The nef cone of toroidal compactifications of ${\cal A}_4$

Abstract

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

A finiteness theorem for Lagrangian fibrations

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Some intersection numbers of divisors on toroidal compactifications of 𝒜_{ℊ}

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“…In doing so, we shall distinguish between the cases when ξ is one of the factors or is not. Note that by (13) we can always assume the power of ξ to be at most one. We first make the following observation.…”

The intersection cohomology of the Satake compactification of $${\mathcal {A}}_g$$ for $$g \le 4$$

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