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 ...

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.

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces.Many moduli spaces in algebraic geometry can be described via period domains as quotients of a symmetric space by a discrete group, or modular group. We shall be concerned with the case of the symmetric space D L associated with a lattice L of signature (2, n), and discrete subgroups of the orthogonal group O(L) that act on D L . Such groups arise in the study of the moduli of K3 surfaces and of other irreducible symplectic manifolds (see [GHS1,GHS3] and the references there), and of polarised abelian surfaces. Orthogonal groups of indefinite forms also appear elsewhere in geometry, for instance in the theory of singularities (see [Br,Eb]). In this paper we study the commutator subgroups and abelianisations of orthogonal modular groups of this kind, especially for lattices of signature (2, n).Notation. For definitions and notation concerning locally symmetric varieties and toroidal compactification we refer to [GHS2].We write X for the group generated by a subset X of some group. If n is an integer n means the rank-1 lattice generated by an element of square n.

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The moduli space A g of principally polarised abelian g-folds is a quasiprojective variety. It has a natural projective compactification, the Satake compactification, which has bad singularities at infinity. By the method of toroidal compactification we can construct other compactifications with milder singularities, at the cost of some loss of uniqueness. Two popular choices of toroidal compactification are the Igusa and the Voronoi compactifications: these agree for g ≤ 3 but for g = 4 they are different. In this paper, we shall be mainly interested in the Voronoi compactification and A Vor 4 . The proofs are inductive in the sense that they involve a reduction to the cases g = 3 and g = 2, where comparable results already exist; but some new techniques are also necessary for the proof. However, the Voronoi compactification for g > 4 is rather complicated and for this reason we are not at present able to extend our results even to g = 5. We also show (Theorem I.15) that the canonical bundle on A Igu 4 (n) is ample for n ≥ 3. The paper is structured as follows. Section I covers the facts we need about the different toroidal compactifications that are available. We describe the Voronoi compactification, in particular, in some detail, and state the main results. In Section II we explain what is known about the partial compactification of Mumford, which we shall need later. In Section III we describe the fine structure of the Voronoi boundary in the case g = 4, which is largely a matter of understanding the behaviour over the lowest stratum of the Satake compactification A Sat 4 . The methods here are toric and much is deduced from the combinatorics of a single cone in a certain 10-dimensional real vector space. The main technical result is that each non-exceptional 1