On the reductive Borel-Serre compactification: $L^p$-cohomology of
  arithmetic groups (for large $p$)
arXiv:0704.1335v1  [math.AG]  11 Apr 2007
On the reductive Borel-Serre compactification:
Lp-cohomology of arithmetic groups (for large p)
Steven Zucker1
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 USA2
Introduction
The main purpose of this article is to give the proof of the following theorem, as
well as some applications of the result.
Theorem 1. Let M be the quotient of a non-compact symmetric space by an
arithmetically-defined group of isometries, and MRBS its reductive Borel-Serre
compactification. Then for p finite and sufficiently large there is a canonical iso-
morphism
H•(p)(M) ≃ H
•(MRBS).
Here, the left-hand side is the Lp-cohomology of M with respect to a (locally)
invariant metric. Though it would be more natural to allow p =∞ in Theorem 1,
this is not generally possible (see (3.2.2)). On the other hand, there is a natural
mapping H•
(M) → H•
(M) when p < ∞, because M has finite volume. The
definition of MRBS is recalled in (1.9).
Theorem 1 can be viewed as an analogue of the so-called Zucker conjecture (in
the case of constant coefficients), where p = 2:
Theorem [L], [SS]. Let M be the quotient of a Hermitian symmetric space of non-
compact type by an arithmetically-defined group of isometries, i.e., a locally sym-
metric variety; let MBB its Baily-Borel Satake compactification. Then there is a
canonical isomorphism
H•(2)(M) ≃ IH
where the right-hand side denotes the middle intersection cohomology of MBB.
However, Theorem 1 is not nearly so difficult to prove, once one senses that it is
true; it follows without much ado from the methods in [Z3] (the generalization to
Lp, p 6= 2, of those of [Z1] for L2).
As far as I know, the reductive Borel-Serre compactification was first used in
[Z1,§4] (where it was called Y ). This space, a rather direct alteration of the
manifold-with-corners constructed in [BS], was introduced there to facilitate the
study of the L2-cohomology of M . It also plays a central role as the natural set-
ting for the related weighted cohomology of [GHM]. It is a principal theme that
MRBS is an important space when M is an algebraic variety over C, despite the
1Support in part by the National Science Foundation, through Grant DMS9820958
2e-mail address: zucker@jhu.edu
http://arxiv.org/abs/0704.1335v1
fact that MRBS is almost never an algebro-geometric, or even complex analytic,
compactification of M .
This work had its origin in my wanting to understand [GP]. It is convenient to
formulate the latter before continuing with the content of this article. Let Y be
a Hausdorff topological space. For any complex vector bundle E on Y , one has
its Chern classes ck(E) ∈ H
2k(Y,Z). If we further assume that Y is connected,
compact, stratified and oriented, then Hd(Y,Z) ≃ Z, where d is the dimension of
Y ; the orientation picks out a generator ζY for this homology group, known as the
fundamental class of Y . We shall henceforth assume that d is even, and we write
d = 2n. Then, if one has positive integers ki for 1 ≤ i ≤ ℓ such that
ki = n, one
can pair ck1(E) ∪ · · · ∪ ckℓ(E) with ζY , and obtain what is called a characteristic
number, or Chern number, of E.
When Y is a C∞ manifold and E is a C∞ vector bundle, the Chern classes
modulo torsion can be constructed from any connection ▽ in E, whereupon they
get represented, via the de Rham theorem, by the Chern forms ck(E,▽) in H
2k(Y ).
(For convenience, we will use and understand C-coefficients here and throughout
the sequel unless it is specified otherwise.) If Y is compact, the Chern numbers can
be computed by integrating ck1(E,▽) ∧ . . . ∧ ckℓ(E,▽) over Y .
For stratified spaces Y , there is a lattice of intersection (co)homology theories,
with variable perversity p as parameter, as defined by Goresky and MacPherson
[GM1]. These range from standard cohomology as minimal object, to standard
homology as maximal, and all coincide when Y is a manifold. They can all be
defined as cohomology with values in some constructible sheaf whose restriction to
the regular locus Y reg of Y is just CY reg . With mappings going in the direction of
increasing perversity, we have the basic diagram
(0.1)
H•(Y ) → . . . → IH•p(Y ) → . . . → H•(Y )
↓ ↑ ↑
H•(Y reg) ←− H•c (Y
reg) ≃ H•(Y
reg).
When Y is compact, ζY lifts to a generator of IH
(Y ) for all p.
From now on, we writeM for Y reg, and start to view the situation in the opposite
way, regarding Y as a topological compactification of the manifold M . For any
vector bundle E on M , and bundle extension of E to E on Y , the functoriality of
Chern classes imply that c•(E) 7→ c•(E) under the restriction mapping H
•(Y )
H•(M). One might think of this as lifting the Chern class of E to the cohomology
of Y , but one should be aware that ρ might have non-trivial kernel, so the lift may
depend on the choice of E.
The case where E = TM , the tangent bundle ofM , is quite fundamental. Finding
a vector bundle on Y that extends TM is not so natural a question when Y has sin-
gularities, and one is often inclined to forget about bundles and think instead about
just lifting the Chern classes. When Y is a complex algebraic variety, one considers
the complex tangent bundle T ′M of M . It is shown in [M] that for constructible Z-
valued functions F on Y , there is a natural assignment of Chern homology classes
c•(Y ;F ) ∈ H•(Y ), such that c•(Y, 1) recovers the usual Chern classes when Y is
smooth. There has been substantial interest in lifting these classes to the lower
intersection cohomology (as in the top row of (0.1)), best to cohomology (the most
difficult lifting problem) for the reason mentioned earlier.
Next, take for M a locally symmetric variety. For Y we might consider any
of the interesting compactifications of M , which include: MBB , the Baily-Borel
Satake compactification ofM as an algebraic variety [BB];MΣ, the smooth toroidal
compactifications of Mumford (see [Mu]); MBS , the Borel-Serre manifold-with-
corners [BS]; MRBS , the reductive Borel-Serre compactification. These fit into a
diagram of compactifications:
(0.2)
MBS −→ MRBS
MΣ −→ M
If one tries to compare MBS and MΣ, one sees that there is a mapping (of
compactifications of M) MBS → MΣ only in a few cases (e.g., G = SU(n, 1)).
However, by a result of Goresky and Tai [GT, 7.3], (if Σ is sufficiently fine) there are
continuous mappingsMΣ −→M
RBS (seldom a morphism of compactifications) such
that upon inserting them in (0.2), the obvious triangle commutes in the homotopy
category. One thereby gets a diagram of cohomology mappings
(0.3)
H•(M) ←− H•(MRBS)
↑ ւ ↑
H•(MΣ) ←− H
•(MBB);
also, the fundamental classes inHd(MΣ,Z) andHd(M
RBS ,Z) are mapped to ζMBB .
Let E be a (locally) homogeneous vector bundle onM (an example of which is the
holomorphic tangent bundle T ′M ). There always exists an equivariant connection
on E, whose Chern forms are L∞ (indeed, of constant length) with respect to the
natural metric on M . In [Mu], Mumford showed that the bundle E has a so-called
canonical extension to a vector bundle EΣ on MΣ, such that these Chern forms,
beyond representing the Chern classes of E in H•(M), actually represent the Chern
classes of EΣ inH
•(MΣ) (see our (3.2.4)). That served the useful purpose of placing
these classes in a ring with Poincaré duality, and implied Hirzebruch proportionality
for M .
In [GP], Goresky and Pardon lift these classes to the cohomology of MBB , min-
imal in the lattice of interesting compactifications of M , so these can be pulled
back to the other compactifications in (0.2). (On the other hand, the bundles
do not extend to MBB in any obvious way.) They achieve this by constructing
another connection in E (see our (5.3.3)), one that has good properties near the
singular strata of MBB , using features from the work of Harris and Harris-Zucker
(see [Z5,App.B]). With this done, the Chern forms lie in the complex of controlled
differential forms on MBB , whose cohomology groups give H•(MBB). In the case
of the tangent bundle, the classes map to one of the MacPherson Chern homology
classes in H•(M
BB), viz., c•(M
BB;χM ), where χM denotes the characteristic (i.e.,
indicator) function of M ⊂MBB [GP, 15.5].
In [GT, 9.2], an extension of E to a vector bundle ERBS toMRBS is constructed;
this does not require M to be Hermitian. There, one finds the following:
Conjecture A [GT, 9.5]. Let MΣ → M
RBS be any of the continuous mappings
constructed in [GT]. Then the canonical extension EΣ is isomorphic to the pullback
of ERBS.
In the absence of a proof of Conjecture A, we derive the “topological” analogue
of Mumford’s result as a consequence of Theorem 1:
Theorem 2. LetM be an arithmetic quotient of a symmetric space of non-compact
type. Then the Chern forms of an equivariant connection onM represent c•(E
in H•(MRBS).
We point out that (0.3) and Conjecture A suggest that this is more basic in the
Hermitian setting than Mumford’s result.
Goresky and Pardon predict further:
Conjecture B [GP]. The Chern classes of ERBS are the pullback of the classes
in H•(MBB) constructed in [GP] via the quotient mapping MRBS →MBB.
Our third main result is the proof of Conjecture B.
The material of this article is organized as follows. In §1 we give a canon-
ical construction of the bundle ERBS along the lines of [BS]. We next discuss
Lp-cohomology, both in general in §2, then on arithmetic quotients of symmetric
spaces in §3, achieving a proof of Theorem 1. We make a consequent observation
in (3.3) that shows how Lp-cohomology can be used to provide definitions of map-
pings between topological cohomology groups when it is unclear how to define the
mappings topologically. In §4, we treat connections and the notion of Chern forms
for a natural class of vector bundles on stratified spaces; this allows for the proof
in §5 of both Theorem 2 and Conjecture B.
