Chern classes of automorphic vectorbundles

Mark Goresky^{1} and William Pardon ^{2}

1. Introduction

1.1. Suppose X is a compact n-dimensional complex manifold. Each partition I = {i_{1}, i_{2}, …, i_{r}} of n corresponds to a Chern number c^{I}(X) = ϵ(c^{i_{1}}(X)∪c^{i_{2}}(X)∪…∪c^{i_{r}}(X)∩[X]) ∈ ℤ where c^{k}(X) ∈ H^{2k}(X; ℤ) are the Chern classes of the tangent bundle, [X] ∈ H_{2n}(X; ℤ) is the fundamental class, and ϵ : H_{0}(X; ℤ) → ℤ is the augmentation. Many invariants of X (such as its complex cobordism class) may be expressed in terms of its Chern numbers ([Mi], [St]). During the last 25 years, characteristic classes of singular spaces have been defined in a variety of contexts: Whitney classes of Euler spaces [Su], [H-T], [Ak], Todd classes of singular varieties [BFM], Chern classes of singular algebraic varieties [Mac], L-classes of stratified spaces with even codimension strata [GM1], Wu classes of singular spaces [Go2], [GP] (to name a few). However, these characteristic classes are invariably homology classes and as such, they cannot be multiplied with each other. In some cases it has been found possible to “lift” these classes from homology to intersection homology, where (some) characteristic numbers may be formed ([BBF], [BW], [Go2],[GP], [T]).

The case of locally symmetric spaces is particularly interesting. Suppose Γ is a torsionfree arithmetic group acting on a complex n-dimensional Hermitian symmetric domain D = G/K, where G is the group of real points of a semisimple algebraic group G defined over ℚ with Γ ⊂ G(ℚ), and where K ⊂ G is a maximal compact subgroup. Then X = Γ\D is a Hermitian locally symmetric space. When X is compact, Hirzebruch’s proportionality theorem [Hr1] says that there is a number v(Γ) ∈ ℚ so that for every partition I = {i_{1}, i_{2}, …, i_{r}} of n, the Chern number satisfies c^{I}(X) = v(Γ)c^{I}(Ď), where Ď = G_{u}/K is the compact dual symmetric space (and G_{u} is a compact real form of G(ℂ) containing K).

If X = Γ\D is noncompact, it has a canonical Baily-Borel (Satake) compactification, \overline{X}. This is a (usually highly singular) complex projective algebraic variety. To formulate a proportionality theorem in the noncompact case, one might hope that the tangent bundle T X extends as a complex vectorbundle over \overline{X}, but this is false. In [Mu1], D. Mumford showed that T X has a particular extension \overline{E}_{Σ} → \overline{X}_{Σ} over any toroidal resolution τ : \overline{X}_{Σ} → \overline{X} of the Baily-Borel compactification and that for any partition I of n, the resulting Chern numbers

c^{I}(\overline{E}_{Σ}) = ϵ(c^{i_{1}}(\overline{E}_{Σ}) ∪ c^{i_{2}}(\overline{E}_{Σ}) ∪ … ∪ c^{i_{r}}(\overline{E}_{Σ}) ∩ [\overline{X}_{Σ}])  (1.1.1)

1School of Mathematics, Institute for Advanced Study, Princeton N.J. Research partially supported by NSF grants # DMS 9626616 and DMS 9900324.

2Dept. of Mathematics, Duke University, Durham N.C. Research partially supported by NSF grant # DMS 9504900.

satisfy the same equation, c^{I}(\overline{E}_{Σ}) = v(Γ)c^{I}(Ď). (The toroidal resolution \overline{X}_{Σ} is constructed in [AMRT]; it depends on a choice Σ of polyhedral cone decompositions of certain selfadjoint homogeneous cones.) Mumford also showed that if Σ′ is a refinement of Σ then there is a natural morphism f : \overline{X}_{Σ′} → \overline{X}_{Σ} and that f^{∗}(\overline{E}_{Σ}) ≅ \overline{E}_{Σ′} (hence f^{∗}c^{i}(\overline{E}_{Σ}) = c^{i}(\overline{E}_{Σ′})). Moreover, it is proven in [Har3] that the coherent sheaf τ_{∗}\overline{E}_{Σ} is independent of the choice of Σ. One is therefore led to suspect the existence of a closer relationship between the characteristic classes of the vectorbundles \overline{E}_{Σ} and the topology of the Baily-Borel compactification \overline{X}. In Theorem 11.6 and Theorem 13.2 we show that, at least for the variety \overline{X}, the original goal of constructing Chern numbers can be completely realized:

Theorem. Every Chern class c^{i}(X) has a canonical lift \overline{c}^{i} ∈ H^{2i}(\overline{X}; ℂ) to the cohomology of the Baily-Borel compactification. Moreover, if τ : \overline{X}_{Σ} → \overline{X} is any toroidal resolution of singularities then

τ^{∗}(\overline{c}^{i}) = c^{i}(\overline{E}_{Σ}) ∈ H^{2i}(\overline{X}_{Σ}; ℂ).

It follows (§13.3) that the lifts \overline{c}^{i} satisfy (1.1.1). In §15 we show that the homology image c̄^{∗} ∩ [X̄] ∈ H_{∗}(X̄) lies in integral homology and coincides with (MacPherson’s) Chern class [Mac] of the constructible function which is 1 on X and is 0 on \overline{X} − X.

1.2. Moreover, a similar result holds for any automorphic vectorbundle. Let λ : K → GL(V) be a representation of K on some finite dimensional complex vectorspace V. By [Mu1], the automorphic vectorbundle E′ = (Γ\G) ×_{K} V on X has a particular extension \overline{E}′_{Σ} over any toroidal resolution \overline{X}_{Σ}. We show that each Chern class c^{i}(E_{Γ}) has a canonical lift c̄^{i}(E′) ∈ H^{2i}(\overline{X}; ℂ) and that these lifts also satisfy the proportionality formula. Moreover, τ^{∗}(c̄^{i}(E′)) = c^{i}(\overline{E}′_{Σ}) and the image of c̄^{i}(E′) in Gr_{2i}^{W} (\overline{X}; ℂ) (the top graded piece of the weight filtration) is uniquely determined by this formula.

1.3. In §16 we consider the subalgebra H_{Chern}^{∗}(\overline{X}; ℂ) of the cohomology of the Baily-Borel compactification that is generated by the (above defined lifts of) Chern classes of certain “universal” automorphic vectorbundles, and show that

Theorem. Suppose the Hermitian symmetric domain D is a product of irreducible factors G_{i}/K_{i} (where K_{i} is a maximal compact subgroup of G_{i}), and that each G_{i} is one of the following: Sp_{n}(ℝ), U(p, q), SO(2n), or SO(2, p) with p odd or p = 2. Then there is a (naturally defined) surjection h : H_{Chern}^{∗}(\overline{X}; ℂ) → H^{∗}(Ď; ℂ) from this subalgebra to the cohomology of the compact dual symmetric space.

This result is compatible with the few known facts about the cohomology of the Baily-Borel compactification. Charney and Lee [CL] have shown, when D = Sp_{2n}(ℝ)/U(n) is the Siegel upper half space, and when Γ = Sp_{2n}(ℤ) that the “stable” cohomology of \overline{X} contains a polynomial algebra which coincides with the “stable” cohomology of the compact dual symmetric space Ď (which is the complex Lagrangian Grassmannian). It is a general fact (cf. §16) that the intersection cohomology IH^{∗}(\overline{X}; ℂ) contains a copy of H^{∗}(Ď; ℂ).

1.4. Here are the main ideas behind the proof of Theorem 11.6. In [Hr2], Hirzebruch shows that the Chern classes of Hilbert modular varieties have lifts to the cohomology of the Baily-Borel compactification because the tangent bundle has a trivialization in a neighborhood of each of the finitely many cusp points, cf. [ADS]. If X = Γ\D is a ℚ-rank 1 locally symmetric space such that \overline{X} is obtained from X by adding finitely many cusps, then the tangent bundle is not necessarily trivial near each cusp, but it admits a connection which is flat near each cusp, and so the Chern forms vanish near each cusp, hence the Chern classes lift to the cohomology of the Baily-Borel compactification. A similar argument applies to arbitrary automorphic vectorbundles (cf. [HZ1] §3.3.9).

In the general ℚ-rank 1 case, the singular set of the Baily-Borel compactification \overline{X} consists of finitely many disjoint smooth compact manifolds (rather than finitely many cusp points). If Y denotes such a singular stratum, then it admits a neighborhood π_{Y} : N_{Y} → Y such that every slice π_{Y}^{−1}(y) ∩ X is diffeomorphic to a neighborhood of a cusp similar to the kind described above. It is then possible to construct a connection ∇ (on the tangent bundle) which is “flat along each fiber π_{Y}^{−1}(y)”. (We call this the “parabolic connection”; it is constructed in Section 10.) Moreover, within the neighborhood N_{Y}, each Chern form σ^{i}(∇) is the pullback π_{Y}^{∗} (σ_{Y}^{i}) of a certain differential form σ_{Y}^{i} on Y. Differential forms with this “π-fiber property” form a complex whose cohomology is the cohomology of \overline{X}, as discussed in Section 4. So the Chern form σ^{i}(∇) determines a class \overline{c}^{i}(∇) ∈ H^{2i}(\overline{X}; ℂ). (In fact, even the curvature form satisfies the π-fiber condition.)

1.5. In higher rank cases, there are more problems. If Y_{1} ⊂ \overline{Y}_{2} ⊂ \overline{X} are singular strata of the Baily-Borel compactification, then it is possible to define a “parabolic” connection in a neighborhood N(Y_{1}) of Y_{1} whose curvature form has the π-fiber property relative to the tubular projection π_{1} : N(Y_{1}) → Y_{1}. It is also possible to construct a “parabolic” connection in a neighborhood N(Y_{2}) of Y_{2} whose curvature form has the π-fiber property relative to the tubular projection π_{2} : N(Y_{2}) → Y_{2}. However these two connections do not necessarily agree on the intersection N(Y_{1})∩N(Y_{2})∩X, nor do their curvature forms. When we patch these two connections together using a partition of unity, the curvature form of the resulting connection fails to have the π-fiber property. Nevertheless it is possible (as explained in Remark 11.3) to patch together connections of this type so as to obtain a connection whose curvature form Ω ∈ End(V) differs from a π-fiber differential form by a nilpotent element n ∈ End(V) which commutes with Ω (cf. §12.10). (Here, V is the representation of K that gives rise to the automorphic vectorbundle E_{Γ} = (Γ\G) ×_{K} V on X = Γ\G/K.) This is enough to guarantee that the Chern forms of this “patched” connection are π-fiber differential forms (cf. Lemma 6.4). A standard argument shows that the resulting cohomology class is independent of the choices that were involved in the construction.

1.6. A number of interesting questions remain. We do not know whether the results on Chern classes which are described in this paper for Hermitian symmetric spaces may be extended to the “equal rank” case (when the real rank of G and of K coincide). We do not know if the lifts \overline{c}^{i}(E′) ∈ H^{2i}(\overline{X}; ℂ) are integer or even rational cohomology classes. We do not know to what extent these lifts are uniquely determined by the properties (11.7), (13.2.1), (15.5). We do not know whether similar techniques can be applied to the Euler class of automorphic vectorbundles (when such a class exists: see §16). We do not know whether the surjection h of Theorem 16.4 admits a natural splitting. We expect that c̄^{∗}(E′) = 0 whenever the automorphic vectorbundle E′ arises from a representation λ : K → GL(V) which extends to a representation of G. If \overline{E}_{Γ}^{RBS} denotes the canonical extension ([GT] §9) of the automorphic vectorbundle E over the reductive Borel-Serre ([Z1] §4.2 p. 190, [GHM] §8) compactification ν : \overline{X}^{RBS} → \overline{X} then it is likely that ν^{∗}(c̄^{∗}(E′)) = c^{∗}(\overline{E}^{RBS}). We expect these results to have interesting applications to the study of the signature defect ([Hr2] §3, [ADS], [Mü], [St1]) and to variations of weight 1 (and some weight 2) Hodge structures ([Gr1], [Gr2]).

1.7. We would like to thank A. Borel, R. Bryant, D. Freed, R. Hain, M. Harris, R. MacPherson, and L. Saper for valuable conversations. We are grateful to S. Zucker for many suggestions and comments on an earlier draft of this paper. We are especially grateful to an anonymous referee for pointing out a mistake in an earlier version of this paper, and for his many helpful comments, suggestions, and corrections. Both authors would like to thank the Institute for Advanced Study in Princeton N.J. for its support and hospitality while this paper was prepared.

2. Control Data

2.1. A weakly stratified space W is a compact Hausdorff space with a decomposition into finitely many smooth manifolds W = Y_{1} ∪ Y_{2} ∪ … ∪ Y_{r} (called the strata of W) which satisfy the axiom of the frontier: If Y and Z are strata and if Z ∩ Ȳ ≠ ϕ then Z ⊂ Ȳ; we write Z < Y and say that Z is incident to Y. The boundary ∂Ȳ = Ȳ −Y = ∪_{Z<Y} Z of the stratum Y is the union of all strata incident to Y. If W = X̄ is the closure of a single stratum X then we say that X is the nonsingular part of W and the other strata Y < X are boundary or singular strata of W. Fix a positive real number ϵ > 0.

2.2. Definition. An ϵ-system of control data on a weakly stratified space W is a collection {T_{Y} (ϵ), π_{Y} , ρ_{Y} } indexed by the boundary strata Y ⊂ W, where

1. T_{Y} (ϵ) ⊂ W is an open subset of W containing Y,

2. T_{Y} (ϵ) ∩ T_{Z}(ϵ) = ϕ unless Y < Z or Z < Y.

3. The tubular projection π_{Y} : T_{Y} (ϵ) → Y is a retraction of T_{Y} (ϵ) to Y which is smooth on each stratum,

4. π_{Z}π_{Y} = π_{Z} whenever Z < Y and both sides of the equation are defined, 5. ρ_{Y} : T_{Y} (ϵ) → [0, ϵ) is a continuous “distance function”, with ρ_{Y}^{−1}(0) = Y, such that the

mapping (ρ_{Y} , π_{Y}) : T_{Y} (ϵ) → [0, ϵ) × Y is proper and its restriction to each stratum is a submersion,

6. ρ_{Z}π_{Y} = ρ_{Z} whenever Z < Y and both sides of the equation are defined,

For τ ≤ ϵ, write T_{Y} (τ) = ρ_{Y}^{−1}([0, τ)). By shrinking the neighborhood T_{Y} (ϵ) and scaling ρ_{Y} if necessary, we may assume that each ρ_{Y} is defined on a slightly larger neighborhood T_{Y} (ϵ′) (where ϵ′ > ϵ) and that the “boundary” of T_{Y} (ϵ) is

∂T_{Y} (ϵ) = \overline{T_{Y} (ϵ)} − T_{Y} (ϵ) = \overline{ρ_{Y}^{−1}(ϵ)} = ρ_{Y}^{−1}(ϵ) ∪ ∂\overline{Y} .

