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''Satake–Baily–Borel compactification''
 
''Satake–Baily–Borel compactification''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300101.png" /> be a semi-simple [[Linear algebraic group|linear algebraic group]] (cf. also [[Semi-simple algebraic group|Semi-simple algebraic group]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300102.png" />, meaning that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300103.png" /> can be embedded as a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300104.png" /> such that each element is diagonalizable (cf. also [[Diagonalizable algebraic group|Diagonalizable algebraic group]]), and that the equations defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300105.png" /> as an [[Algebraic variety|algebraic variety]] have coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300106.png" /> (and that the group operation is an algebraic morphism). Further, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300107.png" /> contains a torus (cf. [[Algebraic torus|Algebraic torus]]) that splits over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300108.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300109.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001010.png" />-rank at least one), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001011.png" /> is of Hermitian type, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001012.png" /> can be given a complex structure with which it becomes a symmetric domain, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001013.png" /> denotes the real points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001015.png" /> is a maximal compact subgroup. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001016.png" /> be an arithmetic subgroup (cf. [[Arithmetic group|Arithmetic group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001017.png" />, commensurable with the integer points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001018.png" />. Then the arithmetic quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001019.png" /> is a [[Normal analytic space|normal analytic space]] whose Baily–Borel compactification, also sometimes called the Satake–Baily–Borel compactification, is a canonically determined projective normal [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001020.png" />, defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001021.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001022.png" /> is Zariski-open (cf. also [[Zariski topology|Zariski topology]]) [[#References|[a1]]] [[#References|[a2]]] [[#References|[a15]]] [[#References|[a16]]].
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Let $G$ be a semi-simple [[Linear algebraic group|linear algebraic group]] (cf. also [[Semi-simple algebraic group|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|Diagonalizable algebraic group]]), and that the equations defining $G$ as an [[Algebraic variety|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|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|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|normal analytic space]] whose Baily–Borel compactification, also sometimes called the Satake–Baily–Borel compactification, is a canonically determined projective normal [[Algebraic variety|algebraic variety]] $V ^ { * }$, defined over $\mathbf{C}$, in which $V$ is Zariski-open (cf. also [[Zariski topology|Zariski topology]]) [[#References|[a1]]] [[#References|[a2]]] [[#References|[a15]]] [[#References|[a16]]].
  
To describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001023.png" /> in the complex topology, first note that the Harish–Chandra realization [[#References|[a6]]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001024.png" /> as a bounded symmetric domain may be compactified by taking its topological closure. Then a rational boundary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001025.png" /> is a boundary component whose [[Stabilizer|stabilizer]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001026.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001027.png" />; based on a detailed analysis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001029.png" />-roots and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001031.png" />-roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001032.png" />, there is a natural bijection between the rational boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001033.png" /> and the proper maximal parabolic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001034.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001036.png" /> denote the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001037.png" /> with all its rational boundary components. Then (cf. [[#References|[a18]]]) there is a unique topology, the Satake topology, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001038.png" /> such that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001039.png" /> extends continuously and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001040.png" /> with its quotient topology compact and Hausdorff. It also follows from the construction that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001041.png" /> is a finite disjoint union of the form
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To describe $V ^ { * }$ in the complex topology, first note that the Harish–Chandra realization [[#References|[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|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. [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001042.png" /></td> </tr></table>
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\begin{equation*} V ^ { * } = V \bigcup V _ { 1 } \bigcup \ldots \bigcup V _ { t }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001043.png" /> for some rational boundary component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001046.png" /> is the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001047.png" /> with the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001048.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001049.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001050.png" /> has a natural structure as a [[Normal analytic space|normal analytic space]]; the closure of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001051.png" /> is the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001052.png" /> with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001053.png" />s of strictly smaller dimension; and it can be proved that every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001054.png" /> has a fundamental system of neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001056.png" /> is connected for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001057.png" />.
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001058.png" /> (cf. also [[Scheme|Scheme]]) with which it becomes a normal analytic space and a projective variety, define an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001060.png" />-function on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001061.png" /> to be a continuous complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001062.png" /> whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001063.png" /> is analytic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001065.png" />. Then, associating to each open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001066.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001067.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001068.png" />-functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001069.png" /> determines the [[Sheaf|sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001070.png" /> of germs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001071.png" />-functions. Further, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001072.png" /> the sheaf of germs of restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001073.png" />-functions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001074.png" /> is the structure sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001075.png" />. Ultimately it is proved [[#References|[a2]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001076.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001077.png" />-functions there are enough automorphic forms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001078.png" />, more specifically, Poincaré-Eisenstein series, which generalize both Poincaré series and Eisenstein series (cf. also [[Theta-series|Theta-series]]), to separate points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001079.png" /> as well as to provide a projective embedding.
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In order to describe the structure sheaf of $V ^ { * }$ (cf. also [[Scheme|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|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 [[#References|[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|Theta-series]]), to separate points on $V ^ { * }$ as well as to provide a projective embedding.
  
