Irreducible holomorphic symplectic manifolds are simply connected compact complex manifolds of Kähler type whose space of global holomorphic \(2\)-forms is generated by a closed non-degenerate \(2\)-form. In complex dimension two, the irreducible holomorphic symplectic manifolds are precisely the \(K3\)-surfaces. Higher dimensional examples are rare. To date, up to deformation equivalence, the only known examples are Hilbert schemes of zero-dimensional subschemes of length \(n\) on a \(K3\)-surface, generalised Kummer varieties and two examples of dimension \(6\) and \(10\) constructed by O’Grady.
In the paper under review the authors study moduli spaces of polarised irreducible holomorphic symplectic manifolds. For their main result they consider irreducible holomorphic symplectic manifolds which are deformation equivalent to Hilb\(^2(S)\) for a \(K3\) surface \(S\), equipped with a primitive polarisation of split type and of Beauville-degree \(2d\geq 24\). They prove that each irreducible component of the 20-dimensional moduli space of such polarised manifolds is of general type.
To prove this, they first show (for general polarised irreducible holomorphic symplectic manifolds) that each component of the moduli space is related by a finite dominant period map to a quotient of a homogeneous domain by an arithmetic group. The proof of the main result is similar to the proof the authors have given for the moduli space of polarised \(K3\) surfaces [Invent.Math.169, No.3, 519–567 (2007; Zbl 1128.14027)].
They use that the existence of a low-weight cusp form with suitable vanishing locus implies that the components of the arithmetic quotient are of general type. Such a cusp form is then constructed as a quasi-pull back of the Borcherds form (a modular form of weight 12 for \(O^+(2U\oplus3E_8(-1))\). This construction requires the existence of a vector of length \(2d\) in the lattice \(E_7\) which is orthogonal to at most 14 roots. The proof of the existence of such a vector is the technical heart of this article. In involves estimating the number of ways certain integers can be represented by various quadratic forms of odd rank.