This article was conceived while I was spending Academic Year 1998–99 on sabbatical at the
Institute for Advanced Study in Princeton. I wish to thank Mark Goresky and John Mather for
helpful discussions.
1. The Borel-Serre construction for homogeneous vector bundles
In this section, we make a direct analogue of the Borel-Serre construction for the
total space of a homogeneous vector bundle on a symmetric space, and then for any
neat arithmetic quotient MΓ thereof. It defines a natural extension of the vector
bundle to the Borel-Serre compactification of the space. That the bundle extends
is clear, for attaching of a boundary-with-corners does not change homotopy type.
Our construction retains at the boundary much of the group-theoretic structure.
The construction is shown to descend to the reductive Borel-Serre compactification
MRBSΓ , reproving [GT, 9.2].
(1.0) Convention. Whenever H is an algebraic group defined over Q, we also let
H denote H(R), taken with its topology as a real Lie group, if there is no danger
of confusion.
(1.1) Standard notions. Let G be a semi-simple algebraic group over Q, and K a
maximal compact subgroup of G, and X = G/K. (Note that this implies a choice
of basepoint for X , namely the point x0 left fixed by K.)
Let E = G ×K E be the homogeneous vector bundle on X determined by the
representation of K on the vector space E. The natural projection π : E → X =
G×K{0} is induced by the projection E → {0}, and isG-equivariant. For Γ ⊂ G(Q)
a torsion-free arithmetic subgroup, letMΓ = Γ\X . Then Γ\E is the total space of a
vector bundle EΓ overMΓ. (The subscript “Γ” was suppressed in the Introduction.)
If P is any Q-parabolic subgroup of G, the action of P on X is transitive. Thus,
one can also describe E → X as P ×KP E → P ×KP {0}, where KP = K ∩ P .
(1.2) Geodesic action. Let UP denote the unipotent radical of P , and AP the lift
to P associated to x0 of the connected component of the maximal Q-split torus Z
of P/UP . Define the geodesic action of AP on E by the formula:
(1.2.1) a ◦ (p, e) = (pa, e)
whenever p ∈ P , e ∈ E and a ∈ AP ; this is well-defined because for k ∈ KP ,
(1.2.2) a ◦ (pk−1, ke) = (pk−1a, ke) = (pak−1, ke),
as AP and KP commute. The geodesic action of AP commutes with the action of
P on E , and it projects to the geodesic action of AP on X as defined in [BS, §3] (in
[BS], the geodesic action is expressed in terms of Z, but the definitions coincide).
(1.2.3) Remark. By taking E to be of dimension zero, the construction of Borel-
Serre can be viewed as a case of ours above. As such, there is no real need to recall
it separately. Conversely, a fair though incomplete picture of our construction can
be seen by regarding E as simply a thickened version of X .
(1.3) Corners. The simple roots occurring in UP set up an isomorphism AP ≃
(0,∞)r(P ), where r(P ) denotes the parabolic Q-rank of P . Let AP be the en-
largement of AP obtained by transport of structure from (0,∞)
r(P ) ⊂ (0,∞]r(P ).
Define the corner associated to P : E(P ) = E×AP AP . There is a canonical mapping
π(P ) : E(P )→ X(P ) = X ×AP AP .
(1.3.1) Remark. ThoughX(P ) is contractible, and hence E(P ) is trivial, (1.2.1) does
not yield a canonical trivialization of E(P ) over X(P ), because of the equivalence
relation (1.2.2) determined by KP .
Let ∞P denote the zero-dimensional AP -orbit in AP , which corresponds to
(∞, ...,∞) ∈ (0,∞]r(P ). The face of E(P ) associated to P is
(1.3.2) E(P ) = E ×AP {∞P} ≃ E/AP .
It maps canonically to X/AP ≃ e(P ) ⊂ X(P ) (from [BS, 5.2]). There are geodesic
projections implicit in (1.3.2), given by the rows of the commutative diagram
(1.3.3)
E(P )
−−−−→ E(P )
X(P )
−−−−→ e(P )
(1.4) Structure of E(P ). There is a natural P -action on E(P ), with AP acting
trivially, projecting to the action of P on e(P ). We know that e(P ) is homogeneous
under 0P (as in [BS, 1.1]), isomorphic to P/AP , which contains KP . We see that
E(P ) is isomorphic to the homogeneous vector bundle on e(P ) determined by the
representation of KP on E.
(1.5) Compatibility. For Q ⊂ P , there is a canonical embedding of E(P ) in E(Q),
given as follows. As in [BS, 4.3], write AQ = AP × AQ,P , with AQ,P ⊂ AQ de-
noting the intersection of the kernels of the simple roots for AP . Then there is an
embedding
E(P ) = E ×AP AP ≃ (E ×AQ,P AQ,P )×AP AP
⊂ (E ×AQ,P AQ,P )×AP AP ≃ E ×AQ AQ = E(Q).
Moreover, this projects to X(P ) ⊂ X(Q) via π(Q).
(1.6) Hereditary property. If Q ⊂ P again, one can view E(Q) as part of the
boundary of E(P ), in the same way that e(Q) is part of the boundary of e(P ). This
is achieved by considering the geodesic action of AQ,P on E(P ) (AP acts trivially),
and carrying out the analogue of (1.3). Thus, E(Q) ≃ E(P )/AQ,P = E(P )/AQ.
(1.7) The bundle with corners. Using the identifications given in (1.5), we recall
that one puts
(1.7.1) X =
X(P ) =
e(P ),
with P ranging over all parabolic subgroups of G/Q, including the improper one
(G itself). With X endowed with the weak topology from the X(P )’s, this is the
manifold-with-corners construction of Borel-Serre for X (see [BS, §7]). As such, it
has a tautological stratification, with the e(P )’s as strata.
We likewise put E =
E(P ), with incidences given by (1.5), and endow it with
the weak topology. There is an obvious projection onto X. Then E is a vector
bundle over X that is stratified by the homogeneous bundles E(P ), given as in
(1.4).
(1.8) Quotient by arithmetic groups. We can see that G(Q) acts as vector bundle
automorphisms on E over its action as homeomorphisms of X (given in [BS, 7.6]);
also, as it is so for X , the action on E of any neat arithmetic subgroup Γ of G(Q)
is proper and discontinuous (cf. [BS, 9.3]). Then EΓ = Γ\E is a vector bundle over
MBSΓ = Γ\X .
Let ΓP = Γ∩P . The action of ΓP (which is contained in
0P of (1.4)) commutes
with the geodesic action of AP . The faces of EΓ are of the form E
′(P ) = ΓP \E(P ),
and are vector bundles over the faces e′(P ) = ΓP \e(P ) of M
Γ . By reduction
theory [BS, §9] (but see also [Z5,(1.3)]), there is a neighborhood of e′(P ) in MBSΓ
on which geodesic projection πP (from (1.3.3)) descends. The same is true for π̃P
and E′(P ) (also from (1.3.3)).
(1.9) The reductive Borel-Serre compactification. We recall the quotient space
XRBS of X. With X given as in (1.7) above, one forms the quotient
(1.9.1) XRBS =
XP , (where XP = UP \e(P )) ,
where UP is, as in (1.2), the unipotent radical of P , and endows it with the quotient
topology from X. Because UQ ⊃ UP whenever Q ⊂ P , X
RBS is a Hausdorff space
(see [Z1,(4.2)]). There is an induced action of G(Q) on XRBS , for which (1.9.1)
is a G(Q)-equivariant stratification; G(Q) takes the stratum XP onto that of a
conjugate parabolic subgroup, with P (Q) preserving XP . For any arithmetic group
Γ ⊂ G(Q), one has a quotient mapping
(1.9.2) q :MBSΓ →M
Γ = Γ\X
RBS =
M̂P ,
with M̂P = ΓP \XP .
(1.10) Descent of E to XRBS. Analogous to the description of X in (1.7), we have
(1.10.1) E =
E(P ) =
E(P ),
and the corresponding quotient
(1.10.2) ERBS =
(UP \E(P )) .
We verify that ERBS is a vector bundle on XRBS . Since {X(P )} is an open cover
of X (see (1.7.1)), it suffices to verify this for E(P )→ X(P ) for each P separately.
Note that UP acts on E(P ) by the formula: u · (p, e, a) = (up, e, a), and this
commutes with the action of KP ·AP . It follows that there is a canonical projection
(1.10.3) UP \E(P )→ UP \X(P ).
This gives a vector bundle on UP \X(P ) because UP ∩ (KP ·AP ) = {1}.
Let XRBS(P ) be the image of X(P ) in XRBS , and ERBS(P ) be the image of
E(P ) in ERBS . These differ from (1.10.3), for the UP quotient there is too coarse
(for instance, there are no identifications on X or E in ERBS → XRBS). Rather,
the pullback of (1.10.3) to XRBS(P ) is ERBS(P ).
When Γ is a neat arithmetic group, ERBSΓ = Γ\E
RBS is a vector bundle on
MRBSΓ . This is verified in the same manner as (1.8).
2. Lp-cohomology
By now, the notion of Lp-cohomology, with 1 ≤ p ≤ ∞, is rather well-established.
The case of p = ∞, though, is visibly different from the case of finite p, and was
neglected in [Z4]. Morally, Theorem 1 is about L∞-cohomology, but for technical
reasons we will have to settle for Lp-cohomology for large finite p. It is our first
goal to prove Theorem 1.
(2.1) Preliminaries. Let M be a C∞ Riemannian manifold. For any C∞ dif-
ferential form φ on M , its length |φ| is a non-negative continuous function on M .
