# Baily-Borel compactification

*Satake–Baily–Borel compactification*

Let $G$ be a semi-simple linear algebraic group (cf. also Semi-simple algebraic group) defined over $\mathbf{Q}$, meaning that $G$ can be embedded as a subgroup of $\operatorname{GL} ( m , \mathbf{C} )$ such that each element is diagonalizable (cf. also Diagonalizable algebraic group), and that the equations defining $G$ as an algebraic variety have coefficients in $\mathbf{Q}$ (and that the group operation is an algebraic morphism). Further, suppose $G$ contains a torus (cf. Algebraic torus) that splits over $\mathbf{Q}$ (i.e., $G$ has $\mathbf{Q}$-rank at least one), and $G$ is of Hermitian type, so that $X : = K \backslash G ( \mathbf{R} )$ can be given a complex structure with which it becomes a symmetric domain, where $G ( \mathbf{R} )$ denotes the real points of $G$ and $K$ is a maximal compact subgroup. Finally, let $\Gamma$ be an arithmetic subgroup (cf. Arithmetic group) of $G ( \mathbf{Q} )$, commensurable with the integer points of $G$. Then the arithmetic quotient $V : = X / \Gamma$ is a normal analytic space whose Baily–Borel compactification, also sometimes called the Satake–Baily–Borel compactification, is a canonically determined projective normal algebraic variety $V ^ { * }$, defined over $\mathbf{C}$, in which $V$ is Zariski-open (cf. also Zariski topology) [a1] [a2] [a15] [a16].

To describe $V ^ { * }$ in the complex topology, first note that the Harish–Chandra realization [a6] of $X$ as a bounded symmetric domain may be compactified by taking its topological closure. Then a rational boundary component of $X$ is a boundary component whose stabilizer in $G ( \mathbf{R} )$ is defined over $\mathbf{Q}$; based on a detailed analysis of the $\mathbf{R}$-roots and $\mathbf{Q}$-roots of $G$, there is a natural bijection between the rational boundary components of $X$ and the proper maximal parabolic subgroups of $G$ defined over $\mathbf{Q}$. Let $X ^ { * }$ denote the union of $X$ with all its rational boundary components. Then (cf. [a18]) there is a unique topology, the Satake topology, on $X ^ { * }$ such that the action of $G ( \mathbf{Q} )$ extends continuously and $V ^ { * } = X ^ { * } / \Gamma$ with its quotient topology compact and Hausdorff. It also follows from the construction that $V ^ { * }$ is a finite disjoint union of the form

\begin{equation*} V ^ { * } = V \bigcup V _ { 1 } \bigcup \ldots \bigcup V _ { t }, \end{equation*}

where $V _ { i } = F _ { i } / \Gamma _ { i }$ for some rational boundary component $F_{i}$ of $X ^ { * }$, and $\Gamma_{i}$ is the intersection of $\Gamma$ with the stabilizer of $F_{i}$. In addition, $V$ and each $V _ { i }$ has a natural structure as a normal analytic space; the closure of any $V _ { i }$ is the union of $V _ { i }$ with some $V _ { j }$s of strictly smaller dimension; and it can be proved that every point $v \in V ^ { * }$ has a fundamental system of neighbourhoods $\{ U _ { s } \}$ such that $U _ { s } \cap V$ is connected for every $s$.

In order to describe the structure sheaf of $V ^ { * }$ (cf. also Scheme) with which it becomes a normal analytic space and a projective variety, define an $\mathcal{A}$-function on an open subset $U \subset V ^ { * }$ to be a continuous complex-valued function on $U$ whose restriction to $U \cap V _ { i }$ is analytic, $0 \leq i \leq t$, where $V _ { 0 } = V$. Then, associating to each open $U$ the $\mathbf{C}$-module of $\mathcal{A}$-functions on $U$ determines the sheaf $\mathcal{A}$ of germs of $\mathcal{A}$-functions. Further, for each $i$ the sheaf of germs of restrictions of $\mathcal{A}$-functions to $V _ { i }$ is the structure sheaf of $V _ { i }$. Ultimately it is proved [a2] that $( V ^ { * } , \mathcal{A} )$ is a normal analytic space which can be embedded in some complex projective space as a projective, normal algebraic variety. The proof of this last statement depends on exhibiting that in the collection of $\mathcal{A}$-functions there are enough automorphic forms for $\Gamma$, more specifically, Poincaré-Eisenstein series, which generalize both Poincaré series and Eisenstein series (cf. also Theta-series), to separate points on $V ^ { * }$ as well as to provide a projective embedding.