 
===History and examples.===
 
===History and examples.===
The simplest example of a Baily–Borel compactification is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001082.png" /> is the complex upper half-plane, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001084.png" /> acts by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001085.png" />. (The bounded realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001086.png" /> is a unit disc, to which the upper half-plane maps by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001087.png" />.) The properly discontinuous action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001088.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001089.png" /> extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001090.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001091.png" /> is a smooth projective curve. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001092.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001093.png" />-rank one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001094.png" /> is a finite set of points, referred to as cusps.
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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 $\left( \begin{array} { l l } { a } &amp; { b } \\ { c } &amp; { d } \end{array} \right)$ 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001096.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001097.png" /> consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001098.png" /> symmetric complex matrices with positive-definite imaginary part; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010100.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010101.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010102.png" />. I. Satake [[#References|[a17]]] was the first to describe a compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010103.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010104.png" /> endowed with its Satake topology (cf. also [[Satake compactification|Satake compactification]]). Then Satake, H. Cartan and others (in [[#References|[a19]]]) and W.L. Baily [[#References|[a13]]] further investigated and exhibited the analytic and algebraic structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010105.png" />, using automorphic forms as mentioned above. Baily [[#References|[a14]]] also treated the Hilbert–Siegel modular group, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010106.png" /> for a totally real [[Number field|number field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010107.png" />.
+
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 $\left( \begin{array} { l l } { A } &amp; { B } \\ { C } &amp; { D } \end{array} \right)$ 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 [[#References|[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|Satake compactification]]). Then Satake, H. Cartan and others (in [[#References|[a19]]]) and W.L. Baily [[#References|[a13]]] further investigated and exhibited the analytic and algebraic structure of $V ^ { * }$, using automorphic forms as mentioned above. Baily [[#References|[a14]]] also treated the Hilbert–Siegel modular group, where $G ({\bf Q }) = \operatorname { Sp } ( 2 n , F )$ for a totally real [[Number field|number field]] $F$.
  
In the meanwhile, under only some mild assumption about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010108.png" />, Satake [[#References|[a18]]] constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010109.png" /> with its Satake topology, while I.I. Piateckii-Shapiro [[#References|[a10]]] described a normal analytic compactification whose topology was apparently weaker than that of the Baily–Borel compactification. Later, P. Kiernan [[#References|[a7]]] showed that the topology defined by Piateckii-Shapiro is homeomorphic to the Satake topology used by Baily and Borel.
+
In the meanwhile, under only some mild assumption about $G$, Satake [[#References|[a18]]] constructed $V ^ { * }$ with its Satake topology, while I.I. Piateckii-Shapiro [[#References|[a10]]] described a normal analytic compactification whose topology was apparently weaker than that of the Baily–Borel compactification. Later, P. Kiernan [[#References|[a7]]] showed that the topology defined by Piateckii-Shapiro is homeomorphic to the Satake topology used by Baily and Borel.
  