This determines a semi-norm:
(2.1.1) ||φ||p =
|φ(x)|p dVM (x)
p if 1 ≤ p <∞;
sup {|φ(x)| : x ∈M} if p =∞,
where dVM (x) denotes the Riemannian volume density of M . One says that φ is
Lp if ||φ||p is finite.
(2.1.2) Definitions. Let w be a positive continuous real-valued function on the
Riemannian manifold M .
i) The [smooth] Lp de Rham complex with weight w is the largest subcomplex of
the C∞ de Rham complex of M consisting of forms φ such that wφ is Lp, viz.
(2.1.2.1) A•(p)(M ;w) = {φ ∈ A
•(M) : wφ and wdφ are Lp}.
ii) The [smooth] Lp-cohomology of M with weight w is the cohomology of
(M ;w). It is denoted H•
(M ;w).
We note that in the above, there is a difference with the notation used elsewhere:
for p 6=∞, w might be replaced with w
p in (2.1.2.1).
When w = 1, one drops the symbol for the weight. Note that the complex
depends on w only through rates of the growth or decay of w at infinity. When
M has finite volume, there are inclusions A•
(M) →֒ A•
(M) whenever 1 ≤ p <
p′ ≤ ∞. The preceding extends to metrized local systems (cf. [Z1, §1]). Smooth
functions are dense in the Banach space Lp for 1 ≤ p <∞, but not in L∞.
We next recall the basic properties of Lp-cohomology. Let M be a compact
Hausdorff topological space that is a compactification ofM . One defines a presheaf
on M by the following rule (cf. [Z4, 1.9]): to any open subset V of M , one assigns
(V ∩M ;w). Because M is compact (see (2.1.5, ii) below), the associated sheaf
(M ;w) satisfies
(2.1.3) A•(p)(M ;w)
−→Γ(M,A•(p)(M ;w)).
It follows from the definition that whenever q : M
→ M is a morphism of com-
pactifications of M , one has for all p:
(2.1.4) q∗A
(p)(M
;w) ≃ A•(p)(M ;w).
(2.1.5) Remarks. i) It is easy to see that the complex A•
(M ;w) consists of fine
sheaves if and only if for every covering of MBS there is a partition of unity subor-
dinate to that covering consisting of functions f whose differential lies in A1
i.e., |df | is a bounded function on M . Thus, (2.1.4) is for q∗ (as written), not for
Rq∗ in general.
ii) Note that in general, the space of global sections of A•
(M ;w), defined in the
obvious way (or equivalently the restriction of A•
(M ;w) toM) is A•
(p),loc
(M ;w) =
A•(M). Without a compact boundary, there is no place to store the global bound-
edness condition.
The following fact makes for a convenient simplification:
(2.1.6) Proposition. LetM be the interior of a Riemannian manifold-with-corners
M (i.e., the metric is locally extendable across the boundary). Let A
(p)(M ;w) be the
sub-complex of A•
(M ;w) consisting of forms that are also smooth at the boundary
of M . Then the inclusion
(p)(M ;w) →֒ A
(p)(M ;w)
is a quasi-isomorphism. �
In other words, one can calculate H•
(M ;w) using only forms with the nicest
behavior along ∂M . Moreover, A
(p)(M ;w) admits a simpler description; for that
and the proof of (2.1.6), see (2.3.7) and (2.3.9) below.
(2.2) The prototype. We compute a simple case of Lp-cohomology, one that will
be useful in the sequel.
(2.2.1) Proposition [Z4, 2.1]. Let R+ denote the positive real numbers, and t the
linear coordinate from R. For a ∈ R, let wa(t) = e
at. Then
i) H0
(R+;wa) ≃
0 if a > 0,
C if a ≤ 0.
ii) H1
(R+;wa) = 0 for all a 6= 0.
Proof. Again, we carry this out here only for p =∞. First, (i) is obvious: it is just
an issue of whether the constant functions satisfy the corresponding L∞ condition.
To get started on (ii), proving that a complex is acyclic can be accomplished by
finding a cochain homotopy operator B (lowering degrees by one), such that φ =
dBφ + Bdφ. For the cases at hand (1-forms on R+), this equation reduces to
φ = dBφ.
When a < 0, one takes
(2.2.2) B(φ)(t) = −
g(x)dx
when φ = g(t)dt (placing the basepoint at ∞ is legitimate, as g decays exponen-
tially). We need to check that (2.2.2) lies in the L∞ complex. By hypothesis,
|g(t)| ≤ Cw−a(t)
for some constant C. This implies that
|B(φ)(t)| ≤
|g(x)|dx ≤ C
w−a(x)dx ∼ w−a(t)
as t→∞. In other words, B(φ)(t)wa(t) ∼ 1, which is what we wanted to show.
When a > 0, one takes instead
(2.2.3) B(φ)(t) =
g(x)dx,
and shows that |B(φ)|(t) ∼ w−a(t), yielding the same conclusion about B(φ) as
before. �
(2.2.4) Remark. One can see that for a = 0, one is talking about H1
(R+), which
is not even finite-dimensional (cf. [Z1, (2.40)]); H1
(R+) contains the linearly in-
dependent cohomology classes of t−νdt, for all 0 ≤ ν ≤ 1. What was essential in
the proof of (2.2.1) was that wa and one of its anti-derivatives had equal rates of
growth or decay when a 6= 0. That is, of course, false for a = 0.
(2.3) Further properties of Lp-cohomology. We begin with
(2.3.1) Proposition (A Künneth formula for Lp-cohomology). Let I be the unit
interval [0, 1], with the usual metric. Then for any Riemannian manifold N and
weight w, the inclusion π∗ : A•
(N ;w) →֒ A•
(I×N ; π∗w) is a quasi-isomorphism;
H•(p)(I ×N ; π
∗w) ≃ H•(p)(N ;w).
Proof. The argument is fairly standard. The formula (2.2.3) defines an operator
on forms on I. Because I has finite length, one has now
(2.3.2) φ = Hφ+ dBφ+Bdφ,
where H is—well—harmonic projection: zero on 1-forms, mean value on 0-forms.
The differential forms on a product of two spaces decompose according to bidegree.
On I ×N , denote the bidegree by (eI , eN ) (thus, for a non-zero form, eI ∈ {0, 1}).
The exterior derivative on I×N can be written as d = dI+σIdN , where σI is given
by (−1)eI . The operators in (2.3.2) make sense for Lp forms on I ×N , taking, for
each q, forms of bidegree (1, q) to forms of bidegree (0, q), and we write them with
a subscript “I”; thus, we have the identity
(2.3.3) φ = HIφ+ dIBIφ+BIdIφ.
It is clear that BIφ is L
p whenever φ is. Note that σI anticommutes with BI . We
can therefore write (2.3.3) as
φ = HIφ+ dBIφ− σIdNBIφ+BIdφ−BIσIdNφ(2.3.4)
= (HIφ+ dBIφ+BIdφ)− (σIdNBIφ+BIσIdNφ).
Since σI and dN commute, the subtracted term equals (σIBI +BIσI)dNφ = 0, so
(2.3.4) is just φ = (HIφ + dBIφ + BIdφ). This implies first that dBIφ is L
p and
then our assertion. �
We next use a standard smoothing argument in a neighborhood of 0 ∈ R. To
avoid unintended pathology, we consider only monotonic weight functions w. Given
a smooth function ψ on R of compact support, let
(2.3.5) (Ψf)(t) = (ψ ∗ f)(t) =
ψ(x)f(t− x)dx =
ψ(t− x)f(x)dx,
defined for those t for which the integral makes sense. The discussion separates
into two cases:
i) w(t) is a bounded non-decreasing function of t. In this case, take ψ to be
supported in R−.
ii) Likewise, when w(t) blows up as t → 0+ take ψ to be supported in R+, and
set f(x) = 0 for x ≤ 0.
(2.3.6) Lemma. If f ∈ Lp(R+, w) (and ψ is chosen as above), then Ψf is also in
Lp(R+, w).
Proof. For p <∞, see [Z4, 1.5]. When p =∞, we consider each of the above cases.
In case (i), we have:
w(t)Ψf(t) =
ψ(t− x)w(t)f(x)dx =
ψ(t− x)w(x)f(x){w(t)w(x)−1}dx.
By hypothesis, the integral involves only those x for which t < x, and there
w(t)w(x)−1 ≤ 1. It follows that w(t)Ψf(t) is uniformly bounded. In case (ii),
when w(t) blows up as t→ 0+ the argument is similar and is left to the reader. �
We use (2.3.6) to prove:
(2.3.7) Proposition. With w restricted as above, let A
(p)(I;w) denote the sub-
complex of A•
(I;w) consisting of forms that are smooth at 0. Then the inclusion
(p)(I;w) →֒ A
(p)(I;w)
is a quasi-isomorphism, with Ψ providing a homotopy inverse.
Proof. There is a well-known homotopy smoothing formula, which is at bottom a
variant of (2.3.2). We use the version given in [Z4,1.5], valid on the level of germs
at 0:
1−Ψ = dE + Ed, E = (1−Ψ)B,
with B as above. Our assertions follow immediately. �
The behavior of w forces the value f(0) of a function f ∈ L∞(I;w)∩A(I) to be
0 precisely in case (ii) above. Thus we have:
(2.3.8) Corollary. Write I for the closed interval [0, 1]. For the two cases pre-
ceding (2.3.6),
A•(∞)(I;w) ≈
A•(I) in case (i),
A•(I, 0) in case (ii).