Such neighborhoods are illustrated in the following diagram.

Z  Z′ Y T^{Z}(ϵ)  T^{Y} (ϵ)  T^{Z′}(ϵ)

Figure 1. Tubular neighborhoods

If W is the closure of a single stratum X we extend this notation by setting T_{X}(ϵ) = X, π_{X}(x) = x and ρ_{X}(x) = 0 for all x ∈ X.

2.3. Any compact real or complex algebraic or analytic variety admits a Whitney stratification ([Ha1], [Ha2]). Any compact Whitney stratified subset of a smooth manifold admits a system of control data (see [Mat] or [Gi] Thm. 2.6). If W is a compact Whitney stratified set and if the mappings {π_{Y} } are preassigned so as to satisfy Conditions (3) and (4) above, then distance functions ρ_{Y} may be found which are compatible with the mappings π_{Y} .

3. Partition of Unity

3.1. Throughout this paper we will fix a choice of a smooth nondecreasing function s : ℝ → [0, 1] so that s(x) = 0 for all x ≤ 1/2 and s(x) = 1 for all x ≥ 3/4. For any ϵ > 0 define s_{ϵ}(ρ) = s(ρ/ϵ).

Fix 0 < ϵ ≤ ϵ_{0}. Let W be a weakly stratified space with an ϵ_{0} system of control data {T_{Y} (ϵ_{0}), π_{Y} , ρ_{Y} }. For each stratum Z ⊂ W define the modified distance function s_{Z}^{ϵ} : T_{Z}(ϵ) → [0, 1] by s_{Z}^{ϵ}(x) = s_{ϵ}(ρ_{Z}(x)). Then s_{Z}^{ϵ} = 0 on T_{Z}(\frac{ϵ}{2}) and s_{Z}^{ϵ} = 1 near the edge ∂T_{Z}(ϵ) of the tubular neighborhood T_{Z}(ϵ).

For each stratum Y ⊂ W define a smooth function t_{ϵ}^{Y} : Y → ℝ as follows: If y ∈ Y is not contained in the tubular neighborhood T_{Z}(ϵ) of any stratum Z < Y then set t_{ϵ}^{Y} (y) = 1.

[figure: ρ; s_{ϵ}; 0; 1; \frac{1}{2} ϵ; \frac{3}{4} ϵ]

Figure 2. The function s_{ϵ}(ρ).

Otherwise, there is a unique maximal collection of boundary strata Z_{1}, Z_{2}, …, Z_{r} such that y ∈ T_{Z_{1}}(ϵ) ∩ T_{Z_{2}}(ϵ) ∩ … ∩ T_{Z_{r}}(ϵ) and in this case, by (2.2) (Condition 2), these boundary strata form a flag Z_{1} < Z_{2} < … < Z_{r} (after possibly relabeling the indices). Define

t_{ϵ}^{Y} (y) = s_{Z_{1}}^{ϵ}(y).s_{Z_{2}}^{ϵ}(y) … s_{Z_{r}}^{ϵ}(y).  (3.1.1)

Then the function t_{ϵ}^{Y} : Y → [0, 1] is smooth and vanishes on

⋃_{Z<Y} T_{Z}(\frac{ϵ}{2}) ∩ Y.

Pull this up to a function π_{Y}^{∗} t_{ϵ}^{Y} : T_{Y} (ϵ) → [0, 1] by setting π_{Y}^{∗} t_{ϵ}^{Y} (x) = t_{ϵ}^{Y} (π_{Y} (x)).

3.2. For each stratum Y ⊂ W, the product

B_{Y}^{ϵ} = (π_{Y}^{∗} t_{ϵ}^{Y}).(1 − s_{Y}^{ϵ}) : T_{Y} (ϵ) → [0, 1]  (3.2.1)

is smooth, vanishes near ∂T_{Y} (ϵ), and also near ∂\overline{Y} :

x ∈ ⋃_{Z<Y} T_{Z}(\frac{ϵ}{2}) ⟹ B_{Y}^{ϵ} (x) = 0.  (3.2.2)

Hence B_{Y}^{ϵ} admits an extension to W which is defined by setting

B_{Y}^{ϵ} (x) = 0 if x ∉ T_{Y} (ϵ).  (3.2.3)

This extension is smooth on each statum of W and satisfies the following conditions whenever Z < Y :

B_{Z}^{ϵ}π_{Y} (x) = B_{Z}^{ϵ}(x) for all x ∈ T_{Y} (ϵ)  (3.2.4)
B_{Y}^{ϵ} (π_{Y} (x)) ≥ B_{Y}^{ϵ} (x) for all x ∈ T_{Y} (ϵ)  (3.2.5)
B_{Y}^{ϵ} π_{Y} (x) = B_{Y}^{ϵ} (x) = t_{ϵ}^{Y} (x) for all x ∈ T_{Y} (ϵ/2)  (3.2.6)
B_{Y}^{ϵ} (x) = 1 for all x ∈ T_{Y} (ϵ/2) − ⋃_{Z<Y} T_{Z}(ϵ).  (3.2.7)

3.3. Lemma. For every stratum Y ⊂ W and for every point y ∈ \overline{Y} we have

B_{Y}^{ϵ} (y) + ∑_{Z<Y} B_{Z}^{ϵ}(y) = 1  (3.3.1)

3.4. Proof. It suffices to verify (3.3.1) for y ∈ Y. Then B_{Y}^{ϵ} (y) = t_{ϵ}^{Y} (y). If y is not in any tubular neighborhood T_{Z}(ϵ)∩Y (for Z < Y) then t_{ϵ}^{Y} (y) = 1 and 1−s_{Z}^{ϵ}(y) = 0. Otherise, let {Z_{1}, Z_{2}, … Z_{r}} be the collection of strata for which Z_{i} < Y and y ∈ T_{Z_{i}}(ϵ). By relabeling the indices, we may assume that Z_{1} < Z_{2} < … Z_{r} < Y form a flag of strata. The nonzero terms in the sum (3.3.1) involve only the functions s_{1}, s_{2}, …, s_{r} (where s_{i} = s_{Z_{i}}^{ϵ}) and can be written:

(1 − s_{1}) + s_{1}(1 − s_{2} + s_{2}(… + s_{r−1}(1 − s_{r} + s_{r}) …)) = 1.  (3.4.1)

Z  ^{Y}^{X}^{B_{Z}+B_{Y} =1}^{B_{Z}=1}^{B_{Z}+B_{X}=1}  ^{B_{Y} =1}^{B_{Y} +B_{X}=1}^{B_{X}=1}^{B_{Z}+B_{Y} +B_{X}=1}^{}^{{ϵ \atop 2}}ϵ^{{ϵ \atop 2}}ϵ^{{ϵ \atop 2}} ϵ

Figure 3. Partition of Unity for fixed ϵ

3.5. In §11.2 we will construct a connection on a modular variety by induction, patching together connections which have been previously defined on neighborhoods of boundary strata. For each step of this induction we will need a different partition of unity, which is obtained from (3.3.1) by shrinking the parameter ϵ. The purpose of this subsection is to construct the family of partitions of unity.

Let W be a weakly stratified space with an ϵ_{0} > 0 system of control data, {T_{Y} (ϵ_{0}), π_{Y} , ρ_{Y} }. Suppose each stratum Y is a complex manifold and define

ϵY = ϵ_{0}/2^{dim_{ℂ}(Y)}.

(The complex structure is irrelevant to this construction and is only introduced so as to agree with the later sections in this paper.) By Lemma 3.3 for every point x ∈ Y we have

∑_{Z≤Y} B_{Z}^{ϵY} (x) = 1.

By (3.2.4) and (3.2.6) the same equation holds, in fact for all x ∈ T_{Y} (\frac{ϵY}{2} ).

Let x ∈ W. Then there is a maximal collection of strata Y_{1}, Y_{2}, …, Y_{r} such that

x ∈ T_{Y_{1}}(ϵ_{0}) ∩ T_{Y_{2}}(ϵ_{0}) ∩ … ∩ T_{Y_{r}}(ϵ_{0}).

These strata form a partial flag which (we may assume) is given by Y_{1} < Y_{2} < ⋯ < Y_{r}. Set

B_{n}^{ϵm} = B_{Y_{n}}^{ϵY_{m}} and π_{m}(x) = π_{Y_{m}}(x).

In the following lemma we assume 1 ≤ m, n, m′, n′ ≤ r, that m ≥ n and that m′ ≥ n′.

3.6. Lemma. If B_{n}^{ϵm}(π_{n}(x)) ≠ 0 then B_{n′}^{ϵm′}(x) = 0 for all n′ < n and for all m′ > m. If B_{n}^{ϵm}(x) ≠ 0 then B_{n′}^{ϵm′}(π_{m′}(x)) = 0 for all n′ > n and for all m′ < m.

3.7. Proof. If B_{n}^{ϵm}(π_{n})(x) ≠ 0 then π_{n}(x) ∉ T_{n′}(ϵm/2) ⊃ T_{n′}(ϵm′) by (3.2.2). Therefore B_{n′}^{ϵm′}(π_{n}(x)) = 0 by (3.2.3). Hence B_{n′}^{ϵm′}(x) = 0 by (3.2.4). The second statement is the contrapositive of the first.

4. π-Fiber Differential Forms

4.1. Suppose W is a stratified space with a fixed ϵ-system of control data {T_{Y} (ϵ), π_{Y} , ρ_{Y} }. Define a π-fiber differential form ω to be a collection ω = {ω_{Y} ∈ A^{∗}(Y; ℂ)} of smooth differential forms (with complex coefficients) on the strata Y of W, which satisfy the following compatibility condition whenever Z < Y: There exists a neighborhood T (ω) ⊂ T_{Z}(ϵ) of Z such that

ω_{Y} |(Y ∩ T (ω)) = π_{Y Z}^{∗}(ω_{Z})|(Y ∩ T (ω)).  (4.1.1)

(Here, π_{Y Z} denotes the restriction π_{Z}|Y ∩ T_{Z}(ϵ).) We refer to equation (4.1.1) as the πfiber condition. If ω = {ω_{Y} } is a π-fiber differential form, define its differential to be the π-fiber differential form dω = {dω_{Y} }. Let A_{π}^{•}(W) denote the complex of π-fiber differential forms and let H_{π}^{∗}(W) denote the resulting cohomology groups. These differential forms are analogous to the π-fiber cocycles in [Go1]. The following result is proven in [V].

4.2. Theorem. The inclusion ℂ → A_{π}^{•}(W) of the constant functions into the complex of π-fiber differential forms induces an isomorphism

H^{i}(W; ℂ) ≅ H_{π}^{i}(W)  (4.2.1)

for all i. The restriction i^{∗}[ω] ∈ H^{∗}(\overline{Y}) of the cohomology class [ω] represented by a closed, π-fiber differential form ω = \{ω_{Y} \}_{Y ⊂W} to the closure \overline{Y} of a single stratum is given by the π-fiber differential form \{ω_{Z}\}_{Z⊂\overline{Y}} (where i : \overline{Y} → W denotes the inclusion).

4.3. Suppose the stratified space W is the closure of a single stratum X. Then a smooth differential form ω_{X} ∈ A^{I}(X) is the X-component of a π-fiber differential form ω ∈ A_{π}^{i}(W) if and only if for each stratum Y there exists a neighborhood U_{Y} of Y such that for each point p ∈ U_{Y} ∩ X and for every tangent vector v ∈ T_{p}X the following condition holds:

If dπ(p)(v) = 0 then i_{v}ω = 0  (4.3.1)

where i_{v} denotes the contraction with v. If (4.3.1) holds, then the π-fiber form ω is uniquely determined by ω_{X}.

Now suppose that W = \overline{X} is a compact subanalytic Whitney stratified subset of some (real) analytic manifold, and that τ : W̃ → W is a (subanalytic) resolution of singularities (cf. [Hi1], [Hi2]). This means that W̃ is a smooth compact subanalytic manifold, the mapping τ is subanalytic, its restriction τ^{−1}(X) → X to τ^{−1}(X) is a diffeomorphism, and τ^{−1}(X) is dense in W̃. Let ω_{X} ∈ A^{i}(X) be a π-fiber differential form on W.

4.4. Lemma. Suppose the differential form τ^{∗}(ω_{X}) is the restriction of a smooth closed differential form ω̃ ∈ A^{i}(W̃). Let [ω̃] ∈ H^{i}(W̃; ℝ) and [ω_{X}] ∈ H^{i}(W; ℝ) denote the corresponding cohomology classes. Then [ω̃] = τ^{∗}([ω_{X}]).

4.5. Proof. The cohomology classes [ω̃] and [ω_{X}] are determined by their integrals over subanalytic cycles by [Ha1], [Ha2]. Any subanalytic cycle ξ ∈ C_{i}(W̃; ℝ) may be made transverse (within its homology class) to the “exceptional divisor” τ^{−1}(W − X) ([GM3] §1.3.6). Then

∫_{ξ} ω̃ = ∫_{ξ∩τ^{−1}(X)} ω̃ = ∫_{τ(ξ)∩X} ω_{X} = ∫_{τ(ξ)} ω_{X}

5. Homogeneous Vectorbundles

5.1. If M is a smooth manifold and E → M is a smooth vectorbundle, let A^{i}(M, E) denote the space of smooth differential i-forms with values in E. Throughout this section, K denotes a closed subgroup of a connected Lie group G with Lie algebras 𝔨 ⊂ 𝔤 and with quotient D = G/K. We fix a representation λ : K → GL(V) on some finite dimensional (real or complex) vectorspace V. Write λ′ : 𝔨 → End(V) for its derivative at the identity, and note that its derivative at a general point g ∈ K is given by

dλ(h)((L_{h})_{∗}(k̇)) = λ(h)λ′(k̇) ∈ End(V)  (5.1.1)

for any k̇ ∈ 𝔨, where L_{h} : K → K is multiplication from the left by h ∈ K. The quotient mapping q : G → D = G/K is a principal K-bundle. The fundamental vertical vectorfields Yk̇(g) = L_{g∗}(k̇) (for k̇ ∈ 𝔨) determine a canonical trivialization, ker(dq) ≅ G × 𝔨.

