Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces

Goresky, Mark; Tai, Yung-sheng

doi:10.1353/ajm.1999.0032

Cited by 18 publications

(28 citation statements)

References 23 publications

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“…The proof of the assertion in (i) about β goes, more or less, like the argument in [4] (for (ii) in Prop. 1 above).…”

Section: Corollary Conjecture 1 Is Truementioning

confidence: 64%

Excentric compactifications

Zucker¹

2005

Pure and Applied Mathematics Quarterly

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The term excentric was coined by the author [6 : §1], [13 : §2]. It is accented on the first syllable, in contrast with the English word "eccentric", and conveys the following idea. For now, let W be a unipotent algebraic group. Then W/W (the trivial group) is the reductive quotient of W . When U ⊆ W is a subgroup that is the center of something, then W/U is the (or an) excentric quotient of W .We present the setting for these notes. Let D be a symmetric space of noncompact type, and Γ an arithmetically defined group of isometries of D; put informally, this means that some algebraic group G over Q has its real points giving the isometry group of D, and Γ is roughly G(Z). If Γ is not too big (i.e., is torsionfree, later neat), then X = Γ\D is a manifold. When D has an invariant complex structure, D is called Hermitian, as is X. The latter is called a locally symmetric variety, for X is a quasi-projective complex algebraic variety [2].Typically, X is non-compact and one soon realizes that it is important to compactify it. There exist too many compactifications of X, so we select one or more to suit a given purpose. It is common enough to attach a Γ-equivariant boundary ∂D to D, and then take the quotient by Γ. Here are two such compactifications of X: i) X = Γ\D, the manifold-with-corners of Borel-Serre [3], ii X Sa = Γ\D Sa , a Satake compactification of X [9] (see also [11]). There are finitely many Satake compactifications. When X is Hermitian, one of these is topologically the Baily-Borel compactification X * , a projective variety over C that is generally quite singular.When X is Hermitian, there are also the smooth toroidal compactifications X tor of Mumford et al. [1], constructed so that ∂X tor is a divisor with normal crossings.

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“…The proof of the assertion in (i) about β goes, more or less, like the argument in [4] (for (ii) in Prop. 1 above).…”

Section: Corollary Conjecture 1 Is Truementioning

confidence: 64%

Excentric compactifications

Zucker¹

2005

Pure and Applied Mathematics Quarterly

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“…By this method, we avoid the application of intersection cohomology to a singular compactifications (e.g. the reductive Borel-Serre compactification [GHM94,GT99]). …”

Section: Topological Trace Formulamentioning

confidence: 99%

A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups

Weselmann¹

2011

Compositio Math.

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For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of Q-rank r, we construct a compact manifoldX by gluing together 2 r copies of the Borel-Serre compactification of X. We apply the classical Lefschetz fixed point formula toX and get formulas for the traces of Hecke operators H acting on the cohomology of X. We allow twistings of H by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G, η). As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.

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“…If {x j } is a sequence converging in both Y 1 and Y 2 , lift it to a sequence {x (1.2) Example: Two compactifications of a simplicial cone. The following is an essential calculation, one that underlies [HZ1,2.3], [HZ2,(1.5)], [GT,§7], and what is to come in this article.…”

Section: An Isomorphism If and Only If The Canonical Mappingmentioning

confidence: 99%

“…We have already seen that this is contractible for all P (Proposition 3.5.6). We invoke the criterion from [GT,§8]: a morphism of compact stratified spaces having contractible fibers is a homotopy equivalence.…”

Section: ])mentioning

confidence: 99%

On the reductive Borel-Serre compactification, II: Excentric quotients and least common modifications

Zucker¹

2008

ajm

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Abstract. Let X be a locally symmetric variety, i.e., the quotient of a bounded symmetric domain by a (say) neat arithmetically-defined group of isometries. Let X exc and X tor,exc denote its excentric Borel-Serre and toroidal compactifications respectively. We determine their least common modification and use it to prove a conjecture of Goresky and Tai concerning canonical extensions of homogeneous vector bundles. In the process, we see that X exc and X tor,exc are homotopy equivalent.

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Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces

Cited by 18 publications

References 23 publications

Excentric compactifications

Excentric compactifications

A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups

On the reductive Borel-Serre compactification, II: Excentric quotients and least common modifications

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