L} invariant. Any subgroup Γ of O^{+}(L) of finite index determines a modular variety

F_{L}(Γ) = Γ \ D_{L}.

By [BB] this is a quasi-projective variety.

For some special lattices L and subgroups Γ < O^{+}(L) one obtains in this way the moduli spaces of polarised abelian or Kummer surfaces (n = 3, see [GH]), the moduli space of Enriques surfaces (n = 10, see [BHPV]), and the moduli spaces of polarised or lattice-polarised K3 surfaces (0 < n ≤ 19, see [Nik1, Dol]). Other interesting modular varieties of orthogonal type include the period domains of irreducible symplectic manifolds: see [GHS3]. It is natural to ask about the birational type of F_{L}(Γ). For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension, but nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. We determine the Kodaira dimension of many quasi-projective varieties associated with two series of even lattices. To explain what these varieties are, we first introduce the stable orthogonal group Õ(L) of a nondegenerate even lattice L. This is defined (see [Nik2] for more details) to be the subgroup of O(L) which acts trivially on the discriminant group A_{L} = L^{∨}/L, where L^{∨} is the dual lattice. If Γ < O(L) then we write Γ̃ = Γ ∩ Õ(L). Note that if L is unimodular then Õ(L) = O(L). The first series of varieties we want to study, which we call the modular varieties of unimodular type, is

F_{II}^{(m)} = O^{+}(II_{2,8m+2})\D_{II_{2,8m+2}}.  (1)

F_{II}^{(m)} is of dimension 8m + 2 and arises from the even unimodular lattice of signature (2, 8m + 2)

II_{2,8m+2} = 2U ⊕ mE_{8}(−1),

where U denotes the hyperbolic plane and E_{8}(−1) is the negative definite lattice associated to the root system E_{8}. The variety F_{II}^{(2)} is the moduli space of elliptically fibred K3 surfaces with a section (see e.g. [CM, Section 2]). The case m = 3 is of particular interest: it arises in the context of the fake Monster Lie algebra [B1]. The second series, which we call the modular varieties of K3 type, is

F_{2d}^{(m)} = Õ^{+}(L_{2d}^{(m)})\D_{L_{2d}^{(m)}}.  (2)

F_{2d}^{(m)} is of dimension 8m + 3 and arises from the lattice

L_{2d}^{(m)} = 2U ⊕ mE_{8}(−1) ⊕ ⟨−2d⟩,

where ⟨−2d⟩ denotes a lattice generated by a vector of square −2d. The first three members of the series F_{2d}^{(m)} have interpretations as moduli spaces. F_{2d}^{(2)} is the moduli space of polarised K3 surfaces of degree 2d. For m = 1 the 11-dimensional variety F_{2d}^{(1)} is the moduli space of lattice-polarised K3 surfaces, where the polarisation is defined by the hyperbolic lattice ⟨2d⟩ ⊕ E_{8}(−1) (see [Nik1, Dol]). For m = 0 and d prime the 3-fold F_{2d}^{(0)} is the moduli space of polarised Kummer surfaces (see [GH]).

Theorem 1.1 The modular varieties of unimodular and K3 type are varieties of general type if m and d are sufficiently large. More precisely:

(i) If m ≥ 5 then the modular varieties F_{II}^{(m)} and F_{2d}^{(m)} (for any d ≥ 1) are of general type.

(ii) For m = 4 the varieties F_{2d}^{(4)} are of general type if d ≥ 3 and d ≠ 4.

(iii) For m = 3 the varieties F_{2d}^{(3)} are of general type if d ≥ 1346.

(iv) For m = 1 the varieties F_{2d}^{(1)} are of general type if d ≥ 1537488.

Remark. The methods of this paper are also applicable if m = 2. Using them, one can show that the moduli space F_{2d}^{(2)} of polarised K3 surfaces of degree 2d is of general type if d ≥ 231000. This case was studied in [GHS2], where, using a different method involving special pull-backs of the Borcherds automorphic form Φ_{12} on the domain D_{II_{2,26}}, we proved that F_{2d}^{(2)} is of general type if d > 61 or d = 46, 50, 54, 57, 58, 60. The methods of [GHS2] do not appear to be applicable in the other cases studied here. Instead, the proof of Theorem 1.1 depends on the existence of a good toroidal compactification of F_{L}(Γ), which was proved in [GHS2], and on the exact formula for the Hirzebruch-Mumford volume of the orthogonal group found in [GHS1]. We shall construct pluricanonical forms on a suitable compactification of the modular variety F_{L}(Γ) by means of modular forms. Let Γ < O^{+}(L) be a subgroup of finite index, which naturally acts on the affine cone D_{L}^{•} over D_{L}. In what follows we assume that dim D_{L} ≥ 3.

Definition 1.2 A modular form of weight k and character χ: Γ → ℂ^{∗} with respect to the group Γ is a holomorphic function

F : D_{L}^{•} → ℂ

which has the two properties

\begin{aligned}F (tz) & = & t^{−k}F (z) & ∀ t ∈ ℂ^{∗}, \\ F (g(z)) & = & χ(g)F (z) & ∀ g ∈ Γ. \end{aligned}

The space of modular forms is denoted by M_{k}(Γ, χ). The space of cusp forms, i.e. modular forms vanishing on the boundary of the Baily–Borel compactification of Γ\D_{L}, is denoted by S_{k}(Γ, χ). We can reformulate the definition of modular forms in geometric terms. Let F ∈ M_{kn}(Γ, det^{k}) be a modular form, where n is the dimension of D_{L}. Then

F (dZ)^{k} ∈ H^{0}(F_{L}(Γ)^{∘}, Ω^{⊗k}),

where dZ is a holomorphic volume form on D_{L}, Ω is the sheaf of germs of canonical n-forms on F_{L}(Γ) and F_{L}(Γ)^{∘} is the open smooth part of F_{L}(Γ) such that the projection π : D_{L} → Γ\D_{L} is unramified over F_{L}(Γ)^{∘}.