5.2. The representation λ : K → GL(V) determines an associated homogeneous vectorbundle E = G ×_{K} V on D = G/K, which consists of equivalence classes [g, v] of pairs (g, v) ∈ G × V under the equivalence relation (gk, v) ∼ (g, λ(k)v) for all g ∈ G, k ∈ K, v ∈ V. It admits the homogeneous G action given by g′ · [g, v] = [g′g, v]. Smooth sections s̃ of E may be identified with smooth mappings s : G → V such that

s(gk) = λ(k^{−1})s(g)  (5.2.1)

by s̃(gK) = [g, s(g)] ∈ E. Then (5.2.1) implies

ds(g)(L_{g∗}(k̇)) = −λ′(k̇)s(g)  (5.2.2)

for all g ∈ G and k̇ ∈ 𝔨.

Similarly we identify smooth differential forms η̃ ∈ A^{i}(D, E) (with values in the vectorbundle E) with differential forms η ∈ A_{bas}^{i}(G, V) which are “basic,” meaning they are both K-equivariant (R_{k}^{∗}(η) = λ(k)^{−1}η for all k ∈ K) and horizontal (i(Yk̇)η = 0 for all k̇ ∈ 𝔨). Here, i(Y) denotes the interior product with the vectorfield Y and R_{k}(g) = gk for g ∈ G.

A connection ∇ on E is determined by a connection 1-form ω ∈ A^{1}(G, End(V)) which satisfies ω(L_{g∗}(k̇)) = λ′(k̇) and R_{k}^{∗}(ω) = Ad(λ(k^{−1}))ω for any k̇ ∈ 𝔨, k ∈ K, and g ∈ G. The covariant derivative ∇_{X} with respect to a vectorfield X on D acts on sections s : G → V satisfying (5.2.1) by

∇_{X}s(g) = ds(g)(X̃(g)) + ω_{g}(X̃(g))(s(g))  (5.2.3)

where X̃ is any lift of X to a smooth vectorfield on G. We write ∇ = d + ω. The curvature form Ω ∈ A^{2}(D, End(E)) takes values in the vectorbundle End(E) = G ×_{K} End(V) and it will be identified with the “basic” 2-form Ω ∈ A_{bas}^{2}(G, End(V)) which assigns to tangent vectors X, Y ∈ T_{g}G the endomorphism

Ω(X, Y) = dω(X, Y) + [ω(X), ω(Y)]  (5.2.4)

If the Lie bracket is extended in a natural way to Lie algebra-valued 1−forms, then it turns out ([BGV] §1.12) that [α, α](X, Y) = 2[α(X), α(Y)] so we may express the curvature form as Ω = dω + \frac{1}{2}[ω, ω].

The connection ∇ = d + ω is G-invariant iff L_{g}^{∗}(ω) = ω, in which case it is determined (on the identity component G^{0}) by its value ω_{0} : 𝔤 → End(V) at the identity. Using [Wa] or [KN] Chapt. II Thm. 11.5. it is easy to verify the following result.

5.3. Proposition. Suppose G is a connected Lie group and K is a closed subgroup. Then the G-invariant connections on the homogeneous vectorbundle E = G ×_{K} V are given by linear mappings ω_{0} : 𝔤 → End(V) such that

1. ω_{0}(k̇) = λ′(k̇) for all k̇ ∈ 𝔨 and

2. ω_{0}([ġ, k̇]) = [ω_{0}(ġ), λ′(k̇)] for all ġ ∈ 𝔤 and all k̇ ∈ 𝔨.

Moreover the curvature Ω ∈ A^{2}(G, End(V)) of such a connection is the left-invariant “basic” differential form whose value Ω_{0} at the identity is given by

Ω_{0}(ġ, ḣ) = [ω_{0}(ġ), ω_{0}(ḣ)] − ω_{0}([ġ, ḣ])  (5.3.1)

for any ġ, ḣ ∈ 𝔤. The connection is flat iff ω_{0} is a Lie algebra homomorphism.

5.4. Example. Suppose the representation λ : K → GL(V) is the restriction of a representation λ̃ : G → GL(V). Then we obtain a flat connection with ω_{0}(ġ) = λ̃′(ġ) for ġ ∈ 𝔤.

5.5. Example. A connection in the principal bundle G → D is given by a connection 1form θ ∈ A^{1}(G, 𝔨) such that R_{k}^{∗}(θ) = Ad(k^{−1})(θ) and θ(Yk̇) = k̇ (for any k ∈ K and any fundamental vectorfield Yk̇). It determines a connection ∇ = d + ω in the associated bundle E = G ×_{K} V by ω(X) = λ′(θ(X)). The principal connection θ is G-invariant iff L_{g}^{∗}(θ) = θ in which case it is determined by its value θ_{0} : 𝔤 → 𝔨 at the identity. Conversely, by [No], any linear mapping θ_{0} : 𝔤 → 𝔨 determines a G-invariant principal connection iff ker(θ_{0}) is preserved under the adjoint action of K. If K is a maximal compact subgroup of G then the Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 determines a canonical G invariant connection in the principal bundle G → G/K, and hence a connection on E which we refer to as the Nomizu connection. It is given by ω_{0}(k̇ + ṗ) = λ′(k̇) for k̇ ∈ 𝔨 and ṗ ∈ 𝔭. By (5.3.1), its curvature is given by

Ω(L_{g∗}(ġ_{1}), L_{g∗}(ġ_{2})) = −λ′([ṗ_{1}, ṗ_{2}])  (5.5.1)

(where ġ_{i} = k̇_{i} + ṗ_{i} ∈ 𝔨 ⊕ 𝔭), since [𝔨, 𝔭] ⊂ 𝔭 and [𝔭, 𝔭] ⊂ 𝔨.

5.6. There is a further description of homogeneous vectorbundles on D which are topologically trivial. Let E = G ×_{K} V be a homogeneous vectorbundle corresponding to a representation λ : K → GL(V) of K . A (smooth) automorphy factor J : G × D → GL(V) for E (or for λ) is a (smooth) mapping such that

1. J(gg′, x) = J(g, g′x)J(g′, x) for all g, g′ ∈ G and for all x ∈ D

2. J(k, x_{0}) = λ(k) for all k ∈ K. It follows (by taking g = 1) that J(1, x) = I. The automorphy factor J is determined by its values J(g, x_{0}) at the basepoint: any smooth mapping j : G → GL(V) such that j(gk) = j(g)λ(k) (for all k ∈ K and all g ∈ G) extends in a unique way to an automorphy factor J : G × D → GL(V) for E, namely

J(g, hx_{0}) = j(gh)j(h)^{−1}.  (5.6.1)

An automorphy factor J, if it exists, determines a (smooth) trivialization

Φ_{J} : G ×_{K} V → (G/K) × V  (5.6.2)

by [g, v] ↦ (gK, J(g, x_{0})v). This trivialization is G-equivariant with respect to the following J-automorphic action of G on (G/K) × V:

g.(x, v) = (gx, J(g, x)v).  (5.6.3)

Conversely, any smooth trivialization Φ : E ≅ (G/K) × V of the homogeneous vectorbundle E determines a unique automorphy factor J such that Φ = Φ_{J}. A trivialization of E (if one exists) allows one to identify smooth sections s of E with smooth mappings r : D → V. If the trivialization is given by an automorphy factor J and the smooth section s of E is given by a K-equivariant mapping s : G → V as in (5.2.1) then the corresponding smooth mapping is

r(gK) = J(g, x_{0})s(g)  (5.6.4)

which is easily seen to be well defined. Sections s which are invariant under γ ∈ G correspond to functions r such that r(γx) = J(γ, x)r(x) for all x ∈ D.

If D = G/K is Hermitian symmetric of noncompact type then the canonical automorphy factor (§8.3) J_{0} : G × D → K(ℂ) determines an automorphy factor J = λ_{ℂ} ∘ J_{0} for every homogeneous vectorbundle E = G ×_{K} V, where λ_{ℂ} : K(ℂ) → GL(V) is the complexification of λ.

5.7. If J_{1}, J_{2} are two automorphy factors for E, then the mapping ϕ : D × V → D × V which is given by ϕ(gx_{0}, v) = (gx_{0}, J_{1}(g, x_{0})J_{2}(g, x_{0})^{−1}v) is a well defined G-equivariant isomorphism of trivial bundles, where G acts on the domain via the J_{1}-automorphic action and G acts on the target via the J_{2}-automorphic action.

5.8. (The following fact will be used in §10.4.) Suppose that J : G × D → GL(V) is an automorphy factor for E. Let ∇ = d + ω be a connection on E. The trivialization Φ_{J} of E (5.6.2) determines a connection ∇^{J} = d + η on the trivial bundle D × V (with η ∈ A^{1}(D, End(V))) as follows: if r(gK) = J(g, x_{0})s(g) as in (5.6.4) then ∇_{q_{∗}X}^{J}r(gK) = J(g, x_{0})∇_{X}s(g) (for any X ∈ T_{g}G). It follows from (5.1.1) that the connection 1-forms are related by

η(q_{∗}(X)) = J(g, x_{0})ω(X)J(g, x_{0})^{−1} − (d_{X}J(g, x_{0}))J(g, x_{0})^{−1}.  (5.8.1)

6. Lemmas on curvature

6.1. Suppose E = G ×_{K} V is a homogeneous vectorbundle over D = G/K arising from a representation K → GL(V) of a closed subgroup K of a Lie group G on a complex vectorspace V. If θ ∈ A^{1}(G, End(V)) is a Lie algebra-valued 1-form, denote by [θ, θ] the Lie algebra valued 2-form (X, Y) ↦ 2[θ(X), θ(Y)] (cf [BGV] §1.12). The proof of the following lemma is a direct but surprisingly tedious computation.

6.2. Lemma. Suppose {f_{1}, f_{2}, …, f_{n}} form a smooth partition of unity on D, that is 0 ≤ f_{i}(x) ≤ 1 and ∑_{i=1}^{n} f_{i}(x) = 1 for all x ∈ D. Let ∇_{i} = d + ω_{1} be connections on E with curvature forms Ω_{i} for 1 ≤ i ≤ n. Let ∇ = ∑_{i=1}^{n} f_{i}∇_{i} be the connection with connection form ω = ∑_{i=1}^{n} f_{i}ω_{i}. Then the curvature Ω of ∇ is given by

Ω = ∑_{i=1}^{n} f_{i}Ω_{i} − \frac{1}{2} ∑_{i<j} f_{i}f_{j}[ω_{i} − ω_{j}, ω_{i} − ω_{j}] + ∑_{i=1}^{n−1} df_{i} ∧ (ω_{i} − ω_{n}).

(Even if the ∇_{i} are flat and the f_{i} are constant, the connection ∇ is not necessarily flat.)

6.3. Let E be a complex vectorspace and let f : E → ℂ be a homogeneous polynomial of degree k. The polarization of f is the unique symmetric k-linear form P : E ×E ×…×E → ℂ such that f(x) = P (x, x, …, x) for all x ∈ E. If N ⊂ E is a vectorsubspace such that f(x + n) = f(x) for all x ∈ E and all n ∈ N then the polarization P satisfies

P (x_{1} + n_{1}, x_{2} + n_{2}, …, x_{k} + n_{k}) = P (x_{1}, x_{2}, …, x_{k})  (6.3.1)

for all x_{1}, …, x_{k} ∈ E and all n_{1}, …, n_{k} ∈ N.

Now let K → GL(V) be a representation on a complex vectorspace V as above, and let f : E = End(V) → ℂ be a homogeneous polynomial of degree k, which is invariant under the adjoint action K → GL(𝔨) → GL(End(V)) of K. Then the polarization P of f is also Ad_{K}-invariant. If ∇ is a connection on E = G×_{K} V with curvature form Ω ∈ A^{2}(G, End(V)) then the characteristic form associated to f is

σ_{∇}^{f}(X_{1}, Y_{1}, …, X_{k}, Y_{k}) = P (Ω(X_{1}, Y_{1}), …, Ω(X_{k}, Y_{k})) ∈ A^{2k}(G, ℂ).  (6.3.2)

It is “basic” (cf. §5.2) and hence descends uniquely to a differential form on D which we also denote by σ_{∇}^{f} ∈ A^{2k}(D, ℂ). Throughout this paper we shall be concerned only with homogeneous polynomials f : End(V) → ℂ which are invariant under the full adjoint action of GL(V) and which we shall refer to as a Ad-invariant polynomials. When f(x) is the i-th elementary symmetric function in the eigenvalues of x, the resulting characteristic form is the i-th Chern form and it will be denoted σ^{i}(∇).

6.4. Lemma. Let V be a complex vectorspace, H = GL(V) and 𝔥 = End(V). Let x, n ∈ 𝔥 and suppose that [x, n] = 0 and that n is nilpotent. Then for any Ad-invariant polynomial f : V → ℂ we have:

f(x + n) = f(x).

6.5. Proof. Let 𝔱 ⊂ 𝔥 be the Lie algebra of a Cartan subgroup T ⊂ H and let W denote its Weyl group. The adjoint quotient mapping χ : 𝔥 → 𝔱/W associates to any a ∈ 𝔥 the W -orbit C(a_{s})∩𝔱 where a = a_{s} +a_{n} is the Jordan decomposition of a into its semisimple and nilpotent parts (with [a_{s}, a_{n}] = 0), and where C(a_{s}) denotes the conjugacy class of a_{s} in 𝔥, cf. [Sp]. (The value χ(a) may be interpreted as the coefficients of the characteristic polynomial of a ∈ End(V).) Since x and n commute, (x + n)_{s} = x_{s} + n_{s} = x_{s}, hence χ(x + n) = χ(x). But every Ad-invariant polynomial f : 𝔥 → ℂ factors through χ, hence f(x+n) = f(x).

7. Hermitian Symmetric Spaces

7.1. Throughout the remainder of this paper, algebraic groups will be denoted by boldface type and the associated group of real points will be denoted in Roman. Fix a reductive algebraic group G which is defined over ℚ and which (for convenience only) is assumed to be connected and simple over ℚ. Suppose G acts as the identity component of the group of automorphisms of a Hermitian symmetric space D. A choice of basepoint x_{0} ∈ D determines a Cartan involution θ : G → G, a maximal compact subgroup K = G^{θ} = Stab_{G}(x_{0}) and a diffeomorphism G/K ≅ D. Let D^{∗} denote the Satake partial compactification of D, in other words, the union of D and all its rational boundary components, with the Satake topology (cf. [BB]). The closure D_{1}^{∗} in D^{∗} of a rational boundary component D_{1} is again the Satake partial compactification of D_{1}.