### History and examples.

The simplest example of a Baily–Borel compactification is when $G = \operatorname{SL} ( 2 , \mathbf{Q} )$, and $\Gamma = \operatorname{SL} ( 2 , \mathbf{Z} )$, and $X$ is the complex upper half-plane, on which $\left( \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right)$ in $\operatorname{SL} ( 2 , \mathbf R )$ acts by $z \mapsto ( a z + d ) ( c z + d ) ^ { - 1 }$. (The bounded realization of $X$ is a unit disc, to which the upper half-plane maps by $z \mapsto ( z - \sqrt { - 1 } ) / ( z + \sqrt { - 1 } )$.) The properly discontinuous action of $\Gamma$ on $X$ extends to $X ^ { * } = X \cup \mathbf{Q} \cup \{ \infty \}$, and $V ^ { * } = X ^ { * } / \Gamma$ is a smooth projective curve. Since $G$ has $\mathbf{Q}$-rank one, $V ^ { * } - V$ is a finite set of points, referred to as cusps.

Historically the next significant example was for the Siegel modular group, with $G = \operatorname { Sp } ( 2 n , \mathbf{Q} )$, and $\Gamma = \operatorname { Sp } ( 2 n , \mathbf{Z} )$, and $X = {\cal H} _ { n }$ consisting of $n \times n$ symmetric complex matrices with positive-definite imaginary part; here $\left( \begin{array} { l l } { A } & { B } \\ { C } & { D } \end{array} \right)$ in $\operatorname { Sp } ( 2 n , \mathbf R )$ acts on $Z \in {\cal H} _ { n }$ by $Z \mapsto ( A Z + B ) ( C Z + D ) ^ { - 1 }$. I. Satake [a17] was the first to describe a compactification of $V _ { n } = \mathcal{H} _ { n } / \Gamma$ as $V _ { n } ^ { * } = V _ { n } \cup \ldots \cup V _ { 0 }$ endowed with its Satake topology (cf. also Satake compactification). Then Satake, H. Cartan and others (in [a19]) and W.L. Baily [a13] further investigated and exhibited the analytic and algebraic structure of $V ^ { * }$, using automorphic forms as mentioned above. Baily [a14] also treated the Hilbert–Siegel modular group, where $G ({\bf Q }) = \operatorname { Sp } ( 2 n , F )$ for a totally real number field $F$.

In the meanwhile, under only some mild assumption about $G$, Satake [a18] constructed $V ^ { * }$ with its Satake topology, while I.I. Piateckii-Shapiro [a10] described a normal analytic compactification whose topology was apparently weaker than that of the Baily–Borel compactification. Later, P. Kiernan [a7] showed that the topology defined by Piateckii-Shapiro is homeomorphic to the Satake topology used by Baily and Borel.

### Other compactifications.

Other approaches to the compactification of arithmetic quotients of symmetric domains to which the Satake and Baily–Borel approach may be compared are the Borel–Serre compactification [a3], see the discussion in [a20], and the method of toroidal embeddings [a12].

### Cohomology.

Zucker's conjecture [a21] that the (middle perversity) intersection cohomology [a4] (cf. also Intersection homology) of the Baily–Borel compactification coincides with its $L^{2}$-cohomology, has been given two independent proofs (see [a8] and [a11]); see also the discussion and bibliography in [a5].

### Arithmetic and moduli.

In many cases $V$ has an interpretation as the moduli space for some family of Abelian varieties (cf. also Moduli theory), usually with some additional structure; this leads to the subject of Shimura varieties (cf. also Shimura variety), which also addresses arithmetic questions such as the field of definition of $V$ and $V ^ { * }$. Geometrically, the strata of $V ^ { * } - V$ parameterize different semi-Abelian varieties, i.e., semi-direct products of algebraic tori with Abelian varieties, into which the Abelian varieties represented by points on $V$ degenerate. For an example see [a9], where this is thoroughly worked out for $\mathbf{Q}$-forms of $ \operatorname{SU} ( n , 1 )$, especially for $n = 2$.