There are several standard consequences and variants of (2.3.7) in higher dimen-
sion. The simplest to state are (2.1.6) and its corollary; we now give the latter:
(2.3.9) Proposition. LetM be the interior of a Riemannian manifold-with-corners
M , and let A•
(M) be the subcomplex of A•
(M) consisting of forms that are
smooth at the boundary. Then the inclusion
A•(p)(M) →֒ A
(p)(M)
induces an isomorphism on cohomology. Thus the Lp-cohomology of M can be
computed as the cohomology of A•
(M), i.e., H•
(M) ≃ H•(M). �
Finally, we will soon need the following generalization of (2.3.1):
(2.3.10) Proposition. Let wM and wN be positive functions on the Riemannian
manifolds M and N respectively. Suppose that on the Riemannian product M ×N ,
one has in the sense of operators on Lp that d = dM ⊗ 1N + σM ⊗ dN , and that
(N ;wN) is finite-dimensional. Then
H•(p)(M ×N ;wM × wN ) ≃ H
(p)(M ;wM)⊗H
(p)(N ;wN ).
Remarks. i) The condition on M × N is asserting that the forms on M ×N that
have separate Lp exterior derivatives along M and along N are dense in the graph
norm (cf. [Z1, pp.178–181] for some discussion of when this condition holds.)
ii) When p = 2, the above proposition recovers only a special case of what is
in [Z1, pp.180–181]; however, the full statement of the latter does generalize to all
values of p, by a parallel argument.
Proof of (2.3.10). The argument is similar to what one finds in [Z1,§2], which
is for the case p = 2, though we cannot use orthogonal projection here. Let
h• = h•p(N ;wN) be any space of cohomology representatives for H
(N ;wN ); by
hypothesis, h• is a finite-dimensional Banach space. It suffices to show that the
inclusion
(2.3.10.1) A•(p)(M ;wM )⊗ h
p(N ;wN)
→֒ A•(p)(M ×N ;wM × wN )
induces an isomorphism on cohomology.
For each i, let Zi denote the closed forms in Ai = Ai
(N ;wN). Then D
dAi−1 is a complement to hi in Zi; it is automatically closed because of the finite-
dimensionality of hi. By the Hausdorff maximal principle, there is a closed linear
complement Ci to Zi in Ai (canonical complements exist when p = 2). Then the
open mapping theorem of functional analysis (applied for the Lp graph norm on
Ai), gives that the direct sum of Banach spaces,
(2.3.10.2) hi ⊕Di ⊕ Ci,
is boundedly isomorphic to Ai. With respect to this decomposition of Ai, dN breaks
into the 0-mapping on Zi and an isomorphism di : Ci → Di+1.
We can now obtain a cochain homotopy for A•. Let Bi denote the inverse of
di−1, and B and d the respective direct sums of these. One calculates that dB+Bd
is equal to 1− q, where q denotes projection onto h• with respect to (2.3.10.2).
Adapting this formula to M × N runs a standard course. First, B defines an
operator BN = 1M ⊗ B on M × N , and likewise does q. We have the identity
1− qN = dNBN +BNdN . Noting that dN commutes with σM and that σ
M = 1M ,
we obtain
(1− qN ) = σMdN (σMBN ) + (σMBN )σMdN ,
and likewise dM (σMBN ) + (σMBN )dM = 0. Adding, we get 1 − qN = dB̃ + B̃d,
with B̃ = σMBN , and this gives what we wanted to know about (2.3.10.1), so we
are done. �
3. Lp cohomology on the reductive Borel-Serre compactification
In this section, we determine the cohomology sheaves of A•
(MRBS) for large
finite values of p, and compare the outcome to that of related calculations.
(3.1) Calculations for MRBS , and the proof of Theorem 1. We first observe that
(MRBS) is a complex of fine sheaves, for the criterion of (2.1.5, i) was verified
in [Z1]. (The analogous statement on MBS is false unless M is already compact;
indeed, this is why the space MRBS was introduced.)
Let y ∈ UP \e(P ) ⊂ M
RBS . The issue is local in nature, so it suffices to work
with q̃ : MBSΓUP
→ XRBS , and therefore we lift y to ỹ ∈ XRBS . The fiber q̃−1(ỹ) is
the compact nilmanifold NP = ΓUP \UP . Since NP is compact, neighborhoods of ỹ
in XRBS give, via q̃−1, a fundamental system of neighborhoods of NP in M
As in [Z1,(3.6)], the intersection with MΓUP of such a neighborhood is of the
(3.1.1) A+
× V ×NP ,
where A+P ≃ (R
+)r(P ) and V is a coordinate cell on M̂P (notation as in (1.9)). After
taking the exponential of the A+
-variable, the metric is given, up to quasi-isometry,
(3.1.2)
dt2i + dv
e−2αduα,
where α runs over the roots in UP . By the Künneth formula (2.3.1), we may replace
V by a point in (3.1.2); we are reduced to determining H•
(A+P ×NP ), where the
metric is
dt2i +
e−2αduα.
The means of computing this runs parallel to the discussion in [Z1,(4.20)]. We
consider the inclusions of complexes
(3.1.3)
A•(p)(A
P ;wβ)⊗H
β(uP ,C)
A•(p)(A
P ;wβ)⊗ ∧
β(uP )
≃ A•(p)(A
P ×NP )
UP →֒ A•(p)(A
P ×NP ).
Here, uP denotes the Lie algebra of UP , and
(3.1.4) wβ(a) = a
pβa−δ = apβ−δ = a
p(β− δ
(ai = e
where δ denotes the sum of the positive Q-roots (cf. (3.1.9, ii) below). We can see
that the contribution of δ (which enters because of the weighting of the volume
form of NP ) is non-zero yet increasingly negligible as p→∞.
The second inclusion in (3.1.3) is that of the “UP -invariant” forms. Note that
this reduces considerations onNP to a finite-dimensional vector space, viz. ∧
•(uP )
Here, one is invoking the isomorphism
(3.1.5) H•(NP ) ≃ H
•(uP , C)
for nilmanifolds, which is a theorem of Nomizu [N]. The exterior algebra decomposes
into non-positive weight spaces for aP , which we write as
(3.1.6) ∧•(uP )
∧•β(uP )
The first inclusion in (3.1.3) is given by Kostant’s embedding [K,(5.7.4)] ofH•(uP ,C)
in ∧•(uP )
∗ as a set of cohomology representatives, and it respects aP weights. Our
main Lp-cohomology computation is based on:
(3.1.7) Proposition. For all p ≥ 1, the inclusions in (3.1.3) are quasi-isomorphisms.
Proof. This is asserted in [Z1,(4.23),(4.25)] for the case p = 2. The proof given
there was presented with p = 2 in mind, though there is no special role of L2 in it
(cf. [Z3,(8.6)]). We point out that [Z1,(4.25)] is about the finite-dimensional linear
algebra described above, and that the proof (4.23) of [Z1] goes through because the
process of averaging a function over a circle (hence a nilmanifold, by iteration) is
bounded in Lp-norm. As such, one sees rather easily that the proof carries over
verbatim for general p, and (3.1.7) is thereby proved. �
Remark. We wish to point out and rectify a small mistake in the argument in
[Z1, §4], one that “corrects itself”. It is asserted that the second terms in (4.37) and
(4.41) there vanish by Uj−1-invariance. This is false in general. However, the two
expressions actually differ only by a sign, and they cancel, yielding the conclusion
of (4.41).
We next show how (3.1.7) yields the determination of H•
(A+P × NP ). We
may use the first complex in (3.1.3) for this purpose. The weights in (3.1.6) are
non-positive, and (3.1.4) shows that once p is sufficiently large, wβ blows up expo-
nentially in some direction whenever β 6= 0, and decays exponentially when β = 0.
Applying (2.2.1) and the Künneth theorem, we obtain:
(3.1.8) Corollary. For sufficiently large p < ∞, H•
(A+P ×NP ) ≃ H
0(uP ,C) ≃
(3.1.9) Remark. i) We can specify what “sufficiently large” means, using (3.1.4).
Write δ as a (non-negative) linear combination of the simple Q-roots: δ =
Then we mean to take p > max{cβ}.
ii) When p = ∞, one runs into trouble with the infinite-dimensionality of the
unweighted H1
(R+) (see (2.2.4)). By using instead large finite p, we effect a
perturbation away from the trivial weight, thereby circumventing the problem.
There is a straightforward globalization of (3.1.8), which we now state:
(3.1.10) Theorem. For sufficiently large p, the inclusion
CMRBS → q∗A
(p)(M
BS) ≃ A•(p)(M
is a quasi-isomorphism. �
From this follows Theorem 1:
(3.1.11) Corollary. For sufficiently large finite p, H•
(M) ≃ H•(MRBS).
(3.2) An example (with enhancement). Take first G = SL(2). Then M is a
modular curve. There are only two distinct interesting compactifications (those in
(0.2)): one is MBS , and the other is MRBS (which is homeomorphic to MBB and
MΣ). A deleted neighborhood of a boundary point (cusp) of M
BB is a Poincaré
punctured disc ∆∗R = {z ∈ C : 0 < |z| < R}, with R < 1, with metric given in
polar coordinates by ds2 = (r | log r|)−2(dr2 + (rdθ)2). Because R < 1, the metric
is smooth along the boundary circle |z| = R. Setting u = log | log r| converts the
metric to ds2 = du2+e−2udθ2 (recall (3.1.3)). One obtains from (2.3.1) and (3.1.7):
(3.2.1) Proposition. Write ∆R for ∆
R ∪ {0}. Then for the Poincaré metric on
H•(p)(∆
R) ≃ H
•(∆R) ≃ C whenever 1 < p <∞. �
(3.2.2) Remark. When p = 1, H2
(∆∗R) is infinite-dimensional, as is H
(∆∗R);
this follows from (2.2.4) and (3.1.7). By using a Mayer-Vietoris argument, in the
same manner as [Z1,§5], we get that when M is a modular curve, we see that
(M) is likewise infinite-dimensional. Thus the assertion in (3.1.11) fails to
hold for p =∞, already when G = SL(2).
