Baily–Borel compactification

Remarks

The theorem was proved by Walter Baily and Armand Borel in 1966. As a topological space the compactification is Satake's: one adds the rational boundary components of $D$ and divides by $\Gamma$. The projective embedding is given by automorphic forms on $D$ of sufficiently high weight, through Poincaré–Eisenstein series; $(\Gamma\backslash D)^*$ is the Proj of the graded ring of automorphic forms for $\Gamma$.

Example: for $D$ the upper half-plane and $\Gamma = SL_2(\Z)$, the boundary is one cusp, and $(\Gamma \backslash D)^* = \Gamma \backslash D \cup \set{\infty}$ is the projective line, with coordinate the $j$-invariant. For $D$ the Siegel upper half-space $\mathbb{H}_g$ and $\Gamma = Sp(2g, \Z)$, the boundary strata are the quotients $Sp(2h, \Z) \backslash \mathbb{H}_h$ with $0 \le h < g$.

Going deeper

For a student

The algebraicity of $\Gamma \backslash D$ is what makes it possible to study these spaces, such as Shimura varieties, with algebraic geometry and to define them over number fields. The Baily–Borel compactification is minimal and usually very singular along the boundary. The toroidal compactifications of Ash, Mumford, Rapoport and Tai can be smooth, and each of them maps onto the Baily–Borel compactification. Borel's extension theorem says that holomorphic maps from a punctured polydisc into $\Gamma\backslash D$, with $\Gamma$ torsion-free, extend across the punctures to $(\Gamma\backslash D)^*$.