The Kodaira dimension of the moduli of K3 surfaces

Abstract

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 ...

Moduli Spaces of Polarized Symplectic O'Grady Varieties and Borcherds Products

“…This is a particular case of Theorem 1.1 in [GHS1]. The dimension of the modular variety is smaller than 26.…”

“…The present case is of dimension 21. The non-split case is similar to the cases considered in [GHS1]- [GHS2] (see also the example at the end of this section) but the split case is different from the previous ones because we need a cusp form with respect to the modular group O G (L A,2d ), which is strictly larger than the stable orthogonal group O + (L A,2d ). For this reason we will concentrate in this paper on the split case.…”

“…[n] -type in [GHS2], where we restricted ourselves to the case of split polarisations (see [GHS1,Example 3.8] for a definition and details).…”

“…The dimension of the modular variety M A,2d is 21, which is larger than 8. Therefore we can use the low weight cusp form trick from [GHS1]. Let L be an even integral lattice of signature (2, n) with n ≥ 3.…”

“…We conjecture that for the three lowest non-split polarisations, of Beauville degrees 2d = 12, 30 and 48, the corresponding moduli spaces are unirational. Using the arithmetic and analytic methods developed in [GHS1]- [GHS2] we hope to prove that for other non-split polarisations the moduli spaces are of general type. In this paper we study the split polarisation because this case is very different and has new phenomena appearing.…”