Using the Künneth formula (2.3.10), it is easy to obtain the corresponding as-
sertion for (∆∗R)
(3.2.3) Corollary. For the Poincaré metric on (∆∗R)
H•(p)((∆
n) ≃ H•(∆nR) ≃ C whenever 1 < p <∞. �
Now, let M be an arbitrary locally symmetric variety. The smooth toroidal
compactifications MΣ are constructed so that they are complex manifolds and the
boundary is a divisor with normal crossings onMΣ. The local pictures ofM →֒MΣ
are (∆∗)k × ∆n−k →֒ ∆n, for 0 ≤ k ≤ n. The invariant metric of M is usually
not Poincaré in these coordinates, not even asymptotically. However, it is easy to
construct other metrics which are. We will use a subscript “P” to indicate that
one is using such a metric instead of the invariant one. We note that such a metric
depends on the choice of toroidal compactification. The global version of (3.2.3)
follows by standard sheaf theory:
(3.2.4) Proposition. For a metric on M that is Poincaré with respect to MΣ,
H•(p),P(M) ≃ H
•(MΣ) whenever 1 < p <∞. �
The above proposition actually gives a reinterpretation of the method in [Mu].
There, Mumford decided to work in the rather large complex of currents that also
gives the cohomology of MΣ. However, he shows that the connection and Chern
forms involved are “of Poincaré growth”, and that is equivalent to saying that
they are L∞ with respect to any metric that is asymptotically Poincaré near the
boundary of MΣ. One thereby sees that his argument for comparing Chern forms
([Mu, p.243], based on (4.3.4) below) in the complex of currents actually takes place
in the subcomplex of Poincaré L∞ forms on M .
Remark. For a convenient exposition of the growth estimates in the latter, see
[HZ2,(2.6)]. Since there is in general no morphism of compactifications between
MRBS and MΣ, the reader is warned that the comparison of their boundaries is a
bit tricky (see [HZ1,(1.5),(2.7)] and [HZ2,(2.5)]).
(3.3) On defining morphisms via Lp-cohomology. We give next an interesting
consequence of Theorem 1. The space M has finite volume, so there is a canonical
morphism (see (2.1))
(3.3.1) H•(p)(M)→ H
(2)(M)
whenever p > 2. For p sufficiently large, the left-hand side of (3.3.1) is naturally
isomorphic to H•(MRBS). For p = 2, there is an analogous assertion: by the
Zucker conjecture, proved in [L] and [SS] (see [Z2]), the right-hand side is naturally
isomorphic to IH•m(M
BB), intersection cohomology with middle perversity m of
[GM1]. These facts transform (3.3.1) into the diagram
(3.3.2)
H•(MRBS)
H•(MBB) → IH•m(M
In other words,
(3.3.3) Proposition. The mapping in (3.3.1) defines a factorization of the canon-
ical mapping
H•(MBB) −→ IH•m(M
through H•(MRBS).
A related assertion had been conjectured by Goresky-MacPherson and Rapoport,
and was proved recently by Saper:
(3.3.4) Proposition. Let h : MRBS → MBB be the canonical quotient mapping.
Then there is a quasi-isomorphism
Rh∗IC
RBS ,Q) ≈ IC•m(M
BB,Q),
where IC denotes sheaves of intersection cochains.
This globalizes to an isomorphism IH•m(M
RBS ,Q)
−→IH•m(M
BB,Q), which un-
derlies (3.3.3), enlarging the triangle into a commutative square defined over Q:
H•(MRBS) −−−−→ IH•m(M
H•(MBB) −−−−→ IH•m(M
4. Chern forms for vector bundles on stratified spaces
In this section, we will treat the de Rham theory for stratified spaces that will
be needed for the proof of Theorem 2. We also develop the associated treatment of
Chern classes for vector bundles.
(4.1) Differential forms on stratified spaces. Let Y be a paracompact space
with an abstract prestratification (in the sense of Mather) by C∞ manifolds. Let
S denote the set of strata of Y . If S and T are strata, one writes T ≺ S whenever
T 6= S and T lies in the closure S of S.
The notion of a prestratification specifies a system C of Thom-Mather control
data (see [GM2, p. 42],[V1],[V2]), and that entails the following. For each stratum
S of Y , there is a neighborhood NS of S in Y , a retraction πS : NS → S, and a
continuous “distance function” ρS : NS → [0,∞) such that ρ
(0) = S, subject to:
(4.1.1) Conditions. Whenever T � S, put NT,S = NT ∩ S, πT,S = πT |NT,S ,
and ρT,S = ρT |NT,S . Then:
i) πT (y) = πT,S(πS(y)) whenever both sides are defined, viz., for y ∈ NT ∩
−1NT,S; likewise ρT (y) = ρT,S(πS(y)).
ii) The restricted mapping πT,S×ρT,S : N
T,S → T×R
+, where N◦T,S = NT,S−T ,
is a C∞ submersion.
(The above conditions will be relaxed after (4.1.4) below.) A prestratified space is,
thus, the triple (Y,S, C).
Let Y ◦ denote the open stratum of Y . One understands that when S = Y ◦,
one has NS = Y
◦, πS = 1Y ◦ and ρS ≡ 0. From (4.1.1), it follows that for all
S ∈ S, πT,S|N◦
is a submersion; moreover, the closure S of S in Y is stratified by
{T ∈ S : T � S}, and CS = {(πT,S, ρT,S) : T ≺ S} is a system of control data for
We also recall the following (see [V2,Def. 1.4]):
(4.1.2) Definition. A controlled mapping of prestratified spaces, f : (Y,S, C) →
(Y ′,S′, C′), is a continuous mapping f : Y → Y ′ satisfying:
i) If S ∈ S, there is S′ ∈ S′ such that f(S) ⊆ S′, and moreover, f |S is a smooth
mapping of manifolds.
ii) For S and S′ as above, f ◦ πS = πS′ ◦ f in a neighborhood of S.
iii) For S and S′ as above, ρS′ ◦ f = ρS in a neighborhood of S.
Let j : Y ◦ →֒ Y denote the inclusion. A subsheaf A•Y,C of j∗A
Y ◦ , the complex of
C-controlled C-valued differential forms on Y , is the sheafification of the following
presheaf: for V open in Y , put
(4.1.3) A•Y,C(V ) = {ϕ ∈ A
•(V ∩ Y ◦) : ϕ|NS∩V∩Y ◦ ∈ imπ
S for all S ∈ S}.
(4.1.4) Remark. From (4.1.1, i), one concludes that the condition in (4.1.3) for T
implies the same for S whenever S ≻ T , as (πT )
∗ϕ = (πS)
∗(πT,S)
We observe that the definition of A•Y,C is independent of the distance functions
ρS . Indeed, all that we will need from the control data for most purposes is the
collection of germs of πS along S. We term this weak control data (these are the
equivalence classes implicit in [V2,Def. 1.3]). In this spirit, one has the notion of a
weakly controlled mapping, obtained from (4.1.2) by discarding item (iii); cf. (5.2.2).
The main role that ρS plays here is to specify a model for the link of S:
(4.1.5) LS = π
(s0) ∩ ρ
for any s0 ∈ VS and sufficiently small ε > 0, but the link is also independent of C;
besides, we will not need that notion in this paper.
The following is well-known:
(4.1.6) Lemma. Let Y be a space with prestratification. For any open covering
V of Y , there is a partition of unity {fV : V ∈ V} subordinate to V that consists
of C-controlled functions. �
This is used in [V1, p. 887] to prove the stratified version of the de Rham theorem:
(4.1.7) Proposition. Let C be a system of (weak) control data on Y . Then the
complex A•Y,C is a fine resolution of the constant sheaf CY. �
(4.1.8) Corollary. A closed C-controlled differential form on Y determines an
element of H•(Y ). �
(4.2) Controlled vector bundles. We start with a basic notion.
(4.2.1) Definition. A C-controlled vector bundle on Y is a topological vector
bundle E, given with local trivializations for all V in some open covering V of Y ,
such that the entries of the transition matrices are C-controlled.
It follows from the definition that a C-controlled vector bundle determines a Cech
1-cocycle forV with coefficients inGL(r,A0Y,C). It thereby yields a cohomology class
in H1(Y,GL(r,A0Y,C)). The latter has a natural interpretation:
(4.2.2) Proposition. The set H1(Y,GL(r,A0Y,C)) is in canonical one-to-one cor-
respondence with the set of isomorphism classes of vector bundles E of rank r on
Y with ES = E|S smooth for all S ∈ S, together with a system {φS : S ∈ S},
{φT,S : S, T ∈ S} of germs of isomorphisms of vector bundles (total spaces) along
each T ∈ S:
i) φT : (πT )
∗ET = ET ×T NT
−→E|NT ,(4.2.2.1)
ii) φT,S : (πT,S)
∗ET = ET ×T NT,S
−→E|NT,S
whenever T ≺ S, satisfying the compatibility conditions φT = φS ◦ φT,S.
(4.2.2.2) Remark. Condition (ii) above is, of course, the restriction of (i) along S.
If we use {ET : T ∈ S}, the stratification of E induced by S, then the natural
projection ET ×T NT → ET gives weak control data for E. Thus we obtain from
(4.2.2):
(4.2.3) Corollary. A vector bundle E on Y is C-controlled (as in (4.2.1)) if and
only if E admits weak control data such that the bundle projection E → Y is a
weakly controlled mapping (as in (4.1)). �
Proof of (4.2.2). Let ξ be a 1-cocycle for the open covering V of Y , with coefficients
in GL(r,A0Y,C). Since the functions in A
Y,C are continuous, ξ determines a vector
bundle of rank r in the usual way; putting E0 for C
r, one takes
E = Eξ =
{(E0 × Vα) : Vα ∈ V}
modulo the identifications on Vαβ = Vα ∩ Vβ :
(4.2.2.3)
E0 × Vαβ →֒ E0 × Vα
1×ξαβ
E0 × Vαβ →֒ E0 × Vβ
It is a tautology that there exist isomorphisms (4.2.2.1) locally on the respective
bases (Y ◦ or S), but we want it to be specified globally.