If D_{1} ⊂ D^{∗} is a rational boundary component of D, its normalizer P is the group of real points of a rationally defined maximal proper parabolic subgroup P ⊂ G. Let U_{P} denote the unipotent radical of P and let L(P) = P/U_{P} be the Levi quotient with projection ν_{P} : P → L(P). It is well known that L(P) is an almost direct product (commuting product with finite intersection) of two subgroups, L(P) = G_{h}G_{ℓ} (we include possible compact factors in the G_{ℓ} factor), where the “hermitian part” G_{h} is semisimple and defined over ℚ. (It may be trivial). The choice of basepoint x_{0} ∈ D determines a unique θ-stable lift L_{P} (x_{0}) ⊂ P of the Levi quotient [BoS] (which, from now on, we shall use without mention), as well as basepoints x_{1} ∈ D_{1} in the boundary component D_{1} such that Stab_{G_{h}}(x_{1}) = K ∩ G_{h}. We obtain a decomposition

P = U_{P} G_{h}G_{ℓ}.  (7.1.1)

The group P acts on the boundary component D_{1} through the projection ν_{h} : P → L(P) → G_{h}/(G_{h} ∩ G_{ℓ}) which also determines a diffeomorphism G_{h}/K_{h} ≅ D_{1} (where K_{h} = K ∩ G_{h}). This projection ν_{h} also gives rise to a P -equivariant canonical projection

π : D → D_{1}  (7.1.2)

by π(ug_{h}g_{ℓ}K_{P}) = g_{h}K_{h}.

The “linear part” G_{ℓ} is reductive and contains the 1-dimensional ℚ-split torus S_{P}(ℝ) in the center of the Levi quotient. If 𝔷 ⊂ 𝔑_{P} denotes the center of the Lie algebra 𝔑_{P} of the unipotent radical of P, then the adjoint action of G_{ℓ} on 𝔷 has a unique open orbit C(P) which is a self adjoint homogeneous cone.

7.2. Let P_{0} ⊂ G be a fixed minimal rational parabolic subgroup and define the standard parabolic subgroups to be those which contain P_{0}. Let S_{0} ⊂ L(P_{0}) be the greatest ℚ-split torus in the center of L(P_{0}) and let Φ = Φ(S_{0}, G) be the (relative) roots of G in S_{0} with positive roots Φ^{+} consisting of those roots which appear in the unipotent radical U_{P_{0}} and resulting simple roots Δ. Each simple root α corresponds to a vertex in the (rational) Dynkin diagram for G and also to a maximal standard parabolic subgroup P such that S_{P} ⊂ ker(β) for each β ∈ Δ − {α}.

7.3. Two maximal parabolic subgroups. For simplicity, let us assume that G is (almost) simple over ℚ. The (rational) Dynkin diagram for G is linear, of type BC, and determines a canonical ordering among the maximal standard rational parabolic subgroups with P_{1} ≺ P_{2} iff D_{1} ≺ D_{2} (meaning that D_{1} ⊂ D_{2}^{∗} ⊂ D^{∗}) where D_{i} is the rational boundary component fixed by P_{i}. Write P_{1} = U_{1}G_{1h}G_{1ℓ} and P_{2} = U_{2}G_{2h}G_{2ℓ} as in (7.1.1). If P_{1} ≺ P_{2} then G_{1h} ⊂ G_{2h} and G_{1ℓ} ⊃ G_{2ℓ}. Let P = P_{1} ∩ P_{2}. In Figure 4, Δ − {α_{1}} and Δ − {α_{2}} denote the simple roots corresponding to P_{1} and P_{2} respectively.

α_{1}  α_{2} G_{1h}  G′_{ℓ}  G_{2ℓ}

Figure 4. Dynkin diagrams for G and P

Then we have a commutative diagram

\begin{aligned}U_{1} & ⊂ & U_{P} & ⊂ & P & ⊂ & P_{1} \\ & & ↓ & & ↓ & & ↓^{ν_{1}} \\ U\overline{P} & ⊂ & \overline{P} & ⊂ & L(P_{1}) & = G_{1h}G_{1ℓ} \end{aligned}  (7.3.1)

where \overline{P} = ν_{1}(P) ⊂ L(P_{1}) is the image of P. Then \overline{P} = G_{1h}P_{ℓ} where P_{ℓ} ⊂ G_{1ℓ} is a parabolic subgroup of G_{1ℓ} whose Levi factor decomposes as a commuting, almost direct product L(P_{ℓ}) = G′_{ℓ}G_{2ℓ}. Writing \overline{U} for the lift of U\overline{P} = U_{P_{ℓ}} we conclude that P has a decomposition

P = U_{1}G_{1h}(\overline{U}G′_{ℓ}G_{2ℓ}) = U_{P} G_{1h}G_{P ℓ}  (7.3.2)

with U_{P} = U_{1}\overline{U}, G_{P ℓ} = G′_{ℓ}G_{2ℓ} and P_{ℓ} = \overline{U}G′_{ℓ}G_{2ℓ} ⊂ G_{1ℓ}. Similarly, we have a diagram

\begin{aligned}U_{2} & ⊂ & U_{P} & ⊂ & P & ⊂ & P_{2} \\ & & ↓ & & ↓ & & ↓^{ν_{2}} \\ U\overline{\overline{P}} & ⊂ & \overline{\overline{P}} & ⊂ & L(P_{2}) & = G_{2h}G_{2ℓ} \end{aligned}  (7.3.3)

where \overline{\overline{P}} = ν_{2}(P) ⊂ L(P_{2}) is the image of P. Then \overline{\overline{P}} = P_{h}G_{2ℓ} with P_{h} ⊂ G_{2h} a parabolic subgroup of G_{2h} whose Levi factor decomposes as a product L(P_{h}) = G_{1h}G′_{ℓ}. Writing \overline{\overline{U}} for the canonical lift of U\overline{\overline{P}} = U_{P_{h}} we obtain another decomposition,

P = U_{2}(\overline{\overline{U}}G_{1h}G′_{ℓ})G_{2ℓ} = U_{P} G_{1h}G_{P ℓ}  (7.3.4)

with P_{h} = \overline{\overline{U}}G_{1h}G′_{ℓ}.

Similarly, an arbitrary standard parabolic subgroup Q may be expressed in a unique way as an intersection Q = P_{1} ∩ P_{2} ∩ … ∩ P_{m} of maximal standard parabolic subgroups, with P_{1} ≺ P_{2} ≺ … ≺ P_{m}. In this case we write Q^{♭} = P_{1} and Q^{♯} = P_{m}. If P_{1} = U_{1}G_{1h}G_{1ℓ} (with projection ν_{1} : P_{1} → G_{1h}G_{1ℓ}) then the Levi factor L(Q) decomposes as an almost direct product of m + 1 factors

L(Q) = G_{1h}(G′_{1}G′_{2} … G′_{m−1}G_{mℓ}) = G_{1h}G_{Qℓ}  (7.3.5)

where G_{1h} is the hermitian part of L(P_{1}), and G_{Qℓ} consists of the remaining factors, including G_{mℓ}, the linear part of L(P_{m}). Each factor in G_{Qℓ} acts as an automorphism group of a certain symmetric cone in the boundary of the cone C(P_{1}). The projection

\overline{Q} = ν_{1}(Q) = G_{1h}U_{P_{1}Q}G_{Qℓ}

is parabolic in L(P_{1}) with unipotent radical U_{P_{1}Q} ⊂ G_{1ℓ} which also has a lift depending on the choice of basepoint. In summary we obtain a decomposition

Q = U_{1}G_{1h}U_{P_{1}Q}G_{Qℓ}.  (7.3.6)

8. Cayley transform

8.1. As in §7, suppose that G is defined over ℚ and simple over ℚ, that G = G(ℝ), and that K is a maximal compact subgroup of G with D = G/K Hermitian. The following proposition is the key technical tool behind our construction of a connection which is flat along the fibers of π. The proof follows from the existence of a “canonical automorphy factor for P_{1}” as defined by M. Harris [Har2] [HZ1] (1.8.7). See also the survey in [Z3]. In this section we will approximately follow [Har2] and derive these results from known facts about the Cayley transform [WK], [Sa] Chapter III.

8.2. Proposition. Let P_{1} = U_{1}G_{1h}G_{1ℓ} be a maximal rational parabolic subgroup of G. Set K_{1} = K ∩ P_{1} = K_{1h}K_{1ℓ}. Let λ : K → GL(V) be a representation of K. Then the restriction λ|K_{1} admits a natural extension λ_{1} : K_{1h}G_{1ℓ} → GL(V).

8.3. Proof. The group K is the set of real points of an algebraic group K defined over ℝ. As in [He] VIII§7, [Sa] II §4, [AMRT] III §2 when the Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 is complexified it gives rise to two abelian unipotent subgroups P^{+} and P^{−} of G(ℂ) such that the complex structure on 𝔤(ℂ) acts with eigenvalue ±i on Lie(P^{±}). The natural mapping P^{+}K(ℂ)P^{−} → G(ℂ) is injective and its image contains G = G(ℝ). Let j : P^{+}K(ℂ)P^{−} → K(ℂ) denote the projection to the middle factor. The group P^{+} is the unipotent radical of the maximal parabolic subgroup P^{+}K(ℂ) and hence is normal in P^{+}K(ℂ); similarly P^{−} is normal in K(ℂ)P^{−}. It follows that, for all h ∈ P^{+}K(ℂ), for all h′ ∈ K(ℂ)P^{−} and for all g ∈ P^{+}K(ℂ)P^{−} we have

j(hgh′) = j(h)j(g)j(h′).  (8.3.1)

In particular, j(gk) = j(g)k whenever k ∈ K(ℂ), so by equation (5.6.1), j determines a unique automorphy factor (the “usual” canonical automorphy factor) J_{0} : G × D → K(ℂ) such that J_{0}(g, x_{0}) = j(g) (for all g ∈ G).

Let c_{1} ∈ G(ℂ) be the Cayley element [Sa],[WK], [BB], [Har2], [Z3] which corresponds to P_{1}. That is, c_{1} is a lift to G(ℂ) of the standard choice of Cayley element in Ad(𝔤)(ℂ). We follow Satake’s convention, rather than that of [WK] (which would associate c_{1}^{−1} to P_{1}.) The element c_{1} satisfies the following properties, [Sa] (Chapt. III (7.8),(7.9),(2.4)), [Har2], [Z3]:

1. c_{1}, c_{1}^{−1} ∈ P^{+}K(ℂ)P^{−} and c_{1}G ⊂ P^{+}K(ℂ)P^{−};

2. c_{1} commutes with G_{1h};

3. c_{1}U_{1}K_{1h}G_{1ℓ}c_{1}^{−1} ⊂ P^{+}K(ℂ);

4. c_{1}K_{1h}G_{1ℓ}c_{1}^{−1} ⊂ K(ℂ), and hence

5. j(c_{1}g) = j(c_{1}gc_{1}^{−1})j(c_{1}) = c_{1}gc_{1}^{−1}j(c_{1}) for all g ∈ K_{1h}G_{1ℓ}.

Harris then defines the canonical automorphy factor (for P_{1}) to be the automorphy factor J_{1} : G × D → K(ℂ) which is determined by its values at the basepoint, J_{1}(g, x_{0}) = j(c_{1})^{−1}j(c_{1}g). This is well defined because j(c_{1})^{−1}j(c_{1}gk) = j(c_{1})^{−1}j(c_{1}g)k by (8.3.1), see (5.6.1). So we may define the canonical extension

λ_{1}(g) = λ_{ℂ}J_{1}(g, x_{0}) = λ_{ℂ}(j(c_{1})^{−1}j(c_{1}g))  (8.3.2)

for any g ∈ K_{1h}G_{1ℓ}, where λ_{ℂ} is the complexification of λ. Then λ_{1}(k) = λ(k) for any k ∈ K_{1}. Moreover λ_{1} is a homomorphism: if k_{1h}g_{1ℓ} and k′_{1h}g′_{1ℓ} are elements of K_{1h}G_{1ℓ} then

J_{1}(k_{1h}g_{1ℓ}k′_{1h}g′_{1ℓ}, x_{0}) = j(c_{1})^{−1}(c_{1}k_{1h}g_{1ℓ}c_{1}^{−1} · c_{1}k′_{1h}g′_{1ℓ}c_{1}^{−1})j(c)
= j(c_{1})^{−1}c_{1}k_{1h}g_{1ℓ}c_{1}^{−1}j(c_{1}) · j(c_{1})^{−1}c_{1}k′_{1h}g′_{1ℓ}c_{1}^{−1}j(c_{1})
= J_{1}(k_{1h}g_{1ℓ}, x_{0})J_{1}(k′_{1h}g′_{1ℓ}, x_{0}).

Verification that J_{1}(k_{1h}^{−1}g_{1ℓ}^{−1}, x_{0}) = J_{1}(k_{1h}g_{1ℓ}, x_{0})^{−1} is similar. We remark, following [Har2] that modifying c_{1} by any element d ∈ P^{+}K(ℂ) will not affect the values of J_{1}(k_{1h}g_{1ℓ}, x_{0}).

Now suppose P_{1} ≺ P_{2} are rational maximal parabolic subgroups of G with P_{i} = U_{i}G_{ih}G_{iℓ} (i = 1, 2) and, as in (7.3.4),

P_{1} ∩ P_{2} = U_{2}(\overline{\overline{U}}G_{1h}G′_{ℓ})G_{2ℓ} with P_{h} = \overline{\overline{U}}G_{1h}G′_{ℓ} ⊂ G_{2h}.

8.4. Proposition. Let λ_{i} : K_{ih}G_{iℓ} → GL(V) be the canonical extensions of λ|K_{i} = K ∩ P_{i} (i = 1, 2) and let λ_{21} : K_{1h}G′_{ℓ} → GL(V) be the canonical extension of λ|K_{2h} corresponding to the canonical automorphy factor J_{21} for P_{h} ⊂ G_{2h}. Then

λ_{1}(g_{2ℓ}) = λ_{2}(g_{2ℓ}) and λ_{1}(g′_{ℓ}) = λ_{21}(g′_{ℓ})

for all g_{2ℓ} ∈ G_{2ℓ} and all g′_{ℓ} ∈ G′_{ℓ}.

