The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treat moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d > 61 and for d = 46, 50, 54, 58, 60. IntroductionModuli spaces of polarised K3 surfaces can be identified with the quotient of a classical hermitian domain of type IV and dimension 19 by an arithmetic group. The general set-up for the problem is the following. Let L be an integral lattice with a quadratic form of signature (2, n) and letbe the associated n-dimensional Hermitian domain (here + denotes one of its two connected components). We denote by O(L) + the index 2 subgroup of the integral orthogonal group O(L) preserving D L . We are, in general, interested in the birational type of the n-dimensional varietywhere Γ is a subgroup of O + (L) of finite index. Clearly, the answer will depend strongly on the lattice L and the chosen subgroup Γ.A compact complex surface S is a K3 surface if S is simply connected and there exists a holomorphic 2-form ω S ∈ H(S, Ω 2 ) without zeros. For example, a smooth quartic in P 3 (C) is a K3 surface and all quartics (modulo projective equivalence) form a (unirational) space of dimension 19.The second cohomology group H 2 (S, Z) with the intersection pairing is an even unimodular lattice of signature (3, 19), more precisely,1 where U is the hyperbolic plane and E 8 (−1) is the negative definite even lattice associated to the root system E 8 . The 2-form ω S , considered as a point of P(L K3 ⊗ C), is the period of S. By the Torelli theorem the period of a K3 surface determines its isomorphism class. The moduli space of all K3 surfaces is not Hausdorff. Therefore it is better to restrict to moduli spaces of polarised K3 surfaces. The moduli of all algebraic K3 surfaces are parametrised by a countable union of 19-dimensional irreducible algebraic varieties. To choose a component we have to fix a polarisation. A polarised K3 surface of degree 2d is a pair (S, H) consisting of a K3 surface S and a primitive pseudo-ample divisor H on S of degree H 2 = 2d > 0. If h is the corresponding vector in the lattice L K3 then its orthogonal complementis a lattice of signature (2, 19).By the global Torelli theorem ([P-SS]) and the surjectivity of the period mapis the coarse moduli space of polarised K3 surfaces of degree 2d. By a result of Baily and Borel [BB], F 2d is a quasi-projective variety. One of the fundamental problems is to determine its birational type. In the other direction there are two results of Kondo and of Gritsenko. Kondo [Ko1] considered the moduli spaces F 2p 2 where p is a prime number. (The reason for this choice is that all these spaces are covers of F 2 .) He ...

Abstract. We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac-Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice. These results are important for Mirror Symmetry and theory of Lorentzian Kac-Moody algebras. §0. IntroductionIn Part I we developed the general theory of reflective automorphic forms and their particular case of Lie reflective automorphic forms on Hermitian symmetric domains of type IV. These automorphic forms are very important in Mirror Symmetry (for K3's and Calabi-Yau's) and for Lorentzian Kac-Moody algebras. In Part I, in particular, we showed that this theory is similar to the theory of hyperbolic root systems (it is its mirror symmetric variant). We explained that reflective automorphic forms are very exceptional. Conjecturally their number is finite similarly to finiteness results for corresponding hyperbolic root systems with some condition of finiteness of volume for fundamental polyhedron (i.e., of elliptic or parabolic type). We believe and hope to show in further publications that classification of reflective automorphic forms and corresponding hyperbolic root systems is the key step in classification of some important class of Calabi-Yau's (see [GN6]). For example, finiteness results for hyperbolic root systems of elliptic and parabolic type and for reflective automorphic forms are related with finiteness of families of these Calabi-Yau's.We demonstrated in Part I a general method of classification of the hyperbolic root systems on the example of symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type of rank 3 and with a lattice Weyl vector. Let us denote the set of these matrices by A. It contains 60 matrices. In this Part II we consider methods of construction of reflective automorphic forms. In particular, for many generalized Cartan matrices A ∈ A we find their mirror symmetric objects -automorphic forms F defining automorphic corrections of Kac-Moody algebras g(A) corresponding to A. These forms F define so-called automorphic Lorentzian Kac-Moody algebras g F containing g(A) and having good automorphic properties. In this paper we consider mainly 3-dimensional automorphic forms F with respect to the full paramodular groups, i.e. automorphic forms on the Siegel upper half-space H 2 of complex dimension 3 with respect to the paramodular groups Γ t ⊂ Sp 4 (Q) (the threefold A t = Γ t \ H 2 is the moduli space of Abelian surfaces with ...

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector.We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T . We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac-Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac-Moody algebras and formulate finiteness Conjecture for these forms.Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2] manifolds with polarisation of degree 2d and split type is of general type if d 12.

The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*t\ℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γt\ℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]t which lies over [Ascr ]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [Ascr ]t and [Ascr ]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [Ascr ]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]t→[Ascr ]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.