Next, let
(4.2.2.4) VS = {V ∈ V : V ∩ S 6= ∅}, V(S) = {V ∩ S : V ∈ VS}.
Then V(S) is an open cover of S. By refining V, we may assume without loss
of generality that ξαβ ∈ im (πS)
∗ on Vαβ ∩ NS whenever Vα, Vβ ∈ VS , and write
ξαβ = (πS)
∗ ξSαβ. The bundle ES = E|S is constructed from the 1-cocycle ξ
S. Let
N ′S = NS ∩
{V : V ∈ VS}.
The relation ξ = (πS)
∗ξS onN ′S determines a canonical isomorphism φS : E|N ′S
−→(πS)
∗ES ,
for the local ones patch together; it is smooth on each stratum R ≻ S. One produces
φS,T by doing the above for the restriction of E to S, along its stratum T .
The consistency condition, φT = φS ◦ φT,S whenever T ≺ S, holds because of
(4.1.4). Replacing V by any refinement of it, only serves to make N ′S smaller, so
the germs of the pullback relations do not change. Also, we must check that the
isomorphisms above remain unchanged when we replace ξ by an equivalent cocycle.
Let ξ′αβ = ψβξαβψ
α , where ψ is a 0-cochain for V with coefficients in GL(r,A
Y,C).
Without loss of generality again, we assume that ψ is of the form (πS)
∗ψS on N ′S.
The isomorphism E(ξ′S) ≃ E(ξS) induced by ψ
S then pulls back to the same for
the restrictions of E(ξ′) and E(ξ) to N ′S, respecting the compatibilities.
Thus, we have constructed a well-defined mapping from H1(Y,GL(r,A0Y,C)) to
isomorphism classes of bundles on Y with pullback data along the strata. We wish
to show that it is a bijection.
Actually, we can invert the above construction explicitly. Given E, φT , etc., as
in (4.2.2.1), let, for each T ∈ S, VT be a covering of T that gives a 1-cocycle ξ
T for
ET (as a smooth vector bundle on T ); N
T a neighborhood of T , contained in NT , on
which the isomorphisms φT and φT,S (for all S ≻ T ) are defined; V(T ) = π
the corresponding covering of N ′S, on which (πT )
∗ξT is a cocycle giving E|N ′
. Then
{VT : T ∈ S}
is a covering of Y , such that for all V ∈ V, E|V has been trivialized.
We claim that the 1-cocycle for E, with respect to these trivializations, has
coefficients in GL(r,A0Y,C). For Vα and Vβ in the same VT , we have seen already
that ξαβ is in im(πT )
∗. Suppose, then, that T ≺ S, and that Vα ∈ VT and Vβ ∈ VS
have non-empty intersection. Then ξαβ is actually in im(πS)
∗, which one sees is a
consequence of the compatibility conditions for (4.2.2.1), and our claim is verified.
That we have described the inverse construction is easy to verify. �
(4.3) Controlled connections on vector bundles. When we speak of a connection
on a smooth vector bundle over a manifold, and write the symbol ▽ for it, we mean
foremost the covariant derivative. Then, the difference of two connections is a 0-th
order operator, given by the difference of their connection matrices with respect to
any one frame.
We can define the notion of a connection on a C-controlled vector bundle:
(4.3.1) Definition. Let E be a C-controlled vector bundle on Y . A C-controlled
connection on E is a connection ▽ on E|Y ◦ for which there is a covering V of Y
such that for each V ∈ V, there is a frame of E|V such that the connection forms
lie in A1V,C ⊗ End(E).
Remark [added]. It is more graceful to define a controlled connection so as to be in
accordance with (4.2.2.1): it is a system of connections {(ET ,▽T )}, with germs of
isomorphisms
(▽S)|NT,S = (πT,S)
▽T whenever T ≺ S.
One sees that (4.1.1) and (4.3.1) imply that a C-controlled connection on E
defines a usual connection on E|S for every S ∈ S. The next observation is evident
from the definition:
(4.3.2) Lemma. The curvature form Θ ∈ j∗(A
Y◦ ⊗ End(E|Y◦)) of a C-controlled
connection ▽ lies in A2Y,C ⊗ End(E). �
It is also obvious that A•Y,C is closed under exterior multiplication. One can thus
define for each k the Chern form ck(E,▽), a closed C-controlled 2k-form on Y , by
the usual formula:
ck(E,▽) = Pk(Θ, . . . ,Θ),
where Pk is the appropriate invariant polynomial of degree k. By (4.1.8), ck(E,▽)
defines a cohomology class in H2k(Y ).
(4.3.3) Proposition. i) Every C-controlled vector bundle E on Y admits a C-
controlled connection.
ii) The cohomology class of ck(E,▽) in H
2k(Y ) is independent of the C-controlled
connection ▽ on E.
Proof. Let {φT , φT,S : T ≺ S} be the data defining a C-controlled vector bundle, as
in (4.2.2.1), and N ′T ⊂ NT a domain for the isomorphisms involving ET . For each
T , let ▽T be any smooth connection on ET , and (πT )
T the pullback connection
on E|N ′
. Then V = {N ′T : T ∈ S} is an open covering of Y . Apply (4.1.6) to get
a C-controlled partition of unity {fT } subordinate to V. Then ▽ =
V is a
C-controlled connection on E. This proves (i).
The argument for proving (ii) is the standard one. For two connections on a
smooth manifold, such as Y ◦, there is an identity:
(4.3.4) ck(E,▽1)− ck(E,▽0) = dηk,
where
(4.3.4.1) ηk = k
Pk(ω,Θt, . . . ,Θt)dt,
ω = ▽1−▽0, ▽t = (1− t)▽0+ t▽1, and Θt denotes the curvature of ▽t. Now, if ▽0
and ▽1 are both C-controlled, one sees easily that ω and ▽t are likewise, and then
so is ηk. It follows that (4.3.4) is an identity in A
Y,C, giving (ii).
We have been leading up to the following:
(4.3.5) Theorem. Let E be a C-controlled vector bundle on the stratified space
Y . Then the cohomology class in (4.3.3, ii) gives the topological Chern class of E
in H2k(Y ); in particular, it is independent of the choice of C.
Proof. This argument, too, follows standard lines. We start by proving the assertion
when E is a line bundle L. On Y , there is the short exact exponential sequence (of
sheaves):
(4.3.5.1) 0→ ZY → A
Y,C → (A
∗ → 1.
The Chern class of L, c1(L), is then the image of any controlled Cech cocycle that
determines L, under the connecting homomorphism
(4.3.5.2) H1(Y, (A0Y )
∗) −→ H2(Y,Z).
To prove the theorem for line bundles, it is convenient to work in the double
complex C•(A•Y,C), where C
• denotes Cech cochains. It has differential D = δ + σd
(i.e., Cech differential plus a sign σ = (−1)a times exterior derivative, where a is
the Cech degree). On a sufficiently fine covering of Y we have a cochain giving L,
ξ ∈ C1((A0Y,C)
∗) (if eα is the specified frame for L on the open subset Vα of Y , one
has on Vα ∩ Vβ that eα = ξαβeβ), with δξ = 1, the connection forms ω ∈ C
0(A1Y,C),
and λ = log ξ in C1(A0Y,C). We know by (4.3.5.2) above that δλ gives c1(L). The
change-of-frame formula for connections gives δω + dλ = 0. Finally, the curvature
(for a line bundle) is Θ = dω, so we wish to show that dω and δλ are cohomologous
in the double complex. By definition, Dλ = δλ−dλ, and Dω = δω+dω = dω−dλ.
This gives δλ− dω = D(λ− ω), and we are done.
To get at higher-rank bundles, we invoke a version of the splitting principle. Let
p : F(E)→ Y be the bundle of total flags for E. As E is locally the product of Y
and a vector space, F(E) is locally on Y just Fr×Y , where Fr is a (smooth compact)
flag manifold. As such, F(E) is a stratified space that is locally no more complicated
than Y ; we take as the set of strata S̃ = p−1(S) = {p−1S : S ∈ S}. For weak control
data, we deduce it from C in the same way it is done for E (see (4.2.2.2)): we take
NFT = F(E|N ′T ), and use the natural projection F(E|N
) → F(ET ) induced by
(4.2.2.1).
It is standard that the vector bundle p∗E on F(E) decomposes (non-canonically)
into a direct sum of line bundles: p∗E =
1≤j≤r Λj . (p
∗E is canonically filtered:
Λ1 = F1 ⊂ F2 . . . Fr = p
∗E, with Λj ≃ Fj/Fj−1.) To obtain this, one starts by
taking Λ1 to be the line bundle given at each point of F(E) by the one-dimensional
subspace from the corresponding flag. Then, one splits the exact sequence
(4.3.5.3) 0→ Λ1 → p
∗E → p∗E/Λ1 → 0,
using a controlled metric on E. By that, we mean a metric that is a pullback via the
isomorphisms (4.2.2.1); these can be constructed by the usual patching argument,
using controlled partitions of unity (4.1.6). One obtains Λj , for j > 1, by recursion.