#### References

[a1] | W.L. Baily, Jr., A. Borel, "On the compactification of arithmetically defined quotients of bounded symmetric domains" Bull. Amer. Math. Soc. , 70 (1964) pp. 588–593 |

[a2] | W.L. Baily, Jr., A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains" Ann. of Math. (2) , 84 (1966) pp. 442–528 |

[a3] | A. Borel, J.P. Serre, "Corners and arithmetic groups" Comment. Math. Helv. , 48 (1973) pp. 436–491 |

[a4] | M. Goresky, R. MacPherson, "Intersection homology, II" Invent. Math. , 72 (1983) pp. 135–162 |

[a5] | M. Goresky, "$L^{2}$-cohomology is intersection cohomology" R.P. Langlands (ed.) D. Ramakrishnan (ed.) , The Zeta Functions of Picard Modular Surfaces , Publ. CRM (1992) pp. 47–63 |

[a6] | Harish-Chandra, "Representations of semi-simple Lie groups. VI" Amer. J. Math. , 78 (1956) pp. 564–628 |

[a7] | P. Kiernan, "On the compactifications of arithmetic quotients of symmetric spaces" Bull. Amer. Math. Soc. , 80 (1974) pp. 109–110 |

[a8] | E. Looijenga, "$L^{2}$-cohomology of locally symmetric varieties" Computers Math. , 67 (1988) pp. 3–20 |

[a9] | "The zeta functions of Picard modular surfaces" R.P. Langlands (ed.) D. Ramakrishnan (ed.) , Publ. CRM (1992) |

[a10] | I.I. Piateckii-Shapiro, "Arithmetic groups in complex domains" Russian Math. Surveys , 19 (1964) pp. 83–109 Uspekhi Mat. Nauk. , 19 (1964) pp. 93–121 |

[a11] | L. Saper, M. Stern, "$L_{2}$-cohomology of arithmetic varieties" Ann. of Math. , 132 : 2 (1990) pp. 1–69 |

[a12] | A. Ash, D. Mumford, M. Rapoport, Y. Tai, "Smooth compactifications of locally symmetric varieties" , Math. Sci. Press (1975) |

[a13] | W.L. Baily, Jr., "On Satake's compactification of $V _ { n }$" Amer. J. Math. , 80 (1958) pp. 348–364 |

[a14] | W.L. Baily, Jr., "On the Hilbert–Siegel modular space" Amer. J. Math. , 81 (1959) pp. 846–874 |

[a15] | W.L. Baily, Jr., "On the orbit spaces of arithmetic groups" , Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) , Harper and Row (1965) pp. 4–10 |

[a16] | W.L. Baily, Jr., "On compactifications of orbit spaces of arithmetic discontinuous groups acting on bounded symmetric domains" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 281–295 |

[a17] | I. Satake, "On the compactification of the Siegel space" J. Indian Math. Soc. (N.S.) , 20 (1956) pp. 259–281 MR0084842 Zbl 0072.30002 |

[a18] | I. Satake, "On compactifications of the quotient spaces for arithmetically defined discontinuous groups" Ann. of Math. , 72 : 2 (1960) pp. 555–580 MR0170356 Zbl 0146.04701 |

[a19] | "Fonctions automorphes" , Sém. H. Cartan 10ième ann. (1957/8) , 1–2 , Secr. Math. Paris (1958) (Cartan) |

[a20] | S. Zucker, "Satake compactifications" Comment. Math. Helv. , 58 (1983) pp. 312–343 MR0705539 Zbl 0565.22009 |

[a21] | S. Zucker, "$L^{2}$-cohomology of warped products and arithmetic groups" Ann. of Math. , 70 (1982) pp. 169–218 MR0684171 Zbl 0508.20020 |

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Baily-Borel compactification.

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