On the Kodaira dimension of the moduli space of K3 surfaces II

SHIGEYUKI KONDŌ^{1}

Graduate school of Polymathematics, Nagoya University, Nagoya, 464-01 Japan

e-mail: kondo@math.nagoya-u.ac.jp

Introduction

Let K_{2n} be the coarse moduli space of K3 surfaces with a primitive polarization of degree 2n. It follows from Torelli theorem for K3 surfaces that K_{2n} is described as an arithmetic quotient D_{2n}/Γ_{2n} where D_{2n} is a 19-dimensional bounded symmetric domain of type IV and Γ_{2n} is an arithmetic subgroup of O(2, 19)_{Q}. Recently Borcherds [2] constructed a remarkable automorphic form Φ of weight 12 on a 26-dimensional bounded symmetric domain of type IV. By dividing Φ by a product of linear functions vanishing on D_{2n} and by restricting it to D_{2n}, we have an automorphic form Φ_{2n} on D_{2n} with respect to Γ_{2n} (see Borcherds et al. [3]). The purpose of this note is to show:

THEOREM. Assume that n = 42, 43, 51, 53, 55, 57, 59, 61, 66, 67, 69, 74, 83, 85, 105, 119 or 133. Then Φ_{2n} is a cusp form of weight 19 on D_{2n} with respect to Γ_{2n}.

An automorphic form of weight 19 on D_{2n} gives a section of the canonical line bundle of the smooth locus of D_{2n}/Γ_{2n}. Moreover it is known that a cusp form of weight 19 gives a global section of the canonical line bundle of a smooth model of a compactification of D_{2n}/Γ_{2n} (Bauermann [1]). Thus we have:

COROLLARY 1. Assume that n is the same as above. Then the Kodaira dimension of K_{2n} = D_{2n}/Γ_{2n} is non negative.

If m = n·l^{2} for some natural number l, then Γ_{2n} ⊃ Γ_{2m} and we have a dominant map from D_{2m}/Γ_{2m} to D_{2n}/Γ_{2n} (O’Grady [12], Kondō [6], Lemma 3.2). Hence we have:

COROLLARY 2. Assume that n is the same as above and m = n·l^{2} for some l. Then the Kodaira dimension of K_{2m} is non negative.

We remark that if 1≤n≤11, 17 or 19, then K_{2n} is unirational and in particular its Kodaira dimension is −∞. For n ≤ 4, this is classical and for other n this was shown by Mukai [7], [8], [9]. On the other hand, in the paper [6], the author proved that if n = p^{2} and p is a sufficiently large prime number, then K_{2n} is of general type, i.e. the Kodaira dimension

1 Supported by Max-Planck-Institut in Bonn of K_{2n} is equal to dim K_{2n} (= 19). However the author did not have an effective estimate of p. Also Gritsenko [5] proved that the Kodaira dimension of the double cover D_{2n}/Γ̄_{2n} of D_{2n}/Γ_{2n} is non-negative if n is not perfect square and positive if n is square free and n > 3 by using the lifting of Jacobi forms, where Γ̄_{2n} = Γ_{2n} ∩ SO(2, 19)_{Q}.

§1. Preliminaries

A lattice L is a free Z-module of finite rank endowed with an integral symmetric bilinear form ⟨ , ⟩. If L_{1} and L_{2} are lattices, then L_{1}⊕L_{2} denotes the orthogonal direct sum of L_{1} and L_{2}. An isomorphism of lattices preserving the bilinear forms is called an isometry. A sublattice S of L is called primitive if L/S is torsion free.

A lattice L is even if ⟨x, x⟩ is even for each x ∈ L. A lattice L is non-degenerate if the discriminant d(L) of its bilinear form is non zero, and unimodular if d(L) = ±1. If L is a non-degenerate lattice, the signature of L is a pair (t_{+}, t_{−}) where t_{±} denotes the multiplicity of the eigenvalues ±1 for the quadratic form on L⊗R.

Let L be a non-degenerate even lattice. The bilinear form of L determines a canonical embedding L ⊂ L^{∗} = Hom(L, Z). The factor group L^{∗}/L, which is denoted by A_{L}, is an abelian group of order |d(L)|. We extend the bilinear form on L to the one on L^{∗}, taking value in Q, and define

q_{L} : A_{L} → Q/2Z, q_{L}(x + L) = ⟨x, x⟩ + 2Z (x ∈ L^{∗}).

We call q_{L} the discriminant quadratic form of L.

