Reductive Borel-Serre Compactification Research Articles

The reductive Borel–Serre compactification as a model for unstable algebraic K-theory

Let A be an associative ring and M a finitely generated projective A-module. We introduce a category RBS(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {RBS}}(M)$$\\end{document} and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories RBS(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {RBS}}(M)$$\\end{document} naturally arise as generalisations of the exit path ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}-category of the reductive Borel–Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications.

The Stratified Homotopy Type of the Reductive Borel–Serre Compactification

Abstract We identify the exit path $\infty $-category of the reductive Borel–Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As immediate consequences, we identify the fundamental group of the reductive Borel–Serre compactification, recovering a result of Ji–Murty–Saper–Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel–Serre compactification as elements in a derived functor category.

Motivic weightless complex and the relative Artin motive

We construct a motivic analogue of the weightless cohomology groups [18] and demonstrate several expected properties – ring structure, Kunneth, functoriality for certain morphisms etc. We also demonstrate the expected relationship with the motivic intersection complex of Wildeshaus [26,25] (whenever it exists). The methods also give another construction of the weightless cohomology groups over any base field. In characteristic 0, this motive coincides with the motive EX of Ayoub–Zucker [3] and recovers the motivic construction of the reductive Borel–Serre compactification of Shimura varieties.

Simplicial volume of moduli spaces of Riemann surfaces

Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space $M_{g,n}$ is equal to $0$ if $g \ge 2$; $g = 1$, $n \ge 3$; or $g = 0$, $n \ge 6$; and the orbifold simplicial volume of $M_{g,n}$ is positive if $g = 1, n = 0, 1; g = 0, n = 4$. We also observe that the simplicial volume of the Deligne-Mumford compactification of $M_{g,n}$ is equal to $0$, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces $\Gamma\backslash X$ and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces $\Gamma\backslash X$ are also equal to $0$ if the $\mathbb{Q}$-rank of $\Gamma\backslash X$ is at least $3$ or if $\Gamma\backslash X$ is irreducible and of $\mathbb{Q}$-rank 2.

Relative Artin motives and the reductive Borel–Serre compactification of a locally symmetric variety

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $\omega^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $\omega^0_X$ and the computation of the motives E_X. In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the projection of its reductive Borel-Serre compactification W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is naturally isomorphic to the Betti realization of the motive E_Z, where Z is viewed as a scheme. In particular, the direct image of E_Z along the projection of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic to the cohomology of W.

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

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 ${\overline X}^{\rm exc}$ and $X^{{\rm tor,exc}}$ 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 ${\overline X}^{\rm exc}$ and $X^{{\rm tor,exc}}$ are homotopy equivalent.

Compactifications of Locally Symmetric Spaces

Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space, whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K. This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact, they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms, in particular Eisenstein series.

The L-class of non-Witt spaces

Characteristic classes for oriented pseudomanifolds can be defined using appropriate self-dual complexes of sheaves. On non-Witt spaces, self-dual complexes compatible to intersection homology are determined by choices of Lagrangian structures at the strata of odd codimension. We prove that the associated signature and L-classes are independent of the choice of Lagrangian structures, so that singular spaces with odd codimensional strata, such as e.g. certain compactifications of locally symmetric spaces, have well-defined L-classes, provided Lagrangian structures exist. We illustrate the general results with the example of the reductive Borel-Serre compactification of a Hilbert modular surface.

L<sup>2</sup>-Cohomology of Locally Symmetric Spaces, I

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a combinatorial invariant that to a great extent characterizes the cohomology of the associated sheaf. The theory has been successfully applied to solve a number of problems concerning the intersection cohomology and weighted cohomology of the reductive Borel-Serre compactification [math.RT/0112251], as well as the ordinary cohomology of X [math.RT/0112250]. In this paper we extend the theory so that it covers L^2-cohomology. In particular we construct an L-module whose cohomology is the L^2-cohomology of X and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.

Self-dual Sheaves on Reductive Borel–Serre Compactifications of Hilbert Modular Surfaces

The main theorem of this article asserts that the category of self-dual sheaves compatible with the intersection chain sheaves (for upper/lower middle perversity) on the reductive Borel—Serre compactification \(\bar X\) of a Hilbert modular surface is nonempty. Also we prove that the direct image of such a sheaf under the canonical map to the Baily—Borel compactification is isomorphic (in the derived category) to the intersection chain sheaf for upper and lower middle perversity. As a consequence of the main theorem, there exist characteristic L-classes of these sheaves in the rational homology of \(\bar X\). In fact, these classes do not depend on the choice of a self-dual sheaf and hence are invariants of the compactification \(\bar X\).

On the Reductive Borel-Serre Compactification, III: Mixed Hodge Structures

We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications. Introduction. In (BS), Borel and Serre gave a construction that compactified the arithmetic quotient X of a symmetric space (also certain more general locally homogeneous spaces) to a manifold-with-corners X. With its faces fibered by nilman- ifolds, it was both an easy and a useful little construction to collapse each fiber to a point, producing what is now called the reductive Borel-Serre compactification X red

On the cohomology of locally symmetric spaces and of their compactifications

This expository article is an expanded version of talks given at the Current Developments in Mathematics, 2002 conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally symmetric space with that of the reductive Borel-Serre compactification. We motivate the conjecture with examples and then give an introduction to the various topics that are involved: intersection cohomology, the derived category, and compactifications of a locally symmetric space, particularly those above. We then give an overview of the theory of L-modules and micro-support (see math.RT/0112251) which was developed to solve the conjecture but has other important applications as well. We end with sketches of the proofs of three main theorems on L-modules that lead to the resolution of the conjecture. The text is enriched with many examples, illustrations, and references to the literature.

