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