Slopes of Siegel cusp forms and geometry of compactified Kuga varieties

The universal family X(Γ) over a moduli space A(Γ) = Γ\Hg of abelian varieties associated with a finite index subgroup of Sp(2g,Z) is known as the Kuga variety. Such families were first studied systematically by M. Kuga, whose 1964 Chicago lecture notes on the subject [12] have been recently published. The construction is given in [12] for Γ < Sp(2g,Z) a torsion-free subgroup of finite index, but the restriction to torsion-free can be removed. However, if −1 ∈ Γ then the fibre of the family is no longer the abelian variety but instead the corresponding Kummer variety. One could also allow Γ to be a subgroup of Sp(2g,Q) commensurable with Sp(2g,Z), for example taking A(Γ) to be the moduli space of abelian varieties with some nonprincipal polarisation, but we shall not pursue this here. A natural generalisation is to consider the n-fold Kuga varieties Xn(Γ), whose general fibre is the n-fold direct product of the corresponding abelian variety A or, if −1 ∈ Γ, of the associated Kummer variety A/± 1. Alternatively one may consider the universal family X(Γ) over the stack A(Γ) := [Γ\Hg]. In this case the fibre is the abelian variety in all cases, but if −1 ∈ Γ then the base has non-trivial stabilisers generically. This is the object that is studied in a particular case in [9]. We shall be concerned with compactifications of the n-fold Kuga variety Xn g = X n(Sp(2g,Z)) associated with the coarse moduli space Ag of principally polarised abelian g-folds over C. By convention, X0 g = Ag. The starting point for our work is Ma’s study [13] ofXn g (Γ). We construct a special compactification, which we call a Namikawa compactification, of Xn g and this, together with recent and less recent results about the slope of Ag, allows us to determine the Kodaira dimension κ(X n g ) of X n g whenever g ≥ 2 and n ≥ 1.