 
===Other compactifications.===
 
===Other compactifications.===
Line 22: Line 30:
  
 
===Cohomology.===
 
===Cohomology.===
Zucker's conjecture [[#References|[a21]]] that the (middle perversity) intersection cohomology [[#References|[a4]]] (cf. also [[Intersection homology|Intersection homology]]) of the Baily–Borel compactification coincides with its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010111.png" />-cohomology, has been given two independent proofs (see [[#References|[a8]]] and [[#References|[a11]]]); see also the discussion and bibliography in [[#References|[a5]]].
+
Zucker's conjecture [[#References|[a21]]] that the (middle perversity) intersection cohomology [[#References|[a4]]] (cf. also [[Intersection homology|Intersection homology]]) of the Baily–Borel compactification coincides with its $L^{2}$-cohomology, has been given two independent proofs (see [[#References|[a8]]] and [[#References|[a11]]]); see also the discussion and bibliography in [[#References|[a5]]].
  
 
===Arithmetic and moduli.===
 
===Arithmetic and moduli.===
In many cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010112.png" /> has an interpretation as the moduli space for some family of Abelian varieties (cf. also [[Moduli theory|Moduli theory]]), usually with some additional structure; this leads to the subject of Shimura varieties (cf. also [[Shimura variety|Shimura variety]]), which also addresses arithmetic questions such as the field of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010114.png" />. Geometrically, the strata of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010115.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010116.png" /> degenerate. For an example see [[#References|[a9]]], where this is thoroughly worked out for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010117.png" />-forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010118.png" />, especially for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010119.png" />.
+
In many cases $V$ has an interpretation as the moduli space for some family of Abelian varieties (cf. also [[Moduli theory|Moduli theory]]), usually with some additional structure; this leads to the subject of Shimura varieties (cf. also [[Shimura variety|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 [[#References|[a9]]], where this is thoroughly worked out for $\mathbf{Q}$-forms of $ \operatorname{SU} ( n , 1 )$, especially for $n = 2$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.L. Baily, Jr.,   A. Borel,   "Compactification of arithmetic quotients of bounded symmetric domains" ''Ann. of Math. (2)'' , '''84''' (1966) pp. 442–528</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Borel,   J.P. Serre,   "Corners and arithmetic groups" ''Comment. Math. Helv.'' , '''48''' (1973) pp. 436–491</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Goresky,   R. MacPherson,   "Intersection homology, II" ''Invent. Math.'' , '''72''' (1983) pp. 135–162</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Goresky,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010120.png" />-cohomology is intersection cohomology" R.P. Langlands (ed.) D. Ramakrishnan (ed.) , ''The Zeta Functions of Picard Modular Surfaces'' , Publ. CRM (1992) pp. 47–63</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Harish-Chandra,   "Representations of semi-simple Lie groups. VI" ''Amer. J. Math.'' , '''78''' (1956) pp. 564–628</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Kiernan,   "On the compactifications of arithmetic quotients of symmetric spaces" ''Bull. Amer. Math. Soc.'' , '''80''' (1974) pp. 109–110</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> E. Looijenga,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010121.png" />-cohomology of locally symmetric varieties" ''Computers Math.'' , '''67''' (1988) pp. 3–20</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> "The zeta functions of Picard modular surfaces" R.P. Langlands (ed.) D. Ramakrishnan (ed.) , Publ. CRM (1992)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> L. Saper,   M. Stern,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010122.png" />-cohomology of arithmetic varieties" ''Ann. of Math.'' , '''132''' : 2 (1990) pp. 1–69</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A. Ash,   D. Mumford,   M. Rapoport,   Y. Tai,   "Smooth compactifications of locally symmetric varieties" , Math. Sci. Press (1975)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> W.L. Baily, Jr.,   "On Satake's compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010123.png" />"  ''Amer. J. Math.'' , '''80''' (1958) pp. 348–364</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> W.L. Baily, Jr.,   "On the Hilbert–Siegel modular space" ''Amer. J. Math.'' , '''81''' (1959) pp. 846–874</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> I. Satake,   "On the compactification of the Siegel space" ''J. Indian Math. Soc. (N.S.)'' , '''20''' (1956) pp. 259–281</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> I. Satake,   "On compactifications of the quotient spaces for arithmetically defined discontinuous groups" ''Ann. of Math.'' , '''72''' : 2 (1960) pp. 555–580</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> "Fonctions automorphes" , ''Sém. H. Cartan 10ième ann. (1957/8)'' , '''1–2''' , Secr. Math. Paris (1958) (Cartan)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> S. Zucker,   "Satake compactifications" ''Comment. Math. Helv.'' , '''58''' (1983) pp. 312–343</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> S. Zucker,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010124.png" />-cohomology of warped products and arithmetic groups" ''Ann. of Math.'' , '''70''' (1982) pp. 169–218</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> W.L. Baily, Jr., A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains" ''Ann. of Math. (2)'' , '''84''' (1966) pp. 442–528</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> A. Borel, J.P. Serre, "Corners and arithmetic groups" ''Comment. Math. Helv.'' , '''48''' (1973) pp. 436–491</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> M. Goresky, R. MacPherson, "Intersection homology, II" ''Invent. Math.'' , '''72''' (1983) pp. 135–162</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> Harish-Chandra, "Representations of semi-simple Lie groups. VI" ''Amer. J. Math.'' , '''78''' (1956) pp. 564–628</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> P. Kiernan, "On the compactifications of arithmetic quotients of symmetric spaces" ''Bull. Amer. Math. Soc.'' , '''80''' (1974) pp. 109–110</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> E. Looijenga, "$L^{2}$-cohomology of locally symmetric varieties" ''Computers Math.'' , '''67''' (1988) pp. 3–20</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> "The zeta functions of Picard modular surfaces" R.P. Langlands (ed.) D. Ramakrishnan (ed.) , Publ. CRM (1992)</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> L. Saper, M. Stern, "$L_{2}$-cohomology of arithmetic varieties" ''Ann. of Math.'' , '''132''' : 2 (1990) pp. 1–69</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> A. Ash, D. Mumford, M. Rapoport, Y. Tai, "Smooth compactifications of locally symmetric varieties" , Math. Sci. Press (1975)</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> W.L. Baily, Jr., "On Satake's compactification of $V _ { n }$" ''Amer. J. Math.'' , '''80''' (1958) pp. 348–364</td></tr><tr><td valign="top">[a14]</td> <td valign="top"> W.L. Baily, Jr., "On the Hilbert–Siegel modular space" ''Amer. J. Math.'' , '''81''' (1959) pp. 846–874</td></tr><tr><td valign="top">[a15]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a16]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a17]</td> <td valign="top"> I. Satake, "On the compactification of the Siegel space" ''J. Indian Math. Soc. (N.S.)'' , '''20''' (1956) pp. 259–281 {{MR|0084842}} {{ZBL|0072.30002}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> I. Satake, "On compactifications of the quotient spaces for arithmetically defined discontinuous groups" ''Ann. of Math.'' , '''72''' : 2 (1960) pp. 555–580 {{MR|0170356}} {{ZBL|0146.04701}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> "Fonctions automorphes" , ''Sém. H. Cartan 10ième ann. (1957/8)'' , '''1–2''' , Secr. Math. Paris (1958) (Cartan)</td></tr><tr><td valign="top">[a20]</td> <td valign="top"> S. Zucker, "Satake compactifications" ''Comment. Math. Helv.'' , '''58''' (1983) pp. 312–343 {{MR|0705539}} {{ZBL|0565.22009}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> S. Zucker, "$L^{2}$-cohomology of warped products and arithmetic groups" ''Ann. of Math.'' , '''70''' (1982) pp. 169–218 {{MR|0684171}} {{ZBL|0508.20020}} </td></tr></table>

Latest revision as of 17:00, 1 July 2020

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
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[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
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[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
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How to Cite This Entry:
Baily-Borel compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baily-Borel_compactification&oldid=14832
This article was adapted from an original article by B. Brent Gordon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article