8.5. Proof. Let c_{1}, c_{2} ∈ G(ℂ) and c_{21} ∈ G_{2h}(ℂ) be the Cayley elements for P_{1}, P_{2} ⊂ G and P_{h} ⊂ G_{2h} respectively. By [Sa] Chapter III (9.5),

c_{21} = c_{1}c_{2}^{−1} = c_{2}^{−1}c_{1}.

The lift G_{2h} ⊂ G is stable under the Cartan involution on G so the corresponding decomposition P_{2h}^{+}K_{2h}(ℂ)P_{2h}^{−} coincides with (G_{2h} ∩ P^{+})(G_{2h} ∩ K(ℂ))(G_{2h} ∩ P^{−}) and in particular j_{2h} : G_{2h} → K_{2h}(ℂ) is the restriction of j : G → K(ℂ). Let c_{21} = c_{21}^{+}c_{21}^{0}c_{21}^{−} be the resulting decomposition. Then c_{2} commutes with each of the factors c_{21}^{∗}. It follows from (8.3.1) that

j(c_{1}) = j(c_{21}c_{2}) = j(c_{21}^{+}c_{21}^{0}c_{2}c_{21}^{−}) = j(c_{21}^{0}c_{2}) = j(c_{21})j(c_{2})

and similarly j(c_{1}) = j(c_{2})j(c_{21}). Since g_{2ℓ} ∈ G_{2ℓ} also commutes with c_{21} ∈ G_{2h}(ℂ) and with j(c_{21}) ∈ K_{2h}(ℂ), we find

λ_{1}(g_{2ℓ})λ_{2}(g_{2ℓ})^{−1} = λ_{ℂ} (j(c_{1})^{−1}c_{1}g_{2ℓ}c_{1}^{−1}j(c_{1})j(c_{2})^{−1}c_{2}g_{2ℓ}^{−1}c_{2}^{−1}j(c_{2}))
= λ_{ℂ} (j(c_{1})^{−1}c_{1}g_{2ℓ}c_{1}^{−1}j(c_{21})c_{2}g_{2ℓ}^{−1}c_{2}^{−1}j(c_{2}))
= λ_{ℂ} (j(c_{1})^{−1}c_{1}g_{2ℓ}c_{1}^{−1}c_{2}g_{2ℓ}^{−1}c_{2}^{−1}j(c_{21})j(c_{2}))
= λ_{ℂ} (j(c_{1})^{−1}c_{1}g_{2ℓ}c_{21}^{−1}g_{2ℓ}^{−1}c_{2}^{−1}j(c_{1}))
= λ_{ℂ} (j(c_{1})^{−1}c_{1}c_{21}^{−1}c_{2}^{−1}j(c_{1})) = 1.

Similarly if g′_{ℓ} ∈ G′_{ℓ} then using (4) above,

λ_{21}(g′_{ℓ}) = λ_{ℂ} (j(c_{21})^{−1}j(c_{21}g′_{ℓ}c_{21}^{−1})j(c_{21}))

= λ_{ℂ} (j(c_{21})^{−1}j(c_{2}^{−1}c_{1}g′_{ℓ}c_{1}^{−1}c_{2})j(c_{21}))

= λ_{ℂ} (j(c_{21})^{−1}j(c_{2})^{−1}j(c_{1}g′_{ℓ}c_{1}^{−1})j(c_{2})j(c_{21}))

= λ_{ℂ} (j(c_{1})^{−1}j(c_{1}g′_{ℓ}c_{1}^{−1})j(c_{1})) = λ_{1}(g′_{ℓ}).

9. Baily-Borel Satake compactification

9.1. As in §7, suppose that G is defined over ℚ and simple over ℚ, that G = G(ℝ), and that K is a maximal compact subgroup of G with D = G/K Hermitian. Let D^{∗} be the Satake partial compactification of D, consisting of D together with its rational boundary components D_{P}, one for each (proper) maximal rational parabolic subgroup P ⊂ G; with the Satake topology [BB]. The action of G(ℚ) on D extends continuously to an action of G(ℚ) on D^{∗}. Let Γ ⊂ G(ℚ) be a neat arithmetic subgroup and let q : D^{∗} → \overline{X} = Γ\D^{∗} denote the quotient mapping. Then \overline{X} is the Baily-Borel compactification of X and it admits the structure of a complex projective algebraic variety with a canonical stratification with a single stratum for every Γ-conjugacy class of rational boundary components as follows. Let D_{1} ⊂ D^{∗} be a rational boundary component with normalizing maximal parabolic subgroup P = U_{P} G_{h}G_{ℓ}. Let ν_{h} : P → G′_{h} = G_{h}/(G_{h} ∩ G_{ℓ}) and let ν_{ℓ} : P → G′_{ℓ} = G_{ℓ}/(G_{h} ∩ G_{ℓ}). The closure D_{1}^{∗} of D_{1} in D^{∗} is the Satake partial compactification of D_{1}. The group P acts on D_{1} through its projection to G′_{h} and the group Γ_{P} = Γ∩P acts on D_{1}^{∗} through its projection Γ_{h} = ν_{h}(Γ_{P}) to G′_{h}. Then X_{1} = Γ_{P} \D_{1}^{∗} = Γ_{h}\D_{1}^{∗} is a stratum of X̄. Its closure X̄_{1} = Γ_{h}\D_{1}^{∗} in X̄ is the Baily-Borel compactification of X_{1}. The stratum X_{1} is also the image of the (infinitely many) rational boundary components D′_{1} which are Γ-conjugate to D_{1}.

9.2. Let D_{1} ⊂ D^{∗} be a rational boundary component which projects to X_{1}. We will say that a neighborhood Ũ ⊂ D^{∗} is a Γ-parabolic neighborhood of D_{1} if the following holds: if x_{1}, x_{2} ∈ Ũ and γ ∈ Γ satisfy x_{2} = γx_{1} then γ ∈ Γ ∩ P. If X_{1} ⊂ X̄ is a stratum in the Baily-Borel compactification of X, we say that a neighborhood U ⊂ X̄ of X_{1} is parabolic if for some (and hence for any) boundary component D_{1} ⊂ D^{∗} with q(D_{1}) = X_{1}, there is a Γ-parabolic neighborhood Ũ ⊂ D^{∗} of D_{1} such that U = q(Ũ). This means that the covering Γ_{P} \D^{∗} → Γ\D^{∗} is one to one on Ũ, and we have a commutative diagram

\begin{aligned}D^{∗} & ⊃ & Ũ & ⊃ & D_{1} \\ ^{q}↓^{Γ} & & ↓^{Γ_{P}} & ↓^{Γ_{h}} \\ \overline{X} & ⊃ & U & ⊃ & X_{1} \end{aligned}.

9.3. Lemma. Each stratum X_{1} ⊂ X̄ has a fundamental system of neighborhoods, each of which is Γ-parabolic.

9.4. Proof. [Sa2] Let D^{BS} be the Borel-Serre partial compactification of D together with its “Satake” topology [BoS]. It is a manifold with corners, having one corner e(P) for each rational parabolic subgroup P. According to [Z2] the identity mapping D → D has a unique continuous extension 𝔴 : D^{BS} → D^{∗}, and it is surjective. If P is standard then 𝔴(e(P)) = D_{P^{♭}}. Let D^{†} denote the quotient topology on the underlying set |D^{∗}| which is induced by 𝔴. Then D^{†} → D^{∗} is a continuous bijection and the quotient mapping Γ\D^{†} → Γ\D^{∗} is a homeomorphism. In [Sa1] Theorem 8.1, Saper constructs a basis of parabolic neighborhoods U_{P} of each corner e(P) in D^{BS}. If P is maximal then 𝔴(U_{P}) is open in D^{∗} as may be shown by verifying the condition at the bottom of page 264 in [AMRT].

We remark that the image of U_{P} is Γ-parabolic and is open in D^{†} by construction, and that the topology D^{†} may be substituted for the Satake topology D^{∗} throughout this paper.

9.5. Fix a standard rational boundary component D_{1} ⊂ D^{∗} normalized by a standard maximal rational parabolic subgroup P_{1}. In the Satake topology (or in the topology D^{†}) there is a natural neighborhood T (D_{1}) = ⋃{D_{2}| D_{1} ≺ D_{2} ≺ D} consisting of the union of all rational boundary components (including D, the nonproper boundary component) whose closures contain D_{1}. The projection π : D → D_{1} (7.1.2) has a unique continuous extension T (D_{1}) → D_{1} to this neighborhood. Its restriction to each intermediate boundary component D_{2} coincides with the canonical projection D_{2} → D_{1} which is obtained by considering D_{2} to be the symmetric space corresponding to the Hermitian part G_{2h} of the Levi factor of P_{2} and by considering D_{1} ⊂ D_{2}^{∗} to be the rational boundary component preserved by the parabolic subgroup P_{h} ⊂ G_{2h} (notation as in §7.3). It follows that π_{1}(x) = π_{1}π_{2}(x) for all x ∈ T (D_{1}) ∩ T (D_{2}). (The above union can be quite large: if D_{20} is a standard boundary component normalized by a standard parabolic subgroup P_{2} ≻ P_{1} then the boundary components D_{2} ⊂ T (D_{1}) which are conjugate to the standard one D_{20} are in one to one correspondence with elements of P_{1}(ℚ)/(P_{1}(ℚ) ∩ P_{2}(ℚ)) cf. (7.3.2).)

9.6. Lemma. For ϵ_{0} > 0 sufficiently small, the Baily-Borel compactification \overline{X} admits an ϵ_{0}-system of control data \{T_{Y} (ϵ_{0}), π_{Y} , ρ_{Y} \} such that for each stratum Y ⊂ ∂\overline{X} and for any choice of boundary component D_{1} ⊂ D^{∗} with q(D_{1}) = Y we have

1. The neighborhood T_{Y} (ϵ_{0}) ⊂ \overline{X} is a parabolic neighborhood of Y; it is the image, say, of some Γ-parabolic neighborhood Ũ(ϵ_{0}) ⊂ D^{∗} of D_{1}, and

2. π_{Y} (q(x)) = q(π(x)) for all x ∈ Ũ(ϵ_{0}) (where π : D → D_{1} is the canonical projection).

9.7. Proof. For any g ∈ P and x ∈ D we have π(gx) = ν_{h}(g)π(x) ∈ D_{1}. It follows that the projection function π passes to the quotient Γ_{P} \D^{∗}, where it may be restricted to a parabolic neighborhood U of Y = q(D_{1}); write π_{Y} : U → Y for the result. If P and P′ are Γ conjugate maximal rational parabolic subgroups corresponding to conjugate boundary components D_{1} and D′_{1} then the projections T (D_{1}) → D_{1} and T (D′_{1}) → D′_{1} are compatible with conjugation, which shows that the resulting projection π_{Y} : U → Y is independent of the choice of lift D_{1} ⊂ D^{∗} of the stratum Y ⊂ X̄. The tubular neighborhood T_{Y} (ϵ_{0}) may be chosen inside U. The compatibility between these projections follows from (9.5). As mentioned in §2.3, by further shrinking the tubular neighborhoods if necessary, control data may be found for which the tubular projections agree with these π_{Y} .

10. Parabolically Induced Connection

10.1. As in §7, 9 we suppose that G is semisimple, defined over ℚ and simple over ℚ; that G = G(ℝ), and K ⊂ G is a maximal compact subgroup with D = G/K Hermitian symmetric. Fix Γ ⊂ G(ℚ) a neat arithmetic subgroup. Let λ : K → GL(V) be a representation of K on some complex vectorspace V and denote by E = G ×_{K} V the associated homogeneous vectorbundle on D.

Let D_{1} be a rational boundary component of D with canonical projection π : D → D_{1}. Let P be the maximal parabolic subgroup of G which preserves D_{1}. Write P = UG_{h}G_{ℓ} as in §7.1 and let K_{h} = K ∩ G_{h} and K_{ℓ} = K ∩ G_{ℓ} be the corresponding maximal compact subgroups. Let 𝔤_{h} = 𝔨_{h} ⊕ 𝔭_{h} and 𝔤_{ℓ} = 𝔨_{ℓ} ⊕ 𝔭_{ℓ} denote the corresponding Cartan decompositions.

The restriction of λ to K_{h} determines a homogeneous vectorbundle E_{1} = G_{h} ×_{K_{h}} V over D_{1}. By Proposition 8.2 the representation λ| K_{h}K_{ℓ} admits an extension to a representation λ_{1} : K_{h}G_{ℓ} → GL(V). This extension determines an action of P on E_{1} which is given by

ug_{h}g_{ℓ}.[g′_{h}, v] = [g_{h}g′_{h}, λ_{1}(g_{ℓ})v].  (10.1.1)

We obtain a vectorbundle mapping (which covers π),

Φ̃ : E = P ×_{K_{P}} V → G_{h} ×_{K_{h}} V = E_{1}  (10.1.2)

by Φ̃([ug_{h}g_{ℓ}, v]) = [g_{h}, λ_{1}(g_{ℓ})v]. Then Φ̃ induces an isomorphism,

Φ : E ≅ π^{∗}(E_{1}); [g, v] ↦ (gK_{P} , Φ̃([g, v])) ∈ D × E_{1}  (10.1.3)

of P -homogeneous vectorbundles (where g ∈ P and v ∈ K).

10.2. Definition. Let ∇_{1} = d+ω_{1} be a connection on E_{1}. The parabolically induced connection ∇ = d+ω on E is defined to be the pullback ∇ = Φ^{∗}(∇_{1}) of ∇^{1} under the isomorphism Φ. It is the unique connection whose covariant derivative (5.2.3) satisfies

∇_{v}(Φ^{∗}(s)) = Φ^{∗}((∇_{1})_{π_{∗}v}s)  (10.2.1)

for any section s of E_{1} and for any tangent vector v ∈ T_{x}D.

10.3. Proposition. Suppose ∇_{1} = d+ω_{1} is a connection on E_{1} = G_{h}×_{K_{h}} V. Let ∇ = d+ω denote the parabolically induced connection on E = P ×_{K_{P}} V. Then

ω(L_{g∗}(u̇ + ġ_{h} + ġ_{ℓ})) = λ′_{1}(ġ_{ℓ}) + Ad(λ_{1}(g_{ℓ}^{−1}))(ω_{1}(L_{g_{h}∗}(ġ_{h}))  (10.3.1)

for any g = ug_{h}g_{ℓ} ∈ P and any u̇ + ġ_{h} + ġ_{ℓ} ∈ Lie(U_{P}) ⊕ 𝔤_{h} ⊕ 𝔤_{ℓ}.

