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

Abstract

We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3 [n] case, which are of dimension 19 and 20 respectively. IntroductionIrreducible symplectic manifolds are simply connected compact Kähler manifolds which have a (up to scalar) unique 2-form, which is non-degenerate. In dimension two these are the K3 surfaces. In higher dimension there are, so far, four known classes of examples. These are deformations of degree n Hilbert schemes of K3 surfaces (the K3[n] case), deformations of generalised Kummer varieties, and two examples of dimensions 6 and 10 due to O'GradyFrom the point of view of the Beauville lattice these examples fall into two series. The first consists of K3 surfaces, the K3[n] case and O'Grady's example of dimension 10. The Beauville lattices are the unimodular K3-lattice L K3 = 3U ⊕ 2E 8 (−1), the lattice L K3 ⊕ −2(n − 1) and L K3 ⊕ A 2 (−1). The moduli spaces of polarised irreducible symplectic manifolds of these classes are of dimensions 19, 20 and 21. The second series consists of generalised Kummer varieties and O'Grady's 6-dimensional variety with Beauville lattices 3U ⊕ −2 and 3U ⊕ −2 ⊕ −2 respectively. Here the dimensions of the moduli spaces of polarised varieties are 4 and 5.In order to describe moduli spaces of irreducible symplectic manifolds one must first classify the possible types of the polarisation. We do this in Section 3 for O'Grady's 10-dimensional example. As in the K3[n] case we find that we have a split and a non-split type. In this paper we shall mostly concentrate on the split case, when the modular group is maximal possible, but we shall also comment on the low degree non-split cases. 1In the non-split case we expect Kodaira dimension −∞ for the three cases of lowest Beauville degree, namely 2d = 12, 30, 48. For the next case of Beauville degree 2d = 66 we prove general type: see Corollary 4.3. The arguments used also suggest that 2d = 12, 30, 48 might be the only degrees of non-split polarisations giving unirational moduli spaces.We should like to comment that there is a natural series consisting of moduli of K3 surfaces of degree 2 (double planes branched along a sextic curve), the non-split K3[2] case of Beauville degree 2d = 6 (corresponding to cubic fourfolds and treated by Voisin in [Vo]) and O'Grady's example of dimension 10 with a non-split polarisation of degree 12. The lattices which are orthogonal to the polarisation vector in this series are 2U ⊕2E 8 (−1)⊕A n (−1) for n = 1, 2, 3. It would be very interesting to find a projective geometric realisation of O'Grady's 10-dimensional irreducible symplectic manifolds with non-split Beauville degree 12.In the split case we prove that the modular variety is...

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