We denote by H the hyperbolic lattice defined by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) which is an even uni-

modular lattice of signature (1,1), and by A_{m}, D_{n} or E_{l} an even negative definite lattice associated with the Dynkin diagram of type A_{m}, D_{n} or E_{l} (m ≥ 1, n ≥ 4, l = 6,7,8). A root of a lattice L is a vector r in L with ⟨r, r⟩ = −2. For an even negative definite lattice L, we denote by R(L) the sublattice of L generated by all roots in L which is called the root sublattice of L and isometric to a direct sum of some A_{m}, D_{n}, E_{l}.

Let X be a K3 surface. Put L = H^{2}(X, Z). Then L admits a canonical structure of a lattice induced from the cup product. It is an even unimodular lattice with signature (3,19) and hence isometric to H ⊕ H ⊕ H ⊕ E_{8} ⊕ E_{8} (e.g. Nikulin [11], Theorem 1.1.1). Let h be a primitive vector of L with ⟨h, h⟩ = 2n. Then the orthogonal complement of h in L is isometric to L_{2n} = H ⊕ H ⊕ E_{8} ⊕ E_{8}⊕ < −2n >. Put Ω_{2n} = {[ω] ∈ P(L_{2n} ⊗ C) : ⟨ω, ω⟩ = 0, ⟨ω, ω̄⟩ > 0}. Then Ω_{2n} consists of two connected components. We denote by D_{2n} either one connected component, which is a bounded symmetric domian of type IV and of dimension 19. Let Γ_{2n} be the group of isometries of L which fix h and preserve the component D_{2n}. Then Γ_{2n} acts on D_{2n} properly discontinuously, and hence by Cartan’s theorem D_{2n}/Γ_{2n} has a canonical structure of normal analytic space. We call D_{2n} the period space of K3 surfaces with a primitive polarization of degree 2n. It follows from Torelli theorem for K3 surfaces that D_{2n}/Γ_{2n} is a coarse moduli space of K3 surface with a primitive polarization of degree 2n.

A boundary component of D_{2n} is a maximal connected complex analytic subset in D̄_{2n} \ D_{2n} where D̄_{2n} is the topological closure of D_{2n} in {[ω] ∈ P(L_{2n} ⊗ C) : ⟨ω, ω⟩ = 0}. A boundary component is called rational if its stabilizer subgroup of O(L_{2n} ⊗R) is defined over Q. It is known that the set of rational boundary components of D_{2n} bijectively corresponds to the set of all primitive totally isotropic sublattices of L_{2n} (Scattone [13], (2.1.7)). If E is a totally isotropic sublattice, then the corresponding rational boundary component is defined by P(E⊗C)∩D̄_{2n}. Since the signature of L_{2n} is (2,19), the dimension of a rational boundary component is either 0 or 1.

For simplicity we now assume that n is square-free. Let F be a rational boundary component and E the corresponding primitive totally isotropic sublattice. If dim F = 0, then F is unique modulo Γ_{2n} and if dim F = 1, then we have an orthogonal decomposition

L_{2n} = H ⊕ H ⊕ K

where H ⊕ H contains E and K ≃ E^{⊥}/E (Scattone [13], Theorem 4.0.1, Lemma 5.2.1). Let {e_{1}, ..., e_{21}} be a base of L_{2n} such that

(⟨e_{i}, e_{j}⟩)_{1≤i,j≤21} = [ \begin{matrix}0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & Q & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix} ]

where {e_{1}, e_{2}} is a base of E and Q is the intersection matrix of K. In case dimF = 0, F is unique modulo Γ_{2n} and hence we may assume that E is generated by e_{1}. Denote by q_{0} (resp. q_{1}) the symmetric bilinear form associated with the matrix (⟨e_{i}, e_{j}⟩)_{2≤i,j≤20} (resp. Q). Also denote by L_{0} the sublattice e_{1}^{⊥}/e_{1} of L_{2n}. Let z = Σz_{i}e_{i} be a homogeneous coordinate of P(L_{2n} ⊗ C). Then we have the following unbounded realization of D_{2n}.

(1.1) In case dimF = 0:

D_{2n} ≃ {z = (z_{2}, ..., z_{20}) ∈ C^{19} : (Im(z_{i})) ∈ C(F)}

where C(F) = {y = (y_{2}, ..., y_{20}) ∈ R^{19} : q_{0}(y, y) > 0, y_{2} > 0}.

