Baily-borel Compactification

Let G = Sp2g be the symplectic group (Chevalley group scheme over Z). Let K be a maximal compact subgroup of G = G(R). The symmetric space G(R)/K is commonly referred to as the Siegel upper half plane, we denote it by hg. It is common to realise hg = {A ∈ Mat g(C) | tA = A,=(A) > 0}. Thus it is a complex analytic space. Let Γ = G(Z) (respectively a congruence subgroup of). Then it is well known (what is the best reference?) that Γ\hg parametrises principally polarised abelian varieties over C (respectively with level structure). If one wants to consider abelian varieties with a non-principal polarisation, then one works with a different Z-form of Sp2g. Our aim is to construct a compactification XB of X = Γ\hg and show that it is a projective complex variety. This is the content of the paper of Baily and Borel. Let us make a few remarks about the generality of this work. Baily and Borel work with an arbitrary arithmetic quotient of a Hermitian symmetric domain. In this document, we will only work with quotients of the Siegel upper half space by a torsion-free Γ (and one can always make Γ torsion-free if one is willing to adopt the cost of passing to a finite index subgroup). This makes the exposition technically simpler, and should not obscure any of the key ideas in the proof. Unfortunately there is still a swathe of notation to wade through.