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, from the point of view of abelian and Kummer surfaces and their moduli, the special quintic threefold known as Nieto's quintic. It is, in some ways, analogous to the Segre cubic and the Burkhardt quartic and can be interpreted as a moduli space of certain Kummer surfaces. It contains 30 planes and has 10 singular points: we describe how some of these arise from bielliptic and product abelian surfaces and their Kummer surfaces.

The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of ${\mathcal A}_3$. The Igusa compactification has a non-${\mathbb Q}$-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor $E$ is a toric Fano variety (of dimension 9): the other boundary divisor, $D$, corresponds to degenerations with corank~1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of $D$ in the Voronoi compactification maps to the Voronoi compactification of ${\mathcal A}_3$. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification.

We show that the moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension and indeed pg ≥ 3. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.

We shall prove below part of a conjecture made by Shigefumi Mori, David Morrison and Ian Morrison in the course of their investigations into the properties of isolated terminal cyclic quotient singularities of prime Gorenstein index in dimension four [1]. The reader of the present paper need have no knowledge of algebraic geometry, because we quickly reduce the problem to one about the geometry of numbers that can be solved by elementary calculations. The calculations are very lengthy and not quite routine, so what the reader does need is either patience, if he intends to check them, or faith, if he does not. We give only part of the calculations below. Full details may be obtained from the author.*

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.