On the reductive Borel-Serre compactification: L p -cohomology of arithmetic groups (for large p )

The L 2 -cohomology of an arithmetic quotient of an Hermitian symmetric space (i.e., of a locally symmetric variety) is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the L p -cohomology of an arithmetic quotient, for p finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified spaces.

Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces

By "Hermitian locally symmetric space" we mean an arithmetic quotient of a bounded symmetric domain. Both the toroidal and the reductive Borel-Serre compactifications of such a space come equipped with canonical mappings to the Baily-Borel Satake compactification. In this article we show that there is a mapping from the toroidal compactification to the reductive Borel-Serre compactification, whose composition with the projection to the Baily-Borel compactification agrees with the canonical projection up to an arbitrarily small homotopy. We also consider arithmetic quotients of a self-adjoint homogeneous cone. There is a canonical mapping from the reductive Borel-Serre compactification to the standard compactification of such a locally symmetric cone. We show that this projection, when restricted to the closure of a polyhedral cone, has contractible fibers.

Weighted Cohomology of Arithmetic Groups

Let G be a semisimple algebraic group defined over the rational numbers, K a maximal compact subgroup ofG = G(R), and Γ ⊂ G(Q) a neat arithmetic subgroup. LetX = Γ\G/K be the locally symmetric space associated to Γ, and E the local system on X constructed out of a finite-dimensional, irreducible, algebraic representation E of G. Fix a maximally Q-split torus S in G; S is assumed to be nontrivial, so that X is noncompact. Let A = S(R)0 and, by a slight abuse of notation, let X∗(A) denote the group of rational characters of S. Choose a minimal rational parabolic subgroup P0 ⊃ S; the choice determines a notion of positivity in X∗(A) and a set of positive roots among the roots of S in G. Let ρ0 be the half-sum of the positive roots. Weighted cohomology is an invariant of Γ introduced by M. Goresky, G. Harder, and R. MacPherson [12] in the study ([15], [16], [12], [14]) of the trace of Hecke operators in the cohomology of Γ. For each p ∈ X∗(A)⊗Q there are weight profiles p and p (upper and lower p; see 1.5) and groups W pH i(Γ, E) and W H i(Γ, E). (The notion of profile here differs from (but is equivalent to) that of [12]; see Remark 1.2 below.) If p+q = −2ρ0 then the profiles p and q are dual and the corresponding groups are in Poincare duality. For very positive p one gets the compactly supported cohomology H i c(X,E) and dually, for very negative p, the full cohomology H i(X,E). The definition of the weighted cohomology groups is somewhat involved, so I will attempt to give some idea of the motivation behind it; detailed definitions are recalled in Section 1. Associated to p and p are complexes of sheaves WpC•(E) and WpC•(E) on the reductive Borel-Serre compactification X of X. Let P be a rational parabolic subgroup, NP its unipotent radical, SP the split centre of P/NP and XP the P-boundary component in X. The idea behind the construction in [12] is as follows: The stalk cohomology of the direct image Ri∗E (by i : X ↪→ X) at x ∈ XP is

The Greatest Common Quotient of Borel-Serre and the Toroidal Compactifications of Locally Symmetric Spaces

In this paper, we identify the greatest common quotient (GCQ) of the Borel-Serre compactification and the toroidal compactifications of Hermitian locally symmetric spaces with a new compactification. Using this compactification, we completely settle a conjecture of Harris-Zucker that this GCQ is equal to the Baily-Borel compactification. We also show that the GCQ of the reductive Borel-Serre compactification and the toroidal compactifications is the Baily-Borel compactification. There are two key ingredients in the proof: ergodicity of certain adjoint action on nilmanifolds and incompatibility between the ambient linear structure and the intrinsic Riemannian structure of homothety sections of symmetric cones.

METRIC COMPACTIFICATIONS OF LOCALLY SYMMETRIC SPACES

We introduce hyperbolic and asymptotic compactifications of metric spaces and apply them to locally symmetric spaces Γ\X. We show that the reductive Borel–Serre compactification [Formula: see text] is hyperbolic and, as a corollary, get a result of Borel, and Kobayashi–Ochiai that the Baily–Borel compactification [Formula: see text] is hyperbolic. We prove that the hyperbolic reduction of the toroidal compactifications [Formula: see text] is equal to [Formula: see text] and use it to derive a result of Kiernan–Kobayashi on extensions of holomorphic maps from Γ\ X to the compactification [Formula: see text]. We also show that the Tits compactification [Formula: see text] is an asymptotic compactification while the asymptotic reduction of both [Formula: see text] and [Formula: see text] is equal to the end compactification Γ\ X, and prove a conjecture of Siegel on some metric properties of Siegel sets.

The fundamental group of reductive Borel–Serre and Satake compactifications

Let $G$ be an almost simple, simply connected algebraic group defined over a number field $k$, and let $S$ be a finite set of places of $k$ including all infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces associated to $G(k_v)$, when $v$ is an infinite place, and the Bruhat-Tits buildings associated to $G(k_v)$, when $v$ is a finite place. The main result of this paper is an explicit computation of the fundamental group of the reductive Borel-Serre compactification of $\Gamma\backslash X$, where $\Gamma$ is an $S$-arithmetic subgroup of $G$. In the case that $\Gamma$ is neat, we show that this fundamental group is isomorphic to $\Gamma/E\Gamma$, where $E\Gamma$ is the subgroup generated by the elements of $\Gamma$ belonging to unipotent radicals of $k$-parabolic subgroups. Analogous computations of the fundamental group of the Satake compactifications are made. It is noteworthy that calculations of the congruence subgroup kernel $C(S,G)$ yield similar results.