(1.2) In case dim F = 1:

D_{2n} ≃ {(z, w, τ) = (z_{2}, z_{3}, ..., z_{19}, z_{20}) ∈ H^{+} × C^{17} × H^{+} : Im(z) − Re(h_{τ} (w, w)) > 0}

where h_{τ} (w, w′) = {−q_{1}(w, w̄′) + q_{1}(w, w′)}/4Im(τ) (see Kondo [6], §2).

Let f be an automorphic form on D_{2n} with respect to Γ_{2n}. Then f has an expansion with respect to (1.1) which is called the Fourier expansion of f at F:

f = Σ_{ρ∈C̄(F)∩L_{0}^{∗}} c_{ρ} · exp(2π\sqrt{−1} · q_{0}(ρ, z))

where C̄(F) is the closure of C(F). Also with respect to (1.2), f has an expansion which is called the Fourier-Jacobi expansion of f at F:

f = Σ_{m≥0} θ_{m}(τ, w) · exp(2π\sqrt{−1}mz).

We call f a cusp form if for any rational boundary components the initial term of the above expansion vanishes, i.e. c_{0} ≡ 0 if dim F = 0 and θ_{0}(τ, w) ≡ 0 if dim F = 0. Note that θ_{0}(τ, w) does not depend on w. For more details we refer the reader to Borcherds [2], Gritsenko [5], Kondo [6].

§2. Automorphic forms on the period domain of K3 surfaces

In the following, as in Borcherds et al. [3], we identify L_{2n} = H ⊕ H ⊕ E_{8} ⊕ E_{8}⊕ < −2n > with the sublattice H ⊕ H ⊕ E_{8} ⊕ E_{8} ⊕ Zv of II_{2,26} = H ⊕ H ⊕ E_{8} ⊕ E_{8} ⊕ E_{8} where v is a primitive vector in E_{8} with ⟨v, v⟩ = −2n. Let U be the orthogonal complement of v in E_{8}. Let D be the 26-dimensional bounded symmetric domain of type IV associated with II_{2,26}. Under the above identification, D_{2n} ⊂ D. In the paper [2], Example 2 of section 10, Borcherds constructed an automorphic form Φ of weight 12 on D with respect to the group of isometries of II_{2,26} which preserve D. The restriction of Φ on D_{2n} is identically 0 whenever U contains a root. So first divide Φ by a product of linear functions vanishing on the divisors of each of these roots in U, and then restrict it on D_{2n}. Then we have an automorphic form Φ_{2n} on D_{2n} with respect to Γ_{2n} which has the following properties (Borcherds et al. [3], also see Borcherds [2], Theorem 13.1 and its proof):

(2.1) Φ_{2n} vanishes on the hyperplanes of D_{2n} each of which is orthogonal to a vector r′ ∈ L_{2n}^{∗} with −2 ≤ ⟨r′, r′⟩ < 0. Here r′ is the projection of a root r of II_{2,26}.

(2.2) The weight of Φ_{2n} is equal to the weight (=12) of Φ plus half the number of roots of U.

Let {e_{i}}_{1≤i≤8} be a set of simple roots of E_{8} which satisfies the following:

⟨e_{i}, e_{i}⟩ = −2, ⟨e_{1}, e_{2}⟩ = ⟨e_{2}, e_{3}⟩ = ⟨e_{3}, e_{4}⟩ = ⟨e_{3}, e_{5}⟩ = ⟨e_{5}, e_{6}⟩ = ⟨e_{6}, e_{7}⟩ = ⟨e_{7}, e_{8}⟩ = 1 and ⟨e_{i}, e_{j}⟩ = 0 for other i, j.

Let R be a root lattice isometric to A_{2} ⊕A_{2} ⊕A_{1} or A_{3} ⊕A_{1}. Note that the number of roots in R is 14. In case R = A_{2} ⊕ A_{2} ⊕ A_{1}, we consider R as one of the following sublattices of E_{8}:

(a) < e_{1}, e_{2}, e_{4}, e_{6}, e_{7} >, (b) < e_{1}, e_{2}, e_{5}, e_{6}, e_{8} >, (c) < e_{1}, e_{2}, e_{5}, e_{7}, e_{8} >,
(d) < e_{1}, e_{3}, e_{4}, e_{6}, e_{7} >, (e) < e_{1}, e_{3}, e_{4}, e_{7}, e_{8} >.