We need a little more than that:
(4.3.6) Lemma. i) The vector bundle p∗E is, in a tautological way, a controlled
vector bundle on F(E).
ii) The line bundles Λj are controlled subbundles of p
Proof. We have that p∗E = E ×Y F, and its strata are (p
∗E)FT = ET ×T FT , for
all T ∈ S. There is natural weak control data for p∗E that we now specify. By
construction, we have a retraction
(4.3.6.1)
π(p∗ET ) : (p
∗E)|NFT = (p
∗E)|F|NT ≃ E|NT ×NT F|NT → ET ×T FT = (p
∗E)FT ,
induced by the weak control data for E (and thus also F), and likewise for the
restriction to (p∗E)|NFT ,FS , when S ≻ T . These provide φFT and φFT ,FS (from
(4.2.2.1)) respectively for p∗E, and (i) is proved.
We show that Λ1 is preserved by φFT and φFT ,FS . (As before, we explain this
only for the former, the other being its restriction to the strata.) Let pT : FT → T
denote the restriction of p to FT . Since pT gives the flag manifold bundle associated
to ET , (p
TET ) contains a tautological line bundle, which we call Λ1,T . We have
from the control data that (Λ1)|NFT ≃ (φFT )
∗Λ1,T .
We claim further that (4.3.6.1) takes Λ1|NFT to Λ1,T , as desired. The explicit
formula for (4.3.6.1), obtained by unwinding the fiber products, is as follows. Let
e be in the vector space ET,t, the fiber of ET over t ∈ T , and f a point the flag
manifold of ET,t. Also, let n ∈ (πT )
−1(t). Then π(p∗ET )(e, f, n) = (e, f), which
implies (ii) for j = 1. The assertion for j > 1 is obtained recursively. �
We return to the proof of (4.3.5). Let ▽0 be the direct sum of C̃-controlled
connections on each Λj ; and take ▽1 = p
▽, where ▽ is a C-controlled connection
on E. Both ▽0 and ▽1 are C̃-controlled connections on F(E). By construction,
∗E,▽0) represents the k-th Chern class of p
∗E in H2k(F(E)). We then apply
(4.3.3, ii) to obtain that ck(p
∗E,▽1) = p
∗ck(E,▽) represents p
∗ck(E) ∈ H
2k(F(E)).
Since p∗ : H2k(Y ) −→ H2k(F(E)) is injective, it follows that ck(E,▽) represents
ck(E) in H
2k(Y ), and (4.3.5) is proved. �
5. Proofs of Theorem 2 and Conjecture B
In this section, we apply the methods of §4 in the case Y =MRBSΓ .
(5.1) Control data for a manifold-with-corners. Let Y be a manifold-with-
corners, with its open faces as strata. For each codimension one boundary stratum
S, let
φS : [0, 1]× S → Y
define the collar NS of S in Y , so that {0} × S is mapped identically onto S.
This determines partial control data (that is, without distance functions) for Y as
follows.
As NS , one takes φ([0, 1) × S), and as πS projection onto S. For a general
boundary stratum T , write
{S : S of codimension one, T ≺ S}.
Let NT =
{NS : S of codimension one, T ≺ S}; given the φS ’s above, this set
is canonically diffeomorphic to [0, 1]r × T , where r is the codimension of T . Then
NT is the subset of NT corresponding to [0, 1)
r × T , in which terms πT is simply
projection onto T .
(5.2) Compatible control data. The existence of natural (partial) control data
for MRBSΓ is, in essence, well-known, as is compatible control data for M
Γ in the
Hermitian case. We give a brief presentation of that here. This will enable us to
determine that Conjecture B is true.
The relevant notions are variants of (4.1.2).
(5.2.1) Definition (see [GM2, 1.6]). Let Y and Y ′ be stratified spaces. A proper
smooth mapping f : Y → Y ′ is said to be stratified when the following two condi-
tions are satisfied:
i) If S′ is a stratum of Y ′, then f−1(S′) is a union of connected components of
strata of Y ;
ii) Let T ⊂ Y be a stratum component as in (i) above. Then f |T : T → S
′ is a
submersion.
It follows that a stratified mapping f is, in particular, open. We assume henceforth,
and without loss of generality, that all strata are connected.
(5.2.2) Definition (cf. [V1: 1.4]). Let f : Y → Y ′ be a stratified mapping, with
(weak) control data C for Y , and C′ for Y ′. We say that f is weakly controlled if
for each stratum S of Y , the equation πS′ ◦ f = f ◦ πS holds in some neighborhood
of S (here f maps S to S′).
(5.2.3) Remark. Note that there is no mention of distance functions in (5.2.2). This
is intentional, and is consistent with our stance in (4.1).
(5.2.4) Lemma. Let f : Y → Y ′ be a stratified mapping. Given partial control
data C for Y , there is at most one system of germs of partial control data C′ for Y ′
such that f becomes weakly controlled. Such C′ exists if and only if for all strata S
of Y , there is a neighborhood of S (contained in NS) in which f(y) = f(z) implies
f(πS(y)) = f(πS(z)). �
When the condition in (5.2.4) is satisfied, one uses the formula
πS′(f(y)) = f(πS(y))
to define C′, and we then write C′ = f∗C. In the usual manner, the mapping f
determines an equivalence relation on Y , viz., y ∼ z if and only if f(y) = f(z). The
condition on C thereby becomes:
(5.2.5) y ∼ z ⇒ πS(y) ∼ πS(z) (near S).
We will use the preceding for the stratified mappings MBSΓ →M
Γ in general,
and MRBSΓ →M
Γ in the Hermitian case. The reason for bringing inM
Γ is that
it is a manifold-with-corners, and it also has natural partial control data.
The boundary strata of X , the universal cover of MBSΓ , are the sets e(P ) of
(1.3), as P ranges over all rational parabolic subgroups of G. Those of MBSΓ itself
are the arithmetic quotients e′(P ) of e(P ), with P ranging over the finite set of
Γ-conjugacy classes of such P . There are projections X → X/AP = e(P ), defined
by collapsing the orbits of the geodesic action of AP to points. This extends to a
P -equivariant smooth retraction (geodesic projection), given in (1.3.3):
(5.2.6) X(P )
−−→ X(P )/AP = e(P ).
Recall from (1.8) that there is a neighborhood of e′(P ) inMBSΓ on which geodesic
projection onto e′(P ), induced by (5.2.6), is defined. We take the restriction of this
geodesic projection over e′(P ) as the definition of πP in our partial control data C
for MBSΓ . We have been leading up to:
(5.2.7) Proposition. The quotient mapping MBSΓ →M
Γ satisfies (5.2.5).
Proof. The mappings e(P ) → e(P )/UP , as P varies, induce the mapping M
MRBSΓ . It is a basic fact ([BS, 4.3]) that for Q ⊂ P , AQ ⊃ AP and πQ ◦ πP = πQ.
This gives e(P )→ e(P )/UP . Since it is also the case that Q ⊂ P implies UQ ⊃ UP ,
we see that (5.2.5) is satisfied. �
Only a little more complicated is:
(5.2.8) Proposition. In the Hermitian case, the quotient mapping MRBSΓ →
MBBΓ satisfies (5.2.5).
Proof. When the symmetric space X is Hermitian, the P -stratum of XRBS , for
each P , decomposes as a product:
(5.2.8.1) e(P )/UP ≃ Xℓ,P ×Xh,P .
This is induced by a decomposition of reductive algebraic groups over Q:
P/UP = Gℓ,P ·Gh,P
(cf. [Mu, p. 254]). Fixing Gh, one sees that the set of Q with Gh,Q = Gh (if non-
empty) is a lattice, whose greatest element is a maximal parabolic subgroup P of
G. The lattice is then canonically isomorphic to the lattice of parabolic subgroups
R of Gℓ,P , whereby Q/UQ ≃ (R/UR)×Gh,P . (Thus Gh,Q = Gh,P . In the language
of [HZ1,(2.2)] such Q are said to be subordinate to P .)
The mapping MRBSΓ →M
Γ is induced, in terms of (5.2.8.1), by
(5.2.8.2) e(Q)/UQ → Xh,Q,
for all Q; perhaps more to the point, the terms can be grouped by lattice, yielding
(5.2.8.3) Xℓ,P ×Xh,P → Xh,P →֒ (Xh,P )
for P maximal (see [GT, 2.6.3]). One sees that (5.2.5) is satisfied. �
We have thereby reached the conclusion:
(5.2.9) Corollary. The natural partial control data for MBSΓ induces compatible
partial control data for MRBSΓ and M
Γ . �
(5.3) Conjecture B. Let EΓ be a homogeneous vector bundle on MΓ, and E
its extension to MRBSΓ from [GT] that was reconstructed in our §1. We select
as partial control data C for MRBSΓ that given in (5.2.9). It is essential that the
following hold:
(5.3.1) Proposition. ERBSΓ is a controlled vector bundle on M
Γ , with the π̃P ’s
of (1.3.3) providing the weak control data.
Proof. This is almost immediate from the construction in §1. Recall that the weak
control data for MRBSΓ consists of the geodesic projections πP , defined in a neigh-
borhood of M̂P . The vector bundle E
Γ also gets local geodesic projections π̃P ,
induced from those of EBS, that are compatible with those of MRBSΓ because of
(1.2). The same holds within the strata of these spaces, by (1.6). We see that the
criterion of (4.2.3) is satisfied. �
We proceed with a treatment of ▽GP, the connection on EΓ constructed in [GP].
For each maximal Q-parabolic subgroup of G, letMP be the corresponding stratum
of MBB ; it is a locally symmetric variety for the group Gh,P . We also use “P” to
label the strata: thus, we have for (4.1.2), πP : NP → MP , etc. Then ▽
GP can be
defined recursively, starting from the strata of lowest dimension (Q-rank zero), and
then increasing the Q-rank by one at each step.
There is, first, the equivariant Nomizu connection for homogeneous vector bun-
dles, whose definition we recall. Homogeneous vector bundles are associated bundles
of the principal K-bundle:
(5.3.2) κ : Γ\G −→MΓ.
When we write the Cartan decomposition g = k ⊕ p, we note that (5.3.2) has
a natural equivariant connection whose connection form lies in the vector space
Hom(g, k); it is given by the projection of g onto k (with kernel p). This is known
as the Nomizu connection. The homogeneous vector bundle EΓ onMΓ is associated
to the principal bundle (5.3.2) via the representation K → GL(E). The connection
induced on EΓ via k → gl(E) = End(E) is also called the Nomizu connection (of
EΓ), and will be denoted ▽
No; its connection form is denoted θ ∈ g∗ ⊗ End(E).
A K-frame for EΓ on an open subset O ⊂ MΓ is given by a smooth cross-section
σ : O → κ−1(O) of κ; the resulting connection matrix is the pullback of θ via σ∗,
an element of A1(O,End(E)).
With that stated, we can start to describe ▽GP. For any maximal Q-parabolic
P , one will be taking expressions of the form
(5.3.3) ▽P = ψPP▽
P,No +
Q,P (▽
[GP, 11.2]. Here, ▽P,No is the Nomizu connection for the homogeneous vector bundle
onMP determined by the restrictionKh,P →֒ K → GL(E), and the functions {ψ
Q � P} form a partition of unity onMP of a selected type, given in [GP, 3.5, 11.1.1];
the function ψ
P is a cut-off function for a large relatively compact open subset VQ,P
of MQ in M
P , with ⋃
VQ,P =M
and can be taken to be supported inside the neighborhood NQ,P of the partial
control data when Q 6= P .
Next, Φ∗Q,P indicates the process of parabolic induction from MQ to NQ,P , by
means of πQ,P . It is defined as follows. Fix a maximal parabolic Q and a rep-
resentation K → GL(E). The latter restricts, of course, to KQ = Kh,Q × Kℓ,Q,
but through the Cayley transform, this actually extends to a representation λ of
Kh,Q × Gℓ,Q. That allows one to define an action of all of Q on Eh,Q [GP, 10.1],
which induces a Q-equivariant mapping
(5.3.4) E = Q×KQ E −→ Gh,Q ×Kh,Q E = Eh,Q,
given by (q, e) 7→ (gh, λ(gℓ)e) for q = ughgℓ ∈ UQGh,QGℓ,Q = Q. That in turn
defines a UQ-invariant isomorphism of vector bundles homogeneous under Q:
E ≃ π∗Q(Eh,Q).
One then takes ▽GP to be ▽G in (5.3.3). Given any connection ▽h,Q on Eh,Q,
the pullback connection ▽ = π∗Q(▽
h,Q) satisfies the same relation for its curvature
form, viz.,
(5.3.5) Θ(▽) = π∗QΘ(▽
h,Q).
It follows that the Chern forms of ▽GP are controlled on MBBΓ [GP, 11.6]. On the
other hand, the connection itself is not. To proceed, weaker information about ▽GP
suffices:
(5.3.6) Proposition. With an appropriate choice of the functions ψ
, the con-
nection ▽GP is a controlled connection when viewed on MRBSΓ .
Proof. TRBS1-2 his is not difficult. Recall from (4.3.1) that the issue is the exis-
tence of local frames at each point of MRBSΓ , with respect to which the connection
matrix is controlled. For each rational parabolic subgroup Q of G, we work in the
corner X(Q). By (5.3.4), one gets local frames for E(Q), the restriction of E to
X(Q) ⊂ X, from local frames for Eh,Q. We can write E(Q) as:
(5.3.6.1) E(Q) ≃ UQ × AQ ×MQ ×KQ E.
This also provides good variables for calculations. We note that Φ∗Q,P is independent
of the UQ-variable. Likewise, ψ
can be chosen to be a function of only (a,mKQ),
constant on the compact nilmanifold fibers NP (i.e., the image of the UP -orbits).
It follows by induction that ▽GP is controlled on MRBSΓ . �
As we said, the Chern forms of ▽GP are controlled differential forms for MBBΓ ,
so are a fortiori controlled for MRBSΓ . It follows from (4.3.5) that
(5.3.7) Proposition. c•(E
GP) represents c•(E
Γ ) ∈ H
•(MRBSΓ ). �
Thereby, Conjecture B is proved.
(5.4) Theorem 2. Let ▽ctrl be any C-controlled connection on ERBSΓ , and ▽
No the
equivariant Nomizu connection on EΓ. By (4.3.5) we know that ck(▽
ctrl) represents
Γ ); we want to conclude the same for ck(▽
No). Toward that, we recall the
standard identity on M satisfied by the Chern forms:
(5.4.1) ck(▽
No)− ck(▽
ctrl) = dηk,
which is a case of (4.3.4). The following is straightforward:
(5.4.2) Lemma. i) A•
is contained in A•
(MRBSΓ ).
ii) A G-invariant form on MΓ is L
Proof. In terms of (3.1.1), a controlled differential form on MRBSΓ is one that is,
for each given P , pulled back from V ⊂ M̂P . Such forms are trivially weighted by
in the metric (3.1.2). It follows that a controlled form is locally L∞ on MRBSΓ .
This proves (i). As for (ii), an invariant form has constant length, so is in particular
L∞. �
(5.4.3) Proposition. The closed forms ck(▽
No) and ck(▽
ctrl) represent the same
class in H2k
(MΓ).
Proof. Since MRBSΓ is compact, a global controlled form on M
Γ is globally L
As such, (5.4.2) gives that the Chern forms for both ▽ctrl and ▽No are in the
complex L•
(MΓ). It remains to verify that ηk in (5.4.1) is likewise L
∞, for then
the relation (5.4.1) holds in the L∞ de Rham complex A•
(MΓ), so ck(▽
No) and
ctrl) are cohomologous in the L∞ complex.
By (4.3.4.1), it suffices to check that the difference ω = ▽No−▽ctrl is L∞. That
can be accomplished by taking the difference of connection matrices with respect
to the same local frame of ERBS , and for that purpose we use, for each Q, frames
pulled back from M̂Q. For that, it is enough to verify the boundedness for ω in
a neighborhood of every point of the boundary of MRBSΓ , and we may as well
calculate on XRBS .
Consider a point in the Q-stratum XQ of X
RBS . As in (3.1.1), we can take as
neighborhood base, intersected with X , sets that decompose with respect to Q as
(5.4.3.1) NQ × A
Q × V,
with V open in XQ. In these terms, πQ is just projection onto V . As in (3.1.2) and
(3.1.4), we use as coordinates (uα, a, v). We also decompose (see the end of (1.1)),
(5.4.3.2) E ≃ Q×KQ E ≃ UQ ×A
Q ×XQ ×KQ E.
We obtain a canonical isomorphism E ≃ π∗QEQ, with EQ a homogeneous vector
bundle on XQ. By (5.4.2, ii), the connection matrix of a connection that is pulled
back from XQ, with respect to a local frame pulled back from XQ is L
∞, so we
wish to do the same for the Nomizu connection.
First, we have:
(5.4.3.3) Lemma. Let Q̂ = Q/AQUQ and consider the diagram
Q −−−−→ Q̂
−−−−→ XQ.
Then Q ≃ Q̂×XQ X, the pullback of Q̂ with respect to πQ.
Proof. We note that both Q̂ and Q are exhibited as principal KQ-bundles. To
prove our assertion, it is simplest use the Langlands decomposition (of manifolds)
Q ≃ Q̂ × AQ × UQ to yield the decomposition X ≃ XQ × AQ × UQ (cf. (5.4.3.1)).
Q̂×XQ X ≃ Q̂×AQ × UQ ≃ Q. �
It follows that if σ : O ⊆ XQ → Q̂ gives a local KQ-frame, then σ̃ : π
Q (O) ⊆
X → Q ≃ Q̂ ×XQ X , defined by σ̃(x) = (σ(πQ(x)), x), gives the pullback frame
π∗Qσ. In other words, π
Qσ takes values in the principal KQ-bundle Q→ X that is
the restriction of structure group of (5.3.2) from K to KQ.
Let ▽No be the Nomizu connection on E . Recall that this is determined by
(5.4.3.4) TX
−→ g −→ k −→ End(E).
As such, ▽No is not a KQ-connection. However, a frame for the restriction to X of
the canonical extension E can be taken to be of the form σ̃ as above (cf. (1.10)). It
follows that for x ∈ π−1
(O), the Nomizu connection is given by
(5.4.3.5) TX,x
→֒ q −→ k −→ End(E),
where q denotes the Lie algebra of Q. This is a mapping that is of constant norm
along the fibers of πQ. It follows that the connection matrix is L
∞. Therefore, we
have:
(5.4.3.6) Proposition. The connection difference ω is L∞. �
This finishes the proof of (5.4.3).
(5.4.4) Remark. The reader may find it instructive to compare, in the case of
G = SL(2), the above argument to the one used in [Mu, pp. 259–260]. The two
discussions, seemingly quite different, are effectively the same.
We now finish the proof of Theorem 2 by demonstrating:
(5.4.5) Proposition. ck(▽
No) and ck(▽
ctrl) represent the same class in H2k(MRBSΓ ).
Proof. Because MΓ has finite volume, there is a canonical mapping
H•(∞)(MΓ)→ H
(p)(MΓ)
for all p (see (3.3.1)). It follows from (5.4.3) that ck(▽
No) and ck(▽
ctrl) represent
the same class in H2k
(MΓ) for all p. Taking p sufficiently large, we apply Theo-
rem 1 (i.e., (3.1.11)) to see that ck(▽
No) and ck(▽
ctrl) represent the same class in
H2k(MRBSΓ ). �
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