10.4. Proof. Let J_{1} : G_{h} × D_{1} → GL(V) be an automorphy factor for E_{1}, corresponding to a trivialization E_{1} ≅ D_{1} × V. Composing this with the isomorphism Φ : E → π^{∗}(E_{1}) determines an automorphy factor J : P × D → GL(V) with

J(ug_{h}g_{ℓ}, x_{0}) = J_{1}(g_{h}, x_{1})λ_{1}(g_{ℓ})  (10.4.1)

where x_{1} = π(x_{0}) ∈ D_{1} denotes the basepoint in D_{1}. To simplify notation we will write j(g) and j_{1}(g_{h}) rather than J(g, x_{0}) and J_{1}(g_{h}, x_{1}).

By (5.8.1), the connection ∇_{1} in E_{1} determines a connection ∇^{J_{1}} = d + η_{1} in the J_{1}- trivialization E_{1} ≅ D_{1} × V with

η_{1}(q_{1∗}(X_{h})) = j_{1}(g_{h})ω_{1}(X_{h})j_{1}(g_{h})^{−1} − d_{X_{h}}(j_{1}(g)) ∘ j_{1}(g)^{−1}  (10.4.2)

for any g_{h} ∈ G_{h} and any X_{h} ∈ T_{g_{h}}G_{h}, where q_{1} : G_{h} → D_{1} denotes the projection.

The parabolically induced connection ∇ in E determines a connection ∇^{J} = d + η in the J-trivialization E ≅ D × V with

η(q_{∗}X) = j(g)ω(X)j(g)^{−1} − d_{X}(j(g)) ∘ j(g)^{−1}.  (10.4.3)

By (10.2.1) and (5.2.3) the connection forms η and η_{1} are related by η(q_{∗}(X)) = η_{1}(π_{∗}q_{∗}(X)) for any X ∈ T_{g}G. Take X = L_{g∗}(u̇ + ġ_{h} + ġ_{ℓ}) ∈ T_{g}G and let X_{h} = L_{g_{h}∗}(ġ_{h}) ∈ T_{g_{h}}G_{h} denote its projection to G_{h}. Then we have

j_{1}(g_{h})ω_{1}(X_{h})j_{1}(g_{h})^{−1} =j_{1}(g_{h})λ_{1}(g_{ℓ})ω(X)λ_{1}(g_{ℓ})^{−1}j_{1}(g_{h})^{−1}
− j_{1}(g_{h})d_{X}(λ_{1}(g_{ℓ}))λ_{1}(g_{ℓ})^{−1}j_{1}(g_{h})^{−1}

or, using (5.1.1)

ω_{1}(L_{g_{h}∗}(ġ_{h})) = λ_{1}(g_{ℓ})ω(X)λ_{1}(g_{ℓ}^{−1}) − λ_{1}(g_{ℓ})λ′_{1}(ġ_{ℓ})λ_{1}(g_{ℓ}^{−1})

10.5. Corollary. Suppose ω_{1} ∈ A^{1}(G, End(V)) commutes with the adjoint action of λ_{1} : G_{ℓ} → GL(V). Then the curvature form Ω of the parabolically induced connection ∇ = Φ^{∗}(∇_{1}) = d + ω satisfies the π-fiber condition,

Ω = π^{∗}(Ω_{1})

where Ω_{1} ∈ A_{bas}^{2}(G_{h}, End(V)) is the curvature form of ∇_{1} = d + ω_{1}.

10.6. Proof. Let us compute Ω(X, Y) where X = L_{g∗}(ẋ) and Y = L_{g∗}(ẏ), where ẋ, ẏ ∈ Lie(P) and where g = ug_{h}g_{ℓ} ∈ P. Set ẋ = u̇_{X} + ġ_{Xh} + ġ_{Xℓ} ∈ Lie(U_{P}) ⊕ 𝔤_{h} ⊕ 𝔤_{ℓ} (and similarly for ẏ). By (10.3.1),

ω(X) = ω_{1}(L_{g_{h}∗}(ġ_{Xh})) + λ′_{1}(ġ_{Xℓ})  (10.6.1)

and similarly for ω(Y). Set X_{h} = L_{g_{h}∗}(ġ_{Xh}) and Y_{h} = L_{g_{h}∗}(ġ_{Y h}). Using the structure equation (5.2.4) and the fact that Lie(U_{P}) is an ideal in Lie(P) gives

Ω(X, Y) = X(ω(Y)) − Y (ω(X)) − ω([X, Y]) + [ω(X), ω(Y)]
= X(ω_{1}(L_{g_{h}∗}(ġ_{Y h}))) + X(λ′_{1}(ġ_{Y ℓ})) − Y (ω_{1}(L_{g_{h}∗}(ġ_{Xh}))) − Y (λ′_{1}(ġ_{Xℓ}))
− ω_{1}(L_{g_{h}∗}([ġ_{Xh}, ġ_{Y h}])) − λ′_{1}([ġ_{Xℓ}, ġ_{Y ℓ}])
+ [ω_{1}(L_{g_{h}∗}(ġ_{Xh})), ω_{1}(L_{g_{h}∗}(ġ_{Y h}))] + [λ′_{1}(ġ_{Xℓ}), λ′_{1}(ġ_{Y ℓ})]
= X_{h}ω_{1}(L_{g_{h}∗}(ġ_{Y h}))) − Y_{h}ω_{1}(L_{g_{h}∗}(ġ_{Xh}))) − ω_{1}([X_{h}, Y_{h}]) + [ω_{1}(X_{h}), ω_{1}(Y_{h})]
= Ω_{1}(X_{h}, Y_{h}).

10.7. If ∇_{1} = d + ω_{1} is a connection on E_{1} which is invariant under a subgroup Γ_{h} ⊂ G_{h}, then by (10.3.1) the induced connection Φ^{∗}(∇_{1}) is invariant under the group ν_{h}^{∗}(Γ_{h}) ⊂ P which is obtained by first projecting Γ_{h} to G′_{h} = G_{h}/(G_{h} ∩ G_{ℓ}) then taking the pre-image under the projection ν_{h} : P → G′_{h} (cf. §7.1, §9.1).

As in §9 let Γ ⊂ G be a neat arithmetic subgroup with X = Γ\D. Write X_{1} = Γ_{h}\D_{1} for the stratum in \overline{X} = Γ\D^{∗} corresponding to the boundary component D_{1}. The homogeneous vectorbundles E → D and E_{1} → D_{1} pass to automorphic vectorbundles E′ → X and E′_{1} → X_{1} respectively. Parabolic induction then passes to an operation on these vectorbundles as follows. Suppose ∇′_{1} is a connection on E′_{1}. It pulls back to a Γ_{h}-invariant connection ∇_{1} on E_{1} → D_{1}. The parabolically induced connection ∇ = Φ^{∗}(∇_{1}) is invariant under Γ_{P} = Γ ∩ P ⊂ ν_{h}^{∗}(Γ_{h}) so it passes to a connection on Γ_{P} \E → Γ_{P} \X. Since T_{X_{1}}(ϵ_{0}) is a Γ-parabolic neighborhood of X_{1} in \overline{X} this defines a connection ∇′ = Φ_{XX_{1}}^{∗}(∇′_{1}) on the restriction E′|(X ∩ T_{X_{1}}(ϵ_{0})).

This procedure may be applied to any pair of strata, say, X_{2} < X_{1} of \overline{X}. Thus, if ∇′_{2} is any connection on the automorphic vectorbundle E′_{2} → X_{2} defined by λ, then we obtain a parabolically induced connection

∇′_{1} = Φ_{X_{1}X_{2}}^{∗}(∇′_{2})

on E′_{1}|(X_{1} ∩T_{X_{2}}(ϵ_{0})). However, if X_{3} < X_{2} < X_{1} are strata of \overline{X} and if ∇′_{3} is a connection on E′_{3} → X_{3} then the parabolically induced connection Φ_{X_{1}X_{3}}^{∗}(∇′_{3}) does not necessarily agree with the connection Φ_{X_{1}X_{2}}^{∗}Φ_{X_{2}X_{3}}^{∗}(∇′_{3}), even in the neighborhood X_{1}∩T_{X_{2}}(ϵ_{0})∩T_{X_{3}}(ϵ_{0}) where they are both defined, cf. Proposition 10.9.

10.8. We will also need the following more technical result concerning parabolic induction for the proof of the main theorem. Suppose P_{1} ≺ P_{2} ≺ … ≺ P_{r} are standard maximal parabolic subgroups with corresponding rational boundary components D_{1} ≺ D_{2} ≺ ⋯ ≺ D_{r}. Let Q = P_{1} ∩ P_{2} ∩ … ∩ P_{r}. Then Q^{♭} = P_{1} = U_{1}G_{1h}G_{1ℓ} so by (7.3.6), Q decomposes as Q = U_{Q}G_{1h}G_{Qℓ}.

Let λ be a representation of K with resulting homogeneous vectorbundles E_{i} → D_{i} (1 ≤ i ≤ r). By Proposition 8.2, λ extends to a representation λ_{1} of K_{1h}G_{1ℓ}. For 1 ≤ i < j ≤ r let Φ_{ji} : E_{j} → π_{ji}^{∗}E_{i} denote the vectorbundle isomorphism of (10.1.2) which covers the canonical projection π_{ji} : D_{j} → D_{i}. Let Φ_{r} : E → π_{r}^{∗}(E_{r}) be the vectorbundle isomorphism (10.1.2) which covers the canonical projection π_{r} : D → D_{r}.

10.9. Proposition. Suppose ∇_{1} = d + ω_{1} is a connection on E_{1} and suppose that the connection form ω_{1} ∈ A^{1}(G_{1h}, End(V)) commutes with the adjoint action of λ_{1}(G_{1ℓ}). Let ∇ = d + ω denote the connection

∇ = Φ_{r}^{∗}Φ_{r,r−1}^{∗} … Φ_{21}^{∗}(∇_{1}).

Then

ω(L_{g∗}(u̇_{Q} + ġ_{1h} + ġ_{Qℓ})) = ω_{1}(L_{g_{1h}∗}ġ_{1h})) + λ′_{1}(ġ_{Qℓ})  (10.9.1)

for any g = u_{Q}g_{1h}g_{Qℓ} ∈ Q and any u̇_{Q} + ġ_{1h} + ġ_{Qℓ} ∈ Lie(Q) = Lie(U_{Q}) + 𝔤_{1h} + 𝔤_{Qℓ}.

10.10. Proof. First we determine the connection form of the connection

∇_{r} = Φ_{r,r−1}^{∗}Φ_{r−1,r−2}^{∗} … Φ_{21}^{∗}(∇_{1}) = d + ω_{r}

on the vectorbundle E_{r} → D_{r}. Set P_{r} = U_{r}G_{rh}G_{rℓ}. The images under the projection ν_{rh} : P → G_{rh} of Q ⊂ P_{1} ∩ P_{r} are parabolic subgroups \overline{Q} ⊂ \overline{P}_{1} ⊂ G_{rh}. In fact, \overline{Q} is the parabolic subgroup corresponding to the flag of rational boundary components D_{1} ≺ D_{2} ≺ ⋯ ≺ D_{r−1} of D_{r}. By (7.3.4) there are compatible decompositions

P_{r} = U_{r}G_{rh}G_{rℓ}
P_{1} ∩ P_{r} = U_{r}(\overline{\overline{U}}G_{1h}G′_{ℓ})G_{rℓ}  with \overline{P}_{1} = \overline{\overline{U}}G_{1h}G′_{ℓ}
Q = U_{r}\overline{\overline{U}}G_{1h}(U_{\overline{Q}ℓ}G_{\overline{Q}ℓ})G_{rℓ} with \overline{Q} = \overline{\overline{U}}U_{\overline{Q}ℓ}G_{1h}G_{\overline{Q}ℓ}.

Corresponding to the maximal parabolic subgroup \overline{P}_{1} ⊂ G_{rh} the representation λ|K_{1h}K′_{ℓ} has a canonical extension

λ_{r1} : K_{1h}G′_{ℓ} → GL(V).

According to Proposition 8.4, λ_{r1}|G′_{ℓ} = λ_{1}|G′_{ℓ} which, by assumption, commutes with ω_{1}. So by induction, for any q̄ = u\overline{Q}g_{1h}g_{\overline{Q}ℓ} ∈ \overline{Q} and for any ˙ \overline{q} = u̇\overline{Q} + ġ_{1h} + ġ_{\overline{Q}ℓ} ∈ Lie(\overline{Q}),

ω_{r}(L_{q̄∗}(u̇\overline{Q} + ġ_{1h} + ġ_{\overline{Q}ℓ})) = ω_{1}(L_{g_{1h}∗}(ġ_{1h})) + λ′_{1}(ġ_{\overline{Q}ℓ}).

Moreover ∇ = Φ_{r}^{∗}(∇_{r}) = d + ω so by Proposition 10.3, for any g = u_{r}g_{rh}g_{rℓ} ∈ P_{r},

ω(L_{g∗}(u̇_{r} + ġ_{rh} + ġ_{rℓ})) = ω_{r}(L_{q̄∗}(ġ_{rh})) + λ′_{1}(ġ_{rℓ})

for any u̇_{r} + ġ_{rh} + q̇_{rℓ} ∈ Lie(P_{r}). Taking g_{rh} = q̄ = u\overline{Q}g_{1h}g_{\overline{Q}ℓ} ∈ \overline{Q} and

ġ_{rh} = ˙ q̄ = u̇\overline{Q} + ġ_{1h} + ġ_{\overline{Q}ℓ} ∈ Lie(\overline{Q})

gives equation (10.9.1):

ω(L_{g∗}(u̇_{r} + u̇\overline{Q} + ġ_{1h} + ġ_{\overline{Q}ℓ} + ġ_{rℓ})) = ω_{1}(L_{g_{1h}∗}(ġ_{1h})) + λ′_{1}(ġ_{\overline{Q}ℓ} + ġ_{rℓ}).