The main question in the proof of Theorem 1.1 is how to extend the form F (dZ)^{k} to F_{L}(Γ) and to a suitable toroidal compactification F_{L}(Γ)^{tor}. There are three possible kinds of obstruction to this, which we call (as in [GHS2]) elliptic, reflective and cusp obstructions. Elliptic obstructions arise if F_{L}(Γ)^{tor} has non-canonical singularities arising from fixed loci of the action of the group Γ. Reflective obstructions arise because the projection π is branched along divisors whose general point is smooth in F_{L}(Γ). Cusp obstructions arise when we extend the form from F_{L}(Γ) to F_{L}(Γ)^{tor}. The problem of elliptic obstructions was solved for n ≥ 9 in [GHS2].

Theorem 1.3 ([GHS2, Theorem 2.1]) Let L be a lattice of signature (2, n) with n ≥ 9, and let Γ < O^{+}(L) be a subgroup of finite index. Then there exists a toroidal compactification F_{L}(Γ)^{tor} of F_{L}(Γ) = Γ\D_{L} such that F_{L}(Γ)^{tor} has canonical singularities and there are no branch divisors in the boundary. The branch divisors in F_{L}(Γ) arise from the fixed divisors of reflections.

Reflective obstructions, that is branch divisors, are a special problem related to the orthogonal group. They do not appear in the case of moduli spaces of polarised abelian varieties of dimension greater than 2, where the modular group is the symplectic group. There are no quasi-reflections in the symplectic group even for g = 3. The branch divisor is defined by special reflective vectors in the lattice L. This description is given in §2. To estimate the reflective obstructions we use the Hirzebruch-Mumford proportionality principle and the exact formula for the Hirzebruch-Mumford volume of the orthogonal group found in [GHS1]. We do the numerical estimation in §4. We treat the cusp obstructions in §3, using special cusp forms of low weight (the lifting of Jacobi forms) constructed in [G2] and the low-weight cusp form trick (see [G2] and [GHS2]).

2 The branch divisors

To estimate the obstruction to extending pluricanonical forms to a smooth projective model of F_{L}(Õ^{+}(L)) we have to determine the branch divisors of the projection

π: D_{L} → F_{L}(Õ^{+}(L)) = Õ^{+}(L) \ D_{L}.  (3)

According to [GHS2, Corollary 2.13] these divisors are defined by reflections σ_{r} ∈ O^{+}(L), where

σ_{r}(l) = l − \frac{2(l, r)}{(r, r)}r,

coming from vectors r ∈ L with r^{2} < 0 that are stably reflective: by this we mean that r is primitive and σ_{r} or −σ_{r} is in Õ^{+}(L). By a (k)-vector for k ∈ ℤ we mean a primitive vector r with r^{2} = k.

Let D be the exponent of the finite abelian group A_{L} and let the divisor div(r) of r ∈ L be the positive generator of the ideal (l, L). We note that r^{∗} = r/ div(r) is a primitive vector in L^{∨}. In [GHS2, Propositions 3.1–3.2] we proved the following.

Lemma 2.1 Let L be an even integral lattice of signature (2, n). If σ_{r} ∈ Õ^{+}(L) then r^{2} = −2. If −σ_{r} ∈ Õ^{+}(L), then r^{2} = −2D and div(r) = D ≡ 1 mod 2 or r^{2} = −D and div(r) = D or D/2.

We need also the following well-known property of the stable orthogonal group.

Lemma 2.2 For any sublattice M of an even lattice L the group Õ(M) can be considered as a subgroup of Õ(L).

Proof. Let M^{⊥} be the orthogonal complement of M in L. We have as usual

M ⊕ M^{⊥} ⊂ L ⊂ L^{∨} ⊂ M^{∨} ⊕ (M^{⊥})^{∨}.

We can extend g ∈ Õ(M) to M ⊕ M^{⊥} by putting g|_{M^{⊥}} ≡ id. It is clear that g ∈ Õ(M ⊕ M^{⊥}). For any l^{∨} ∈ L^{∨} we have g(l^{∨}) ∈ l^{∨} + (M ⊕ M^{⊥}). In particular, g(l) ∈ L for any l ∈ L and g ∈ Õ(L). □

We can describe the components of the branch locus in terms of homogeneous domains. For r a stably reflective vector in L we put

H_{r} = {[w] ∈ ℙ(L ⊗ ℂ) | (w, r) = 0},

and let N be the union of all hyperplane sections H_{r} ∩ D_{L} over all stably reflective vectors r.

Proposition 2.3 Let r ∈ L be a stably reflective vector: suppose that r and L do not satisfy D = 4, r^{2} = −4, div(r) = 2. Let K_{r} be the orthogonal complement of r in L. Then the associated component π(H_{r} ∩ D_{L}) of the branch locus N is of the form Õ^{+}(K_{r})\D_{K_{r}}.

Proof. We have H_{r} ∩ D_{L} = ℙ(K_{r}) ∩ D_{L} = D_{K_{r}}. Let

Γ_{K_{r}} = {φ ∈ Õ^{+}(L) | φ(K_{r}) = K_{r}}.  (4)

Γ_{K_{r}} maps to a subgroup of O^{+}(K_{r}). The inclusion of Õ(K_{r}) in Õ(L) (Lemma 2.2) preserves the spinor norm (see [GHS1, §3.1]), because K_{r} has signature (2, n − 1) and so Õ^{+}(K_{r}) becomes a subgroup of Õ^{+}(L). Therefore the image of Γ_{K_{r}} contains Õ^{+}(K_{r}) for any r. Now we prove that this image coincides with Õ^{+}(K_{r}) for all r, except perhaps if D = 4, r^{2} = −4 and div(r) = 2.

Let us consider the inclusions

⟨r⟩ ⊕ K_{r} ⊂ L ⊂ L^{∨} ⊂ ⟨r⟩^{∨} ⊕ K_{r}^{∨}.

By standard arguments (see [GHS2, Proposition 3.6]) we see that

| det K_{r}| = \frac{| det L| · |r^{2}|}{div(r)^{2}} and [L : ⟨r⟩ ⊕ K_{r}] = \frac{|r^{2}|}{div(r)} = 1 or 2.

If the index is 1, then it is clear that the image of Γ_{K_{r}} is Õ^{+}(K_{r}). Let us assume that the index is equal to 2. In this case the lattice ⟨r⟩^{∨} is generated by r^{∨} = −r/(r, r) = r^{∗}/2, where r^{∗} = r/ div(r) is a primitive vector in L^{∨}. In particular r^{∨} represents a non-trivial class in ⟨r⟩^{∨} ⊕ K_{r}^{∨} modulo L^{∨}. Let us take k^{∨} ∈ K_{r}^{∨} such that k^{∨} ∉ L^{∨}. Then k^{∨} + r^{∨} ∈ L^{∨} and

φ(k^{∨}) − k^{∨} ≡ r^{∨} − φ(r^{∨}) mod L.

We note that if φ ∈ Γ_{K_{r}} then φ(r) = ±r. Hence

φ(k^{∨}) − k^{∨} ≡ \begin{cases}0 mod L & if φ(r) = r \\ r^{∗} mod L & if φ(r) = −r \end{cases}

Since φ(r^{∗}) ≡ r^{∗} mod L, we cannot have φ(r) = −r unless div(r) = 1 or 2. Therefore we have proved that φ(k^{∨}) ≡ k^{∨} mod K_{r} (K_{r} = K_{r}^{∨} ∩ L), except possibly if D = 4, r^{2} = −4, div(r) = 2. □

The group Õ^{+}(L) acts on N. We need to estimate the number of components of Õ^{+}(L) \ N. This will enable us to estimate the reflective obstructions to extending pluricanonical forms which arise from these branch loci. For the even unimodular lattice II_{2,8m+2} any primitive vector r has div(r) = 1. Consequently r is stably reflective if and only if r^{2} = −2. For L_{2d}^{(m)} the reflections and the corresponding branch divisors arise in two different ways, according to Lemma 2.1. We shall classify the orbits of such vectors.

Proposition 2.4 Suppose d is a positive integer.

(i) Any two (−2)-vectors in the lattice II_{2,8m+2} are equivalent modulo O^{+}(II_{2,8m+2}), and the orthogonal complement of a (−2)-vector r is isometric to

K_{II}^{(m)} = U ⊕ mE_{8}(−1) ⊕ ⟨2⟩.

(ii) There is one Õ^{+}(L_{2d}^{(m)})-orbit of (−2)-vectors r in L_{2d}^{(m)} with div(r) = 1. If d ≡ 1 mod 4 then there is a second orbit of (−2)-vectors, with div(r) = 2. The orthogonal complement of a (−2)-vector r in L_{2d}^{(m)} is isometric to

K_{2d}^{(m)} = U ⊕ mE_{8}(−1) ⊕ ⟨2⟩ ⊕ ⟨−2d⟩, if div(r) = 1, and to

N_{2d}^{(m)} = U ⊕ mE_{8}(−1) ⊕ (\begin{matrix}1 & 2 \\ \frac{1−d}{2} & 1 \end{matrix}) ,

if div(r) = 2.

(iii) The orthogonal complement of a (−2d)-vector r in L_{2d}^{(m)} is isometric to

II_{2,8m+2} = 2U ⊕ mE_{8}(−1)

if div(r) = 2d, and to

K_{2}^{(m)} = U ⊕mE_{8}(−1)⊕⟨2⟩⊕⟨−2⟩ or T_{2,8m+2} = U ⊕U(2)⊕mE_{8}(−1)

if div(r) = d.

(iv) Suppose d > 1. The number of Õ(L_{2d}^{(m)})-orbits of (−2d)-vectors with div(r) = 2d is 2^{ρ(d)}. The number of Õ(L_{2d}^{(m)})-orbits of (−2d)-vectors with div(r) = d is

2^{ρ(d)}  if d is odd or d ≡ 4 mod 8;
2^{ρ(d)+1} if d ≡ 0 mod 8;
2^{ρ(d)−1} if d ≡ 2 mod 4.

Here ρ(d) is the number of prime divisors of d.

Proof. If the lattice L contains two hyperbolic planes then according to the well-known result of Eichler (see [E, §10]) the Õ^{+}(L)-orbit of a primitive vector l ∈ L is completely defined by two invariants: by its length (l, l) and by its image l^{∗} + L in the discriminant group A_{L}, where l^{∗} = l/ div(l).

i) If u is a primitive vector of an even unimodular lattice II_{2,8m+2} then div(u) = 1 and there is only one O(II_{2,8m+2})-orbit of (−2)-vectors. Therefore we can take r to be a (−2)-vector in U, and the form of the orthogonal complement is obvious. ii) In the lattice L_{2d}^{(m)} we fix a generator h of its ⟨−2d⟩-part. Then for any r ∈ L_{2d}^{(m)} we can write r = u + xh, where u ∈ II_{2,8m+2} and x ∈ ℤ. It is clear that div(r) divides r^{2}. If f| div(r), where f = 2, d or 2d, then the vector u is also divisible by f. Therefore the (−2)-vectors form two possible orbits of vectors with divisor equal to 1 or 2. If r^{2} = −2 and div(r) = 2 then u = 2u_{0} with u_{0} ∈ 2U ⊕mE_{8}(−1) and we see that in this case d ≡ 1 mod 4. This gives us two different orbits for such d. In both cases we can find a (−2)-vector r in the sublattice U ⊕ ⟨−2d⟩. Elementary calculation gives us the orthogonal complement of r. iii) This was proved in [GHS2, Proposition 3.6] for m = 2. For general m the proof is the same. iv) To find the number of orbits of (−2d)-vectors we have to consider two cases.

a) Let div(r) = 2d. Then r = 2du + xh and r^{∗} ≡ (x/2d)h mod L, where u ∈ II_{2,8m+2} and x is modulo 2d. Moreover (r, r) = 4d^{2}(u, u) − x^{2}2d = −2d. Thus x^{2} ≡ 1 mod 4d. This congruence has 2^{ρ(d)} solutions modulo 2d. For any such x mod 2d we can find a vector u in 2U ⊕ mE_{8}(−1) with (u, u) = (x^{2} − 1)/2d. Then r = 2du + xh is primitive (because u is not divisible by any divisor of x) and (r, r) = −2d.

b) Let div(r) = d. Then r = du+xh, where u is primitive, r^{∗} ≡ (x/d)h mod L and x is modulo d. We have (r^{∗}, r^{∗}) ≡ −2x^{2}/d mod 2ℤ and x^{2} ≡ 1 mod d. For any solution modulo d we can find as above u ∈ 2U ⊕ mE_{8}(−1) such that r = du + xh is primitive and (r, r) = −2d. It is easy to see that the number of solutions {x mod d | x^{2} ≡ 1 mod d} is as stated. □

Remark. To calculate the number of the branch divisors arising from vectors r with r^{2} = −2d one has to divide the corresponding number of orbits found in Proposition 2.4(iv) by 2 if d > 2. This is because ±r determine different orbits but the same branch divisor. For d = 2 the proof shows that there is one divisor for each orbit given in Proposition 2.4(iv).

3 Modular forms of low weight

In this section we let L = 2U ⊕ L_{0} be an even lattice of signature (2, n) with two hyperbolic planes. We choose a primitive isotropic vector c_{1} in L. This vector determines a 0-dimensional cusp and a tube realisation of the domain D_{L}. The tube domain (see the definition of H(L_{1}) below) is a complexification of the positive cone of the hyperbolic lattice L_{1} = c_{1}^{⊥}/c_{1}. If div(c_{1}) = 1 we call this cusp standard (as above, by [E] there is only one standard cusp). In this case L_{1} = U ⊕ L_{0}. In [GHS2, §4] we proved that any 1-dimensional boundary component of Õ^{+}(L) \ D_{L} contains the standard 0-dimensional cusp if every isotropic (with respect to the discriminant form: see [Nik2, §1.3]) subgroup of A_{L} is cyclic. Let us fix a 1-dimensional cusp by choosing two copies of U in L. (One has to add to c_{1} a primitive isotropic vector c_{2} ∈ L_{1} with div(c_{2}) = 1). Then L = U ⊕ L_{1} = U ⊕ (U ⊕ L_{0}) and the construction of the tube domain may be written down simply in coordinates. We have

H(L_{1}) = H_{n} = {Z = (z_{n}, …, z_{1}) ∈ ℍ_{1} × ℂ^{n−2} × ℍ_{1}; (Im Z, Im Z)_{L_{1}} > 0},

where Z ∈ L_{1} ⊗ ℂ and (z_{n−1}, …, z_{2}) ∈ L_{0} ⊗ ℂ. (We represent Z as a column vector.) An isomorphism between H_{n} and D_{L} is given by

p: H_{n} ⟶ D_{L}  (5)
Z = (z_{n}, …, z_{1}) ⟼ ( − \frac{1}{2}(Z, Z)_{L_{1}} : z_{n} : ⋯: z_{1} : 1).

The action of O^{+}(L ⊗ ℝ) on H_{n} is given by the usual fractional linear transformations. A calculation shows that the Jacobian of the transformation of H_{n} defined by g ∈ O^{+}(L ⊗ ℝ) is equal to det(g)j(g, Z)^{−n}, where j(g, Z) is the last ((n+2)-nd) coordinate of g(p(Z)) ∈ D_{L}. Using this we define the automorphic factor

J: O^{+}(L ⊗ ℝ) × H_{n+2} → ℂ^{∗}
(g, Z) ↦ (det g)^{−1} · j(g, Z)^{n}.

The connection with pluricanonical forms is the following. Consider the form

dZ = dz_{1} ∧ ⋯ ∧ dz_{n} ∈ Ω^{n}(H_{n}).

F (dZ)^{k} is a Γ-invariant k-fold pluricanonical form on H_{n}, for Γ a subgroup of finite index of O^{+}(L), if F (g(Z)) = J(g, Z)^{k}F (Z) for any g ∈ Γ; in other words if F ∈ M_{nk}(Γ, det^{k}) (see Definition 1.2). To prove Theorem 1.1 we need cusp forms of weight smaller than the dimension of the corresponding modular variety.

Proposition 3.1 For unimodular type, cusp forms of weight 12 + 4m exist: that is

dim S_{12+4m}(O^{+}(II_{2,8m+2})) > 0.

For K3 type we have the bounds

dim S_{11+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d > 1;
dim S_{10+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d ≥ 1;
dim S_{7+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d ≥ 4;
dim S_{6+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d = 3 or d ≥ 5;
dim S_{5+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d = 5 or d ≥ 7;
dim S_{2+4m}(Õ^{+}(L_{2d}^{(m)})) > 0 if d > 180.

Proof. For any F (Z) ∈ M_{k}(Õ^{+}(L)) we can consider its Fourier-Jacobi expansion at the 1-dimensional cusp fixed above

F (Z) = f_{0}(z_{1}) + ∑_{m≥1} f_{m}(z_{1}; z_{2}, … z_{n−1}) exp(2πimz_{n}).

A lifting construction of modular forms F (Z) ∈ M_{k}(Õ^{+}(L)) with trivial character by means of the first Fourier–Jacobi coefficient is given in [G1], [G2]. We note that f_{1}(z_{1}; z_{2}, …, z_{n−1}) ∈ J_{k,1}(L_{0}), where J_{k,1}(L_{0}) is the space of the Jacobi forms of weight k and index 1. A more general construction of the additive lifting was given in [B2] but for our purpose the construction of [G2] is sufficient.

The dimension of J_{k,1}(L_{0}) depends only on the discriminant form and the rank of L_{0} (see [G2, Lemma 2.4]). In particular, for the special cases of L = II_{2,8m+2} and L = L_{2d}^{(m)} we have

J_{k+4m,1}^{cusp}(mE_{8}(−1)) ≅ S_{k}(SL_{2}(ℤ))

and

J_{k+4m,1}^{cusp}(mE_{8}(−1) ⊕ ⟨−2d⟩) ≅ J_{k,d}^{cusp},

where J_{k,d}^{cusp} is the space of the usual Jacobi cusp forms in two variables of weight k and index d (see [EZ]) and S_{k}(SL_{2}(ℤ)) is the space of weight k cusp forms for SL_{2}(ℤ). The lifting of a Jacobi cusp form of index one is a cusp form of the same weight with respect to O^{+}(II_{2,8m+2}) or Õ^{+}(L_{2d}^{(m)}) with trivial character. The fact that we get a cusp form was proved in [G2] for maximal lattices, i.e., if d is square-free. In [GHS2, §4] we extended this to all lattices L for which the isotropic subgroups of the discriminant A_{L} are all cyclic, which is true in all cases considered here. To prove the unimodular type case of Proposition 3.1 we can take the Jacobi form corresponding to the cusp form Δ_{12}(τ). Using the Jacobi lifting construction we obtain a cusp form of weight 12 + 4m with respect to O^{+}(II_{2,8m+2}). For the K3 type case we need the dimension formula for the space of Jacobi cusp forms J_{k,d}^{cusp} (see [EZ]). For a positive integer l one sets

\{l\}_{12} = \{ \begin{matrix}⌊\frac{l}{12}⌋ & if l ≢ 2 mod 12 \\ ⌊\frac{l}{12}⌋ − 1 & if l ≡ 2 mod 12. \end{matrix}

Then if k > 2 is even

dim J_{k,d}^{cusp} = ∑_{j=0}^{d} (\{k + 2j\}_{12} − ⌊\frac{j^{2}}{4d}⌋) ,

and if k is odd

dim J_{k,d}^{cusp} = ∑_{j=1}^{d−1} (\{k − 1 + 2j\}_{12} − ⌊\frac{j^{2}}{4d}⌋) .

This gives the bounds claimed. For k = 2, using the results of [SZ] one can also calculate dim J_{2,d}^{cusp}: there is an extra term, ⌈σ_{0}(d)/2⌉, where σ_{0}(d) denotes the number of divisors of d. This gives dim J_{2,d}^{cusp} > 0 if d > 180 and for some smaller values of d. □

4 Kodaira dimension results

In this section we prove Theorem 1.1. We first explain the geometric background. Let F_{L}(Γ)^{tor} be a toroidal compactification as in Theorem 1.3. In particular all singularities are canonical and there is no ramification divisor which is contained in the boundary. Then the canonical divisor (as a ℚ-divisor) is given by K_{F_{L}(Γ)^{tor}} = nM − V − D where M is the line bundle of modular forms of weight 1, n is the dimension of F_{L}(Γ), V is the branch locus (which is given by reflections) and D is the boundary. Hence in order to construct k-fold pluricanonical forms we must find modular forms of weight kn which vanish of order k along the branch divisor and the boundary. This also suffices since F_{L}(Γ)^{tor} has canonical singularities. Our strategy is the following. For Γ ⊆ Õ^{+}(L) we choose a cusp form F_{a} ∈ S_{a}(Γ) of low weight a, i.e. a strictly less than the dimension. Then we consider elements F ∈ F_{a}^{k}M_{k(n−a)}(Γ, det^{k}): for simplicity we assume that k is even. Such an F vanishes to order at least k on the boundary of any toroidal compactification. Hence if dZ is the volume element on D_{L} defined in §3 it follows that F (dZ)^{k} extends as a k-fold pluricanonical form to the general point of every boundary component of F_{L}(Γ)^{tor}. Since we have chosen the toroidal compactification so that all singularities are canonical and that there is no ramification divisor which is contained in the boundary the only obstructions to extending F (dZ)^{k} to a smooth projective model are the reflective obstructions, coming from the ramification divisor of the quotient map π: D_{L} → F_{L}(Γ) studied in §2. Let D_{K} be an irreducible component of this ramification divisor. Recall from Proposition 2.3 that D_{K} = ℙ(K ⊗ ℂ) ∩ D_{L} where K = K_{r} is the orthogonal complement of a stably reflective vector r. For the lattices chosen in Theorem 1.1 all irreducible components of the ramification divisor are given in Proposition 2.4.

Proposition 4.1 We assume that k is even and that the dimension n ≥ 9. For Γ ⊆ Õ^{+}(L), the obstruction to extending forms F (dZ)^{k} where F ∈ F_{a}^{k}M_{k(n−a)}(Γ) to F_{L}(Γ)^{tor} lies in the space

B = ⨁_{K} B(K) = ⨁_{K} ⨁_{ν=0}^{k/2−1} M_{k(n−a)+2ν}(Γ ∩ Õ^{+}(K)),

where the direct sum is taken over all irreducible components D_{K} of the ramification divisor of the quotient map π: D_{L} → F(Γ).

Proof. Let σ ∈ Γ be plus or minus a reflection whose fixed point locus is D_{K}. We can extend the differential form provided that F vanishes of order k along every irreducible component D_{K} of the ramification divisor. If F_{a} vanishes along D_{K} then K gives no restriction on the second factor of the modular form F. Now let {w = 0} be a local equation for D_{K}. Then σ^{∗}(w) = −w (this is independent of whether σ or −σ is the reflection). For every modular form F ∈ M_{k}(Γ) of even weight we have F (σ(z)) = F (z). This implies that if F (z) ≡ 0 on D_{K}, then F vanishes to even order on D_{K}.

We denote by M_{2b}(Γ)(−νD_{K}) the space of modular forms of weight 2b which vanish of order at least ν along D_{K}. Since the weight is even we have M_{2b}(Γ)(−D_{K}) = M_{2b}(Γ)(−2D_{K}). For F ∈ M_{2b}(Γ)(−2νD_{K}) we consider (F/w^{2ν}) as a function on D_{K}. From the definition of modular form (Definition 1.2) it follows that this function is holomorphic, Γ ∩ Γ_{K}-invariant (see equation (4)) and homogeneous of degree 2b + 2ν. Thus (F/w^{2ν})|_{D_{K}} ∈ M_{2(b+ν)}(Γ ∩ Γ_{K}). In Proposition 2.3 we saw that, Γ_{K} contains Õ^{+}(K) as sub-

group of Õ^{+}(L)(with equality in almost all cases), so we may replace Γ ∩ Γ_{K} by Γ ∩ Õ^{+}(K). In this way we obtain an exact sequence

0 → M_{2b}(Γ)(−(2 + 2ν)D_{K}) → M_{2b}(Γ)(−2νD_{K}) → M_{2(b+ν)}(Γ ∩ Õ^{+}(K)),

where the last map is given by F ↦ F/w^{2ν}. This gives the result. □

Now we proceed with the proof of Theorem 1.1. Let L be a lattice of signature (2, n) and Γ < Õ^{+}(L): recall that k is even. According to Proposition 4.1 we can find pluricanonical differential forms on F_{L}(Γ)^{tor} if

C_{B}(Γ) = dim M_{k(n−a)}(Γ) − ∑_{K} dim B(K) > 0,  (6)

where summation is taken over all irreducible components of the ramification divisor (see the remark at the end of §2). It now remains to estimate the dimension of B(K) for each of the finitely many components of the ramification locus in the cases we are interested in, namely Γ = O^{+}(II_{2,8m+2}) and Γ = Õ^{+}(L_{2d}^{(m)}). According to the Hirzebruch-Mumford proportionality principle

dim M_{k}(Γ) = \frac{2}{n!} vol_{HM}(Γ)k^{n} + O(k^{n−1}).

The exact formula for the Hirzebruch-Mumford volume vol_{HM} for any indefinite orthogonal group was obtained in [GHS1]. It depends mainly on the determinant and on the local densities of the lattice L. Here we simply quote the estimates of the dimensions of certain spaces of cusp forms. The case of II_{2,8m+2} is easier because the branch divisor has only one irreducible component defined by any (−2)-vector r. According to Proposition 2.4 the orthogonal complement K_{r} is K_{II}^{(m)}. This lattice differs from the lattice L_{2}^{(m)}, whose Hirzebruch-Mumford volume was calculated in [GHS1, §3.5], only by one copy of the hyperbolic plane. Therefore

vol_{HM} Õ^{+}(L_{2}^{(m)}) = (B_{8m+4}/(8m + 4)) vol_{HM} Õ^{+}(K_{II}^{(m)}),

and hence, for even k,

dim M_{k}(Õ^{+}(K_{II}^{(m)})) = \frac{2^{1−4m}}{(8m + 1)!} · \frac{B_{2} … B_{8m+2}}{(8m + 2)!!} k^{8m+1} + O(k^{8m}),

where the B_{i} are Bernoulli numbers. Assume that m ≥ 3. Let us take a cusp form

F ∈ S_{4m+12}(O^{+}(II_{2,8m+2}))

from Proposition 3.1. In this case the dimension of the obstruction space B of Proposition 4.1 for the pluricanonical forms of order k = 2k_{1} is given by

∑_{ν=0}^{k_{1}−1} dim M_{(4m−10)k+2ν}(Õ^{+}(K_{II}^{(m)})) =

\frac{2^{4m+2}}{(8m + 2)!} · \frac{B_{2} … B_{8m+2}}{(8m + 2)!!} ((1 + \frac{1}{4m − 10})^{8m+2} − 1)((4m − 10)k_{1})^{8m+2}

+O(k^{8m+1})

In [GHS1, §3.3] we computed the leading term of the dimension of the space of modular forms for O^{+}(II_{2,8m+2}). Comparing these two we see that the constant C_{B}(O^{+}(II_{2,8m+2})) in the obstruction inequality (6) is positive if and only if

\frac{B_{4m+2}}{4m + 2} > (1 + \frac{1}{4m − 10})^{8m+2} − 1.  (7)

Moreover F_{II}^{(m)} is of general type if C_{B}(O^{+}(II_{2,8m+2})) > 0. From Stirling’s formula

5\sqrt{πn}(\frac{n}{πe})^{2n} > |B_{2n}| > 4\sqrt{πn}(\frac{n}{πe})^{2n}.  (8)

Using this estimate we easily obtain that (7) holds if m ≥ 5. Therefore we have proved Theorem 1.1 for the lattice II_{2,8m+2}. Next we consider the lattice L_{2d}^{(m)} of K3 type. For this lattice the branch divisor of F_{2d}^{(m)} is calculated in Proposition 2.4. It contains one or two (if d ≡ 1 mod 4) components defined by (−2)-vectors and some number of components defined by (−2d)-vectors. To estimate the obstruction constant C_{B}(Γ) in (6) we use the dimension formulae for the space of modular forms with respect to the group Õ^{+}(M), where M is one of the following lattices from Proposition 2.4: L_{2d}^{(m)} (the main group); K_{2d}^{(m)} and N_{2d}^{(m)} (the (−2)-obstruction); M_{2,8m+2}, K_{2}^{(m)} and T_{2,8m+2} (the (−2d)-obstruction). The corresponding dimension formulae were found in [GHS1] (see §§3.5, 3.6.1–3.6.2, 3.3 and 3.4). The branch divisor of (−2d)-type appears only if d > 1. We note that

vol_{HM}(Õ^{+}(T_{2,8m+2})) > vol_{HM}(Õ^{+}(K_{2}^{(m)})).  (9)

Therefore in order to estimate C_{B}(Γ) we can assume that all (−2d)-divisors defined by stably reflective (−2d)-vectors r with div(r) = d (see Proposition 2.4) are of the type T_{2,8m+2}. We put k = 2k_{1}, w = n − a and n = 8m + 3. For the obstruction constant in (6) we obtain

C_{B}(Õ^{+}(L_{2d}^{(m)})) > dim M_{2k_{1}w}(Õ^{+}(L_{2d}^{(m)})) − B_{(−2)} − B_{(−2d)}  (10)

where

B_{(−2)} = dim B(K_{2d}^{(m)}) + dim B(N_{2d}^{(m)}),
B_{(−2d)} = 2^{ρ(d)−1}(dim B(M_{2,8m+2}) + 2^{h_{d}} dim B(T_{2,8m+2}))

and B(K) is the obstruction space from Proposition 4.1. By h_{d} we denote the sum δ_{0,d(8)} − δ_{2,d(4)}, where d(n) is d mod n and δ is the Kronecker delta (see Proposition 2.4 and the remark following it). For any lattice considered above

B(K) = ∑_{ν=0}^{k_{1}−1} dim M_{2(k_{1}w+ν)}(Õ^{+}(K))

= \frac{2^{8m+3}}{(8m + 3)!}E_{w}(8m + 3)vol_{HM}(Õ^{+}(K))(k_{1}w)^{8m+3} + O(k_{1}^{8m+2})

where E_{w}(8m + 3) = (1 + \frac{1}{w})^{8m+3}. All terms in (10) contain a common factor. First

dim M_{2k_{1}w}(Õ^{+}(L_{2d}^{(m)})) = C_{m,d}^{k_{1},w} |\frac{B_{8m+4}}{B_{4m+2}} | \sqrt{d} + O(k_{1}^{8m+2}),  (11)

where

C_{m,d}^{k_{1},w} =

\frac{2^{4m+1+δ_{1,d}}}{(8m + 3)!} \frac{|B_{2} … B_{8m+2}|}{(8m + 2)!!} \frac{|B_{4m+2}|}{4m + 2}d^{4m+ \frac{3}{2}} ∏_{p|d}(1 − p^{−(4m+2)})(k_{1}w)^{8m+3}.

We note that 2^{4m+1} \frac{B_{4m+2}}{4m+2} = π^{−(4m+2)}Γ(4m + 2)ζ(4m + 2). From [GHS1, (16)] it follows that

vol_{HM}(Õ^{+}(K_{2d}^{(m)})) =

2^{δ_{1,d}−δ_{4,d(8)}} \frac{B_{2} … B_{8m+2}}{(8m + 2)!!} d^{4m+ \frac{3}{2}} π^{−(4m+2)}Γ(4m + 2)L(4m + 2, (\frac{4d}{∗})).

We can use the formula for the volume of N_{2d}^{(m)} in the following form:

vol_{HM}(Õ^{+}(N_{2d}^{(m)})) =

2^{1+δ_{1,d}−(8m+4)}d^{4m+ \frac{3}{2}} \frac{B_{2} … B_{8m+2}}{(8m + 2)!!} π^{−(4m+2)}Γ(4m + 2)L(4m + 2, (\frac{d}{∗}))

(see [GHS1, 3.6.2]). It follows that

B_{(−2)} = C_{m,d}^{k_{1},w}E_{w}(8m+3)(2^{8m+3−δ_{4,d(8)}}P_{K}(4m+2)+P_{N}(4m+2))+O(k_{1}^{8m+2}) where

P_{K}(n) = (1 − 2^{−n})^{δ_{0,d(2)}} \frac{L(n, (\frac{4d}{∗}))}{L(n, χ_{0,4d})} ∏_{p|d} \frac{1 − p^{−n}}{1 + p^{−n}}

and

P_{N}(n) = \frac{L(n, (\frac{d}{∗}))}{L(n, χ_{0,d})} ∏_{p|d} \frac{1 − p^{−n}}{1 + p^{−n}} .

Here χ_{0,f} denotes the principal Dirichlet character modulo f. We note that |P_{K}(n)| < 1 and |P_{N}(n)| < 1 for any d. We conclude that

B_{(−2)} < C_{m,d}^{k_{1},w}E_{w}(8m + 3)b_{(−2)}

where b_{(−2)} = 2^{8m+3} − 1. The (−2d)-contribution is calculated according to [GHS1, 3.3–3.4]. We note that Õ^{+}(T_{2,8m+2}) is a subgroup of Õ^{+}(M_{2,8m+2}). We obtain

B_{(−2d)} ≤ C_{m,d}^{k_{1},w}E_{w}(8m + 3)b_{(−2d)}

where for d > 2

b_{(−2d)} = \frac{2^{ρ(d)}}{d} (\frac{4}{d})^{4m+ \frac{1}{2}} 4(2^{h_{d}}(1 + 2^{−(4m+2)} − 2^{−(8m+3)}) + 2^{−(8m+3)}).

As a result we see that that the obstruction constant C_{B}(Õ^{+}(L_{2d}^{(m)})) is positive if

β_{m,d}^{(w)} = |\frac{B_{4m+2}}{B_{8m+4}} | E_{w}(8m + 3)(b_{(−2)} + b_{(−2d)}) < \sqrt{d}.

Using (8) we get

|\frac{B_{4m+2}}{B_{8m+4}} | < \frac{5}{4\sqrt{2}} (\frac{πe}{2m + 1})^{4m+2} \frac{1}{2^{8m+4}} .

For m ≥ 5 we choose a cusp form F_{a} of weight a = 4m + 10, i.e. we take w = 4m − 7 in Proposition 4.1. Such a cusp form exists for all d ≥ 1 by Proposition 3.1. Using the fact that β_{(−2d)} ≤ β_{(−4)} for any d ≥ 2 and the value b_{(−4)} = 2^{4m+ \frac{5}{2}} , we see that

β_{m,d}^{(4m−7)} < (1 + \frac{1}{4m − 7})^{8m+3} \frac{5}{8\sqrt{2}} (\frac{πe}{2m + 1})^{4m+2} \frac{2^{8m+3} + 2^{4m+ \frac{5}{2}} + 1}{2^{8m+3}},

which is smaller than 1 if m ≥ 5. This proves Theorem 1.1 for m ≥ 5. For m = 4 there exists a cusp form F_{a} of weight 4m + 6 if d ≠ 1, 2, 4, i.e. we take w = 4m − 3. To see that β_{4,d}^{(13)} < \sqrt{d} we need check this only for d = 3 because b_{(−2d)} < b_{(−6)} for d > 3. One can do it by direct calculation.

For m ≤ 3 we choose F_{a} of weight 4m + 2, i.e. we take w = 4m + 1. Such a cusp form exists if d > 180 according to Proposition 3.1. For such d we see that β_{(−2d)} < 1. Then the obstruction constant C_{B}(Õ^{+}(L_{2d}^{(m)})) is positive if

|\frac{B_{4m+2}}{B_{8m+2}} | (1 + \frac{1}{4m + 1})^{8m+3}(2^{8m+3} + 2) < \sqrt{d}.

This inequality gives us the bound on d in Theorem 1.1. This completes the proof of Theorem 1.1.

In the proof of Theorem 1.1 above we have seen that the (−2)-part of the branch divisor forms the most important reflective obstruction to the extension of the Õ^{+}(L_{2d}^{(m)})-invariant differential forms to a smooth compact model of F_{2d}^{(m)}. Let us consider the double covering SF_{2d}^{(m)} of F_{2d}^{(m)} for d > 1 determined by the special orthogonal group:

SF_{2d}^{(m)} = S̃O^{+}(L_{2d}^{(m)}) \D_{L_{2d}^{(m)}} → F_{2d}^{(m)}.

Here the branch divisor does not contain the (−2)-part. Theorem 4.2 below shows that there are only five exceptional varieties SF_{2d}^{(m)} with m > 0 and d > 1 that are possibly not of general type. The variety SF_{2d}^{(2)} can be interpreted as the moduli space of K3 surfaces of degree 2d with spin structure: see [GHS2, §5]. The three-fold SF_{2d}^{(0)} is the moduli space of (1, t)-polarised abelian surfaces.

Theorem 4.2 The variety SF_{2d}^{(m)} is of general type for any d > 1 if m ≥ 3. If m = 2 then SF_{2d}^{(2)} is of general type if d ≥ 3. If m = 1 then SF_{2d}^{(1)} is of general type if d = 5 or d ≥ 7.

Proof. The case m = 2 is [GHS2, Theorem 5.1], and the result for m ≥ 5 is immediate from Theorem 1.1. For m = 1, 3 and 4 we can prove more than what follows from Theorem 1.1. The branch divisor of SF_{2d}^{(m)} is defined by the reflections in vectors r ∈ L_{2d}^{(m)}

such that −σ_{r} ∈ S̃O^{+}(L_{2d}^{(m)}), because the rank of L_{2d}^{(m)} is odd. Therefore r^{2} = −2d, by Proposition 2.1. If F ∈ M_{2k+1}(S̃O^{+}(L_{2d}^{(m)})) is a modular form (note that the character det is trivial), d > 1 and z ∈ D_{L_{2d}^{(m)}}^{•} is such that (z, r) = 0, then

F (z) = F (−σ_{r}(z)) = F (−z) = (−1)^{2k+1}F (z).

Therefore any modular form of odd weight for S̃O^{+}(L_{2d}^{(m)}) vanishes on the branch divisor.

To apply the low-weight cusp form trick used in the proof of Theorem 1.1 one needs a cusp form of weight smaller than dim SF_{2d}^{(m)} = 8m + 3. By

Proposition 3.1 there exists a cusp form F_{11+4m} ∈ S_{11+4m}(S̃O^{+}(L_{2d}^{(m)})). For m ≥ 3 we have that 11 + 4m < 8m + 3. Therefore the differential forms F_{11+4m}^{k}F_{(4m−8)k}(dZ)^{k}, for arbitrary F_{(4m−8)k} ∈ M_{(4m−8)k}(S̃O^{+}(L_{2d}^{(m)})), extend to the toroidal compactification of SF_{2d}^{(m)} constructed in Theorem 1.3. This proves the cases m ≥ 3 of the theorem. For the case m = 1 we use a cusp form of weight 9 with respect to S̃O^{+}(L_{2d}^{(1)}) constructed in Proposition 3.1. □

We can obtain some information also for some of the remaining cases.

Proposition 4.3 The spaces SF_{8}^{(1)} and SF_{12}^{(1)} have non-negative Kodaira dimension.

Proof. By Proposition 3.1 there are cusp forms of weight 11 for S̃O^{+}(L_{8}^{(1)}) and S̃O^{+}(L_{12}^{(1)}). The weight of these forms is equal to the dimension. By the well-known observation of Freitag [F, Hilfssatz 2.1, Kap. III] these cusp forms determine canonical differential forms on the 11-dimensional varieties SF_{8}^{(1)} and SF_{12}^{(1)}. □

These varieties may perhaps have intermediate Kodaira dimension as it seems possible that a reflective modular form of canonical weight exists for L_{8}^{(1)} and L_{12}^{(1)}. In [GHS2] we used pull-backs of the Borcherds modular form Φ_{12} on D_{II_{2,26}} to show that many moduli spaces of K3 surfaces are of general type. We can also use Borcherds products to prove results in the opposite direction.

Theorem 4.4 The Kodaira dimension of F_{II}^{(m)} is −∞ for m = 0, 1 and 2.

Proof. For m = 0 we can see immediately that the quotient is rational: a straightforward calculation gives that F_{II}^{(0)} = Γ\ℍ_{1} × ℍ_{1} where ℍ_{1} is the usual upper half plane and Γ is the degree 2 extension of SL(2, ℤ) × SL(2, ℤ) by the involution which interchanges the two factors. Compactifying this, we obtain the projective plane ℙ_{2}. For m = 1, 2 we argue differently. There are modular forms similar to Φ_{12} for the even unimodular lattices II_{2,10} and II_{2,18}. They are Borcherds products Φ_{252} and Φ_{127} of weights 252 and 127 respectively, defined by the automorphic functions

Δ(τ)^{−1}(τ)E_{4}(τ)^{2} = q^{−1} + 504 + q(…)

and Δ(τ)^{−1}(τ)E_{4}(τ) = q^{−1} + 254 + q(…), where q = exp(2πiτ) and Δ(τ) and E_{4}(τ) are the Ramanujan delta function and the Eisenstein series of weight 4 (see [B1]). The divisors of Φ_{252} and Φ_{127} coincide with the branch divisors of F_{II}^{(1)} and F_{II}^{(2)} defined by the (−2)-vectors. Moreover Φ_{252} and Φ_{127} each vanishes with order one along the respective divisor. Therefore if F_{10k}(dZ)^{k} (or F_{18k}(dZ)^{k}) defines a pluricanonical differential form on a smooth model of a toroidal compactification of F_{II}^{(1)} or F_{II}^{(2)}, then F_{10k} (or F_{18k}) is divisible by Φ_{252}^{k} (or Φ_{127}^{k}), since F_{10k} or F_{18k} must vanish to order at least k along the branch divisor. This is not possible, because the quotient would be a holomorphic modular form of negative weight. □

We have already remarked that the space F_{II}^{(2)} is the moduli space of K3 surfaces with an elliptic fibration with a section. Using the Weierstrass equations it is then clear that this moduli space is unirational (such K3 surfaces are parametrised by a linear system in the weighted projective space ℙ(4, 6, 1, 1)). In fact it is even rational: see [Le].

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V. A. Gritsenko Université Lille 1 Laboratoire Paul Painlevé F-59655 Villeneuve d’Ascq, Cedex France valery.gritsenko@math.univ-lille1.fr K. Hulek Institut für Algebraische Geometrie Leibniz Universität Hannover D-30060 Hannover Germany hulek@math.uni-hannover.de

G. K. Sankaran

Department of Mathematical Sciences University of Bath Bath BA2 7AY England gks@maths.bath.ac.uk 20

Documenta Mathematica 13 (2008)