In case R = A_{3} ⊕ A_{1}, we consider R as one of the following sublattices of E_{8}:

(f) < e_{1}, e_{2}, e_{3}, e_{6} >, (g) < e_{1}, e_{2}, e_{3}, e_{7} >, (h) < e_{1}, e_{2}, e_{3}, e_{8} >,
(i) < e_{2}, e_{3}, e_{4}, e_{7} >, (j) < e_{2}, e_{3}, e_{5}, e_{7} >, (k) < e_{2}, e_{3}, e_{5}, e_{8} >,
(l) < e_{3}, e_{4}, e_{5}, e_{7} >, (m) < e_{1}, e_{5}, e_{6}, e_{7} >, (n) < e_{4}, e_{5}, e_{6}, e_{7} >,
(o) < e_{1}, e_{6}, e_{7}, e_{8} >, (p) < e_{2}, e_{6}, e_{7}, e_{8} >, (q) < e_{4}, e_{6}, e_{7}, e_{8} >.

If R =< e_{α}, e_{β}, e_{γ}, e_{δ}, e_{κ} > (resp. R =< e_{α}, e_{β}, e_{γ}, e_{δ} >), then we take a vector v as a sum

v = e_{λ}^{∗} + e_{μ}^{∗} + e_{ν}^{∗} (resp. v = e_{κ}^{∗} + e_{λ}^{∗} + e_{μ}^{∗} + e_{ν}^{∗})

where {α, β, ..., ν} = {1, 2, ..., 8} and e_{α}^{∗} is the dual of e_{α} (see Bourbaki [4], Planche VII). Since E_{8} is unimodular, v ∈ E_{8}. A direct calculation shows that the number −⟨v, v⟩/2 is equal to:

(a) 61, (b) 55, (c) 67, (d) 43, (e) 66, (f) 57, (g) 69, (h) 83, (i) 59, (j) 42, (k) 53, (l) 51,

(m) 85, (n) 74, (o) 133, (p) 105, (q) 119.

Since ⟨v, v⟩/2 is square free, v is primitive in E_{8}. Obviously any root in U (= v^{⊥} in E_{8}) is contained in R. Hence the number of roots of U is 14. Thus we have an automorphic form Φ_{2n} of weight 19 (= 12 + 14/2) by (2.2).

THEOREM. Φ_{2n} is a cusp form of weight 19.

Proof. It suffices to see that Φ_{2n} vanishes on the top dimensional (= 1-dim.) rational boundary components of D_{2n}. Since n is square free, the set of primitive totally isotropic sublattices of rank two corresponds to the set of isomorphy classes of even negative definite lattices K such that L_{2n} ≃ H⊕H⊕K (Scattone [13], Theorem 5.0.2). Here totally isotropic sublattice is contained in H ⊕ H. Since q_{K} ≃ q_{L_{2n}} ≃ −q_{U}, we have an even negative definite unimodular lattice N of rank 24 such that K and U are primitive sublattices of N, K^{⊥} ≃ U and U^{⊥} ≃ K (Nikulin [11], Corollary 1.6.2). Thus each K is obtained as the orthogonal complement of a primitive sublattice U of some N. Let U ⊂ N be a primitive embedding. Since U contains R, N also contains a root. Such lattices N are classified into 23 isomorphism classes which are characterized by its root sublattice R(N) (Niemeier [10]). Since rank R(N) = 24, there exists a root r of N whose projection r′ into K^{∗} has a negative norm : −2 ≤ ⟨r′, r′⟩ < 0. Note that the totally isotropic primitive sublattice of rank 2 corresponding to K is contained in the orthogonal complement of r′. It follows from (2.1) that Φ_{2n} vanishes the hyperplane orthogonal to r′. Let

Φ_{2n} = Σθ_{m}(τ, w) · exp(2π\sqrt{−1}mz)

be the Fourier-Jacobi expansion of Φ_{2n} at this boundary component. Since

θ_{0}(τ, w) = lim_{Im(z)→∞} Φ_{2n}

and θ_{0}(τ, w) does not depend on w, the above implies that θ_{0}(τ, w) ≡ 0. q.e.d.

Remark. There are another embeddings of R into E_{8}. In the above we take embeddings such that the number ⟨v, v⟩/2 is square free. Also we can take v as a linear combination of e_{λ}^{∗}, e_{μ}^{∗}, e_{ν}^{∗} (resp. e_{κ}^{∗}, e_{λ}^{∗}, e_{μ}^{∗}, e_{ν}^{∗}) with positive coefficients. Then Φ_{2n} is a cusp form of weight 19 whenever ⟨v, v⟩/2 is square free and Corollaries 1, 2 hold for these n.

Acknowledgements. This paper was written during the author’s stay at the Max-Planck-Institut für Mathematik in Bonn 1996-1997. He express his thanks to the Institut for giving him the opportunity of this work in the stimulating working atmosphere.

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