11. The Patched Connection

11.1. As §7, 9, suppose that D = G/K is a Hermitian symmetric space which is irreducible over ℚ, and that Γ ⊂ G(ℚ) is a neat arithmetic group. Let \overline{X} = Γ\D^{∗} denote the Baily Borel compactification of X = Γ\D with projection q : D^{∗} → \overline{X}. By lemma 9.6, for any sufficiently small ϵ_{0} > 0 there exists an ϵ_{0}-system of control data {T_{Y} (ϵ_{0}), π_{Y} , ρ_{Y} } (which we now fix) on X̄, so that π_{Y} is obtained from the canonical projection D → D_{1} whenever q(D_{1}) = Y and so that T_{Y} (ϵ) is a Γ-parabolic neighborhood of Y in \overline{X}. Applying §3.5 to this system of control data yields a partition of unity on each stratum Y of \overline{X},

B_{Y}^{ϵY} (y) + ∑_{Z<Y} B_{Z}^{ϵY} (y) = 1  (11.1.1)

for all y ∈ Y, where ϵY = ϵ_{0}/2^{dim Y} .

A choice of representation λ : K → GL(V) on some complex vectorspace V determines homogeneous vectorbundles E = G ×_{K} V on D and E_{1} = G_{h} ×_{K_{h}} V on D_{1} which pass to automorphic vectorbundles E′ = Γ\E on X and E′_{Y} → Y on Y. Here, D_{1} is a rational boundary component (with q(D_{1}) = Y), normalized by some maximal parabolic subgroup P = U_{P} G_{h}G_{ℓ}; and K_{h} = K ∩ G_{h}; cf. §7.1, 9.1. The Nomizu connections ∇_{D}^{Nom} (on E) and ∇_{1}^{Nom} (on E_{1}) pass to connections ∇_{X}^{Nom} on E′ → X and ∇_{Y}^{Nom} on E′_{Y} → Y respectively. We use an inductive procedure to define the patched connection ∇_{Y}^{p} on the vectorbundle E′_{Y} → Y, for any stratum Y ≤ X as follows. If Y ⊂ \overline{X} is a minimal stratum set ∇_{Y}^{p} = ∇_{Y}^{Nom}. Now suppose that the patched connection ∇_{Z}^{p} has been constructed on every stratum Z < Y.

11.2. Definition. The patched connection ∇_{Y}^{p} on E′_{Y} → Y is the connection

∇_{Y}^{p} = B_{Y}^{ϵY} ∇_{Y}^{Nom} + ∑_{Z<Y} B_{Z}^{ϵY} Φ_{Y Z}^{∗}(∇_{Z}^{p})  (11.2.1)

(where the sum is taken over all strata Z < Y in the Baily Borel compactification of X).

11.3. Remarks. The idea behind this construction may be explained when there are two singular strata Z < Y < X. A simpler candidate for a connection on X whose Chern forms might satisfy the π-fiber condition is

∇′_{X} = B_{Z}^{ϵX}Φ_{XZ}^{∗}∇_{Z}^{Nom} + B_{Y}^{ϵX}Φ_{XY}^{∗} ∇_{Y}^{Nom} + B_{X}^{ϵX}∇_{X}^{Nom}.  (11.3.1)

In the region T_{Y} (ϵX/2) only the first two terms contribute to ∇′_{X}. Both connection Φ_{XZ}^{∗}∇_{Z}^{Nom} and Φ_{XY}^{∗} ∇_{Y}^{Nom} have curvature forms which satisfy the π-fiber condition with respect to Y. However the curvature form of (and even the Chern forms of) any affine combination of these fails to satisfy the π-fiber condition. (cf. Figure 3.4: this occurs in the region where B_{Z} + B_{Y} = 1.) The remedy is to create a connection on X for which no nontrivial affine combination of Φ_{XZ}^{∗}∇_{Z}^{Nom} and Φ_{XY}^{∗} ∇_{Y}^{Nom} ever occurs. Replacing ∇_{Y}^{Nom} by ∇_{Y}^{p} in (11.3.1) gives

∇_{X}^{p} = B_{Z}^{ϵX}Φ_{XZ}^{∗}∇_{Z}^{Nom} + B_{Y}^{ϵX}B_{Z}^{ϵY} Φ_{XY}^{∗} Φ_{Y Z}^{∗}∇_{Z}^{Nom} + B_{Y}^{ϵX}B_{Y}^{ϵY} Φ_{XY}^{∗} ∇_{Y}^{Nom} + B_{X}^{ϵX}∇_{X}^{Nom}

Within the region T_{Y} (ϵX/2) only the first three terms appear: the first term alone appears in T_{Z}(ϵX/2); the first and second terms appear in T_{Z}(ϵX)−T_{Z}(ϵX/2); the second term alone appears in T_{Z}(ϵY/2) − T_{Z}(ϵX); the second and third terms appear in T_{Z}(ϵY) − T_{Z}(ϵY/2) and the third term alone appears outside T_{Z}(ϵY). In the region T_{Z}(ϵY) − T_{Z}(ϵY/2),

∇_{X}^{p} = Φ_{XY}^{∗} (B_{Z}^{ϵY} Φ_{Y Z}^{∗}∇_{Z}^{Nom} + B_{Y}^{ϵY} ∇_{Y}^{Nom}).

So ∇_{X}^{p} is parabolically induced from a connection on Y, and by Corollary 10.5 its curvature form satisfies the π-fiber condition relative to Y. In the region T_{Z}(ϵX) − T_{Z}(ϵX/2),

∇_{X}^{p} = (B_{Z}^{ϵX}Φ_{XZ}^{∗} + B_{Y}^{ϵX}B_{Z}^{ϵY} Φ_{XY}^{∗} Φ_{Y Z}^{∗})(∇_{Z}^{Nom}).

In this region, the curvature form still does not satisfy the π-fiber condition however we show in §12.10 that the difference (Φ_{XZ}^{∗} − Φ_{XY}^{∗} Φ_{Y Z}^{∗})ω_{Z}^{Nom} is nilpotent and commutes with the curvature form. This turns out to be enough (Lemma 6.4) to imply that the Chern forms of ∇_{X}^{p} satisfy the π-fiber condition with respect to Y.

11.4. Returning to the general case, suppose Z = Z(1) < Z(2) < ⋯ < Z(r) is a chain of strata in \overline{X}. Write Z ≤ Y if Z(r) = Y, that is, if the chain ends in Y. Write Y ≤ Z if Z(1) = Y, that is, if the chain begins at Y. Suppose Z is such a chain of strata and suppose x ∈ T_{Z(i)}(ϵ_{0}) for 1 ≤ i ≤ r. Denote by

ϵi = ϵ(Z(i)) = ϵ_{0}/2^{dim Z(i)}
B_{i}^{ϵ} = B_{Z(i)}^{ϵ}
Φ_{ji}^{∗} = Φ_{Z(j)Z(i)}^{∗} for j > i
π_{i}(x) = π_{Z(i)}(x).

Define

B_{Z}(x) = B_{r−1}^{ϵr}(π_{r}(x)) … B_{2}^{ϵ3}(π_{3}(x))B_{1}^{ϵ2}(π_{2}(x))  (11.4.1)
Φ_{Z}^{∗} = Φ_{r,r−1}^{∗} … Φ_{32}^{∗}Φ_{21}^{∗}.  (11.4.2)

If Z = {Y } consists of a single element, set B_{Z}(x) = 1 and Φ_{Z}^{∗} = Id. The following lemma is easily verified by induction.

11.5. Lemma. The patched connection may be expressed as follows,

∇_{Y}^{p} (x) = ∑_{Z} B_{Z}(x)Φ_{Z}^{∗} (B_{Z(1)}^{ϵZ(1)}(π_{Z(1)}(x))∇_{Z(1)}^{Nom})  (11.5.1)

where the sum is over all chains of strata Z ≤ Y ending in Y.

(One checks that, although the projection functions x ↦ π_{Z(i)}(x) are not everywhere defined, they occur in (11.5.1) with coefficient 0 unless x lies in the region of definition.)

Definition 11.2 constructs a patched connection ∇_{X}^{p} on each of the automorphic vectorbundles E′_{Y} → Y. The proof of the following theorem will appear in §12.

11.6. Theorem. The Chern forms {σ^{j}(∇_{Y}^{p} )}_{Y ≤X}of the patched connection ∇_{X}^{p} constitute a closed π-fiber differential form.

11.7. Corollary. For each j, the Chern form σ^{j}(∇_{X}^{p}) ∈ A_{π}^{2j}(\overline{X}; ℂ) of the patched connection determines a lift

c̄^{j}(E′) = [σ^{j}(∇_{X}^{p})] ∈ H^{2j}(\overline{X}; ℂ)  (11.7.1)

of the Chern class c^{j}(E′) ∈ H^{2j}(X; ℂ) which is independent of the choices that were made in its construction. For any stratum closure i : \overline{Y} ↪ \overline{X} the restriction i^{∗}c̄^{j}(E′) is equal to the Chern class c̄^{j}(E′_{Y}) ∈ H^{2j}(\overline{Y}; ℂ) of the automorphic vectorbundle E′_{Y} → Y.

11.8. Proof of Corollary 11.7. The restriction map i^{∗} : H^{2j}(\overline{X}) → H^{2j}(X) associates to any π-fiber differential form ω ∈ A_{π}^{2j}(\overline{X}) the cohomology class [ω_{X}] of the differential form ω_{X} ∈ A^{2j}(X) on the nonsingular part. Hence i^{∗}(c̄^{j}(E′)) = c^{j}(E′) ∈ H^{2j}(X; ℂ) since the latter is independent of the connection.

The patched connection ∇_{X}^{p} depends on the choice of a pair (partition of unity, control data which is subordinate to the canonical projections {π_{Z}}) (see §3 and §2). It is tedious but standard to check that two such choices are connected by a smooth 1-parameter family of choices (partition of unity, control data subordinate to {π_{Z}}). The resulting patched connections ∇_{0}^{p} and ∇_{1}^{p} are therefore connected by a smooth 1-parameter family of patched connections ∇_{t}^{p}, each of whose Chern forms is a π-fiber differential form. So the usual argument (e.g. [KN] Chapt. XII lemma 5; [MiS]) produces a differential 2j − 1 form Ψ such that σ^{j}(∇_{1}^{p}) − σ^{j}(∇_{0}^{p}) = dΨ. It is easy to see that Ψ ∈ A_{π}^{2j−1}(\overline{X}; ℂ) is a π-fiber differential form. Consequently the π-fiber cohomology classes coincide: [σ^{j}(∇_{1}^{p})] = [σ^{j}(∇_{0}^{p})] ∈ H^{2j}(\overline{X}; ℂ). The second statement follows from the analogous statement in Theorem 4.2

11.9. Remarks. Theorem 11.6 and Corollary 11.7 extend to the case that G is semisimple over ℚ. The restriction map H^{2j}(\overline{X}; ℂ) → H^{2j}(X; ℂ) factors as follows,

H^{2j}(\overline{X}; ℂ) → IH^{2j}(\overline{X}; ℂ) → H_{2n−2j}(\overline{X}; ℂ) → H_{2n−2j}(\overline{X}, ∂\overline{X}; ℂ) ≅ H^{2j}(X; ℂ)

where 2n = dim_{ℝ}(X), and where ∂\overline{X} denotes the singular set of the Baily-Borel compactification \overline{X}. The Chern class c^{j}(E′) ∈ H^{2j}(X; ℂ) lives in the last group. For any toroidal resolution of singularities τ : \overline{X}_{Σ} → \overline{X}, the pushdown

c_{n−j}(E′) = τ_{∗}(c^{j}(Ē′_{Σ}) ∩ [\overline{X}_{Σ}]) ∈ H_{2n−2j}(\overline{X}; ℤ)

of the Chern class of Mumford’s canonical extension ([Mu1]) Ē′_{Σ} of E′ gives a canonical lift of c^{j}(E′) to the homology of the Baily-Borel compactification. In §15 (in the case of the tangent bundle) we identify this with the (homology) Chern class of the constructible function 1_{X}. In [BBF] it is shown that every algebraic homology class (including c_{n−j}(E′)) admits a (non-canonical) lift to middle intersection homology with rational coefficients.

12. Proof of Theorem 11.6

12.1. Preliminaries. As in §9.1, let q : D^{⋆} → \overline{X} denote the projection. If ϵ_{0} > 0 is sufficiently small, then for each stratum Z of \overline{X} the preimage

q^{−1}(T_{Z}(ϵ_{0})) = ∐_{q(D_{1})=Z} U_{D_{1}}(ϵ_{0})

is a disjoint union of Γ-parabolic neighborhoods U_{D_{1}}(ϵ_{0}) of those boundary components D_{1} such that q(D_{1}) = Z. For such a boundary component define

χ_{D_{1}}(x) = \begin{cases}1 if x ∈ U_{D_{1}}(ϵ_{0}) \\ 0 otherwise \end{cases}

to be the characteristic function of U_{D_{1}}.

Fix a stratum Y and a choice D_{2} of rational boundary component such that q(D_{2}) = Y.. Denote by P_{2} = U_{2}G_{2h}G_{2ℓ} the rational maximal parabolic subgroup of G which normalizes D_{2} and by ν_{h} : P_{2} → G_{2h} the projection as in §9.1. By Proposition 8.2 the representation λ|K ∩ P_{2} extends to a representation λ_{2} of K_{2h}G_{2ℓ}. Set Γ_{h} = ν_{h}(Γ ∩ P_{2}).

The partition of unity B_{Y}^{ϵY} + ∑_{Z<Y} B_{Z}^{ϵY} = 1 on Y pulls back to a Γ_{h}-invariant locally finite partition of unity on D_{2},

B_{D_{2}}^{ϵY} + ∑_{D_{1}≺D_{2}} B_{D_{1}}^{ϵY} = 1

where the sum is over all rational boundary components D_{1} ≺ D_{2}, where B_{D_{1}}^{ϵY} = q^{∗}(B_{Z}^{ϵY})χ_{D_{1}} (and similarly for B_{D_{2}}^{ϵY}, however χ_{D_{2}} = 1 on D_{2}). The patched connection ∇_{Y}^{p} on the vectorbundle E′_{Y} → Y pulls back to a Γ_{h}-invariant connection ∇_{2}^{p} = q^{∗}(∇_{Y}^{p} ) on the homogeneous vectorbundle E_{2} = G_{2h} ×_{K_{2h}} V. This connection may also be described as the affine locally finite combination

∇_{2}^{p} = B_{D_{2}}^{ϵY}∇_{2}^{Nom} + ∑_{D_{1}≺D_{2}} B_{D_{1}}^{ϵY}Φ_{21}^{∗}(∇_{1}^{p})  (12.1.1)

where, for each rational boundary component D_{1} ≺ D_{2} the obvious notation holds: ∇_{1}^{p} is the patched connection on E_{1} → D_{1} and Φ_{21} : E_{2} → π_{21}^{∗}(E_{1}) is the vectorbundle isomorphism which is obtained from (10.1.3) upon replacing G by G_{2h}.

Denote by ω_{2}^{p} ∈ A^{1}(G_{2h}, End(V)) the connection form of ∇_{2}^{p}. The curvature form Ω_{Y}^{p} ∈ A^{2}(Y, End(E′_{Y})) of ∇_{Y}^{p} coincides with the curvature form Ω_{2}^{p} ∈ A_{bas}^{2}(G_{2h}, End(V)) of ∇_{2}^{p} under the canonical isomorphism

A^{2}(Y, End(E′_{Y})) ≅ A^{2}(D_{2}, End(E_{2}))^{Γ_{h}} ≅ A_{bas}^{2}(G_{2h}, End(V))^{Γ_{h}}  (12.1.2)

where the superscript Γ_{h} denotes the Γ_{h}-invariant differential forms.

12.2. Proposition. Let D_{2} be a rational boundary component of D = G/K. Then the connection form ω_{2}^{p} ∈ A^{1}(G_{2h}, End(V)) and the curvature form Ω_{2}^{p} ∈ A_{bas}^{2}(G_{2h}, End(V)) commute with the adjoint action of λ_{2}(G_{2ℓ}) ⊂ GL(V).

12.3. Proof. The proof uses a double induction over boundary components D_{2} in D^{∗}. However, so as to avoid the horribly complicated notation which would arise in the proof, we rephrase the double induction as follows:

1. By induction we assume the proposition has been proven for every rational boundary component D′_{2} of any Hermitian symmetric domain D′ = G′/K for which dim(D′) < dim(D). (The case dim(D′) = 0 is trivial.)

2. For our given domain D, we assume the proposition has been proven for every rational boundary component D_{1} of D such that dim(D_{1}) < dim(D_{2}). (The case D_{1} = ϕ is trivial.)

To prove Proposition 12.2 for D_{2} ⊂ D^{∗}, it suffices to verify that λ_{2}(G_{2ℓ}) commutes with the connection form of each of the connections appearing in the linear combination (12.1.1). The connection form ω_{2}^{Nom} of the Nomizu connection ∇_{2}^{Nom} is given by

ω_{2}^{Nom}(L_{g_{2h}∗})(ġ_{2h}) = λ′(k̇_{2h}) = λ′_{2}(k̇_{2h})

for any g_{2h} ∈ G_{2h} and for any ġ_{2h} ∈ 𝔤_{2h}, where ġ_{2h} = k̇_{2h} + ṗ_{2h} is its Cartan decomposition. This commutes with λ_{2}(G_{2ℓ}) since G_{2h} and G_{2ℓ} commute.

Now consider any boundary component D_{1} ≺ D_{2} which appears in the sum (12.1.1). Let P_{1} = U_{1}G_{1h}G_{1ℓ} be the rational parabolic subgroup which normalizes D_{1}. Decompose the intersection

P = P_{1} ∩ P_{2} = U_{2}(\overline{\overline{U}}G_{1h}G′_{ℓ})G_{2ℓ}

according to (7.3.4), setting P_{h} = \overline{\overline{U}}G_{1h}G′_{ℓ} ⊂ G_{2h}. Let ∇_{1}^{p} be the patched connection on E_{1} → D_{1}. According to Proposition 10.3, the connection form ω_{21} of the parabolically induced connection Φ_{21}^{∗}(∇_{1}^{p}) is given by

ω_{21}(L_{g∗}( ˙¯ū) + ġ_{1h} + ġ′_{ℓ}) = λ′_{21}(ġ′_{ℓ}) + Ad(λ_{21}(g′_{ℓ})^{−1})(ω_{1}^{p}(L_{g_{1h∗}}(ġ_{1h}))).

Here, g = ¯ ūg_{1h}g′_{ℓ} ∈ P_{h} and ˙ ¯ ū + ġ_{1h} + ġ′_{ℓ} ∈ Lie(P_{h}) and λ_{21} : K_{1h}G′_{ℓ} → GL(V) is as in Proposition 8.2. Since dim Y < dim X we may apply the first induction hypothesis and conclude that the adjoint action of λ_{21}(G′_{ℓ}) commutes with the connection form ω_{1}^{p} ∈ A^{1}(G_{1h}, End(V)). Hence, using Proposition 8.4,

ω_{21}(L_{g∗}( ˙¯ū) + ġ_{1h} + ġ′_{ℓ}) = λ′_{1}(ġ′_{ℓ}) + ω_{1}^{p}(L_{g_{1h∗}}(ġ_{1h})).

The group G′_{ℓ} commutes with G_{2ℓ} so the first term λ′_{1}(ġ′_{ℓ}) commutes with λ_{1}(G_{2ℓ}). By the second induction hypothesis, the connection form ω_{1}^{p} ∈ A^{1}(G_{1h}, End(V)) also commutes with λ_{1}(G_{1ℓ}). But λ_{1}(G_{1ℓ}) ⊃ λ_{1}(G_{2ℓ}) = λ_{2}(G_{2ℓ}) by Proposition 8.4 again, which completes the proof that the connection form of the patched connection commutes with λ_{2}(G_{2ℓ}).

12.4. Let x ∈ X be a point near the boundary of \overline{X}. Then there is a maximal collection of strata Y_{1}, Y_{2}, …, Y_{t} such that x ∈ T_{ϵ_{0}}(Y_{i}) for each i, and we may assume they form a partial flag,

Y_{1} < Y_{2} < ⋯ < Y_{t} = X.  (12.4.1)

Let W ≤ X be the largest stratum in this collection such that B_{W}^{ϵW} (π_{W} (x)) ≠ 0. Such a stratum exists since B_{Y_{1}}^{ϵY_{1}}(π_{Y_{1}}(x)) = 1. (Choosing W in this way guarantees that, if W ≠ X, then for every stratum Z > W in this partial flag, at the point π_{Z}(x) the connection ∇_{Z}^{p} is an affine combination of connections induced from smaller strata and contains no contribution from ∇_{Z}^{Nom}, because B_{Z}^{ϵZ}(π_{Z}(x)) = 0.)

12.5. Proposition. At the point x ∈ X, ∑_{W ≤S≤X} B_{S}(x) = 1 and

∇_{X}^{p} (x) = ( ∑_{W ≤S≤X} B_{S}(x)Φ_{S}^{∗}) (∇_{W}^{p} (π_{W} (x)))  (12.5.1)

where the sum is over sub-chains S in the partial flag (12.4.1) which begin at W and end at X.

12.6. Proof. By Lemma 11.5 applied to ∇_{W}^{p} , we need to show that

∇_{X}^{p}(x) = ( ∑_{W ≤S≤X} B_{S}(x)Φ_{S}^{∗}) ( ∑_{R≤W} B_{R}(π_{W} (x))Φ_{R}^{∗} (B_{R(1)}^{ϵR(1)}∇_{R(1)}^{Nom}(π_{R(1)}(x))))) . (12.6.1)

By Lemma 11.5, ∇_{X}^{p} (x) is a sum over chains Z ≤ X of terms

B_{Z}(x)Φ_{Z}^{∗}B_{Z(1)}^{ϵZ(1)}∇_{Z(1)}^{Nom}(π_{Z(1)}(x)).

For any ϵ ≤ ϵ_{0}, B_{Z}^{ϵ}(x) = 0 unless Z occurs in the collection {Y_{1}, Y_{2}, …, Y_{t} = X}. By assumption the term B_{Z(1)}^{ϵZ(1)}π_{Z(1)}(x) also vanishes unless Z(1) ≤ W. Therefore each chain Z = Z(1) < ⋯ < X appearing in the sum may be assumed to occur as a sub-chain of Y_{1} < Y_{2} < ⋯ < Y_{t} = X, and we may also assume the chain begins at Z(1) ≤ W. We claim that if such a chain Z occurs with nonzero coefficient, then the stratum W must appear in the chain. For if not, then Z(k) < W < Z(k + 1) for some k. But the term B_{Z}(x) contains a factor

B_{Z(k)}^{ϵZ(k+1)}(π_{Z(k+1)}(x)).

Since B_{W}^{ϵW} (π_{W} (x)) ≠ 0, this factor vanishes by Lemma 3.6, which proves the claim. Summarizing, every chain Z which occurs with nonzero coefficient in the sum may be described as

R(1) < R(2) < ⋯ < R(r) = W = S(1) < S(2) < ⋯ < S(s) = X.

The contribution to ∇_{X}^{p}(x) in (11.5.1) from such a chain is the product of

B_{S(s−1)}^{ϵX}(π_{X}(x)) … B_{W}^{ϵS(2)}(π_{S(2)}(x))Φ_{XS(s−1)}^{∗} … Φ_{S(2)W}^{∗}

with

B_{R(r−1)}^{ϵW}(π_{W} (x)) … B_{R(1)}^{ϵR(2)}(π_{R(2)}(x))Φ_{W R(r−1)}^{∗} … Φ_{R(2)R(1)}^{∗}

applied to

B_{R(1)}^{ϵR(1)}∇_{R(1)}^{Nom}(π_{R(1)}(x)).

However this product is exactly a single term in (12.6.1) and every such product occurs exactly once, which verifies (12.5.1). Since the coefficients in (11.5.1) sum to 1, so also does ∑_{W ≤S≤X} B_{S}(x).

12.7. We must prove that the Chern forms of the patched connection satisfy the π-fiber condition near each stratum of \overline{X}. Let Y be such a stratum and let x ∈ X ∩ T_{Y} (ϵX/2). We will verify the π-fiber condition relative to the stratum Y at the point x. The point x ∈ X lies in an intersection of ϵ_{0}-tubular neighborhoods of a maximal collection of strata, which (we may assume) form a partial flag Z_{1} < Z_{2} < ⋯ < X. Let W be the largest stratum in this chain such that B_{W}^{ϵW} (π_{W} (x)) ≠ 0. Then W ≤ Y by (3.2.2). Consider the subchain lying between W and Y, which we shall denote by

W = Y_{1} < Y_{2} < ⋯ < Y_{t} = Y.

Fix corresponding boundary components D_{1} ≺ D_{2} ≺ ⋯ ≺ D_{t} and let P_{1} ≺ P_{2} ≺ … ≺ P_{t} be their normalizing maximal parabolic subgroups. Set P = P_{1} ∩ P_{2} ∩ … ∩ P_{t}. Let ∇_{W}^{p} be the patched connection on E′_{W} → W. According to Proposition 12.5,

∇_{X}^{p}(x) = ( ∑_{W ≤S≤X} B_{S}(x)Φ_{S}^{∗}) ∇_{W}^{p} (π_{W} (x)).  (12.7.1)

However the only chains S = {S(1) < ⋯ < S(s)} which occur with nonzero coefficient in this sum satisfy

{S(1), S(2), …, S(s − 1)} ⊂ {Y_{1}, Y_{2}, …, Y_{t}}, S(1) = W = Y_{1}, S(s) = X  (12.7.2)

for the following reason. Suppose a chain S contains a stratum larger than Y (but not equal to X). Let Z be the largest such stratum occurring in S. Then the first factor in B_{S}(x) is B_{Z}^{ϵX}(x), cf. (11.4.1). By assumption, x ∈ T_{Y} (ϵX/2). So by (3.2.2) (with ϵ = ϵX and where the roles of Y and Z are reversed), B_{Z}^{ϵX}(x) = 0.

If W = Y then (12.7.1) becomes ∇_{X}^{p} = Φ_{XY}^{∗} (∇_{Y}^{p} ) so by Corollary 10.5 and Proposition 12.2 the curvature form Ω_{X}^{p} of ∇_{X}^{p} satisfies the π-fiber condition with respect to Y. So the same is true of every Chern form which proves Theorem 11.6 in this case.

Therefore we may assume that W < Y. As in §12.1 the connection ∇_{W}^{p} on E′_{W} → W pulls back to a Γ_{h}-invariant connection ∇_{1}^{p} on E_{1} → D_{1} and the connection ∇_{X}^{p} on E′ → X pulls back to a Γ-invariant connection ∇^{p} on E → D. As in (12.1.2) we identify the curvature form Ω_{X}^{p} of ∇_{X}^{p} with the curvature form Ω^{p} of ∇^{p} under the canonical isomorphism

A^{2}(X, End(E′)) ≅ A^{2}(D, End(E))^{Γ} ≅ A_{bas}^{2}(G, End(V))^{Γ}.

Similarly identify the curvature Ω_{1}^{p} of ∇_{1}^{p} with the curvature Ω_{W}^{p} of ∇_{W}^{p} . Choose a lift x̃ ∈ D of x, which lies in the intersection

U_{D_{1}}(ϵ_{0}) ∩ U_{D_{2}}(ϵ_{0}) ∩ … ∩ U_{D_{t}}(ϵ_{0})

of Γ-parabolic neighborhoods of the boundary components D_{1} ≺ D_{2} ≺ … ≺ D_{t}. Let π : D → D_{1} denote the canonical projection.

12.8. Lemma. For any tangent vectors U, V ∈ Tx̃D,

Ω^{p}(U, V) = π^{∗}Ω_{1}^{p}(U, V) + n ∈ End(V)

for some nilpotent element n ∈ λ′_{1}(𝔤_{1ℓ}). (Here, P_{1} = U_{P_{1}}G_{1h}G_{1ℓ}; 𝔤_{1ℓ} = Lie(G_{1ℓ}); and λ_{1} is the extension (Proposition 8.2) of the representation λ|K ∩ P_{1}.)

12.9. Proof. We will compute the curvature Ω_{X}^{p} of ∇_{X}^{p}. Each chain S satisfying (12.7.2) corresponds also to a chain of rational boundary components D_{1} = D_{S(1)} ≺ D_{S(2)} ≺ … ≺ D_{S(s)} = D. Let B̃_{S} = q^{∗}B_{S} denote the pullback to D. It follows from Proposition 12.5 that ∑_{S} B̃_{S}(x̃) = 1 (where the sum is over all chains S which appear in (12.7.1)). Choose any ordered labeling of the chains S which appear in (12.7.1), say S_{1}, S_{2}, …, S_{M}. By §2.1 (4),

∑_{i=1}^{M} B_{S_{i}}(x)π_{S_{i}}^{∗}(Ω_{W}^{p} ) = ∑_{i=1}^{M} B_{S_{i}}(x)π_{XW}^{∗} Ω_{W}^{p} = π_{XW}^{∗} Ω_{W}^{p} .

Let U, V ∈ T_{x}X. Pulling back the equation (12.7.1) to D and using Lemma 6.2 gives:

Ω^{p}(U, V) =π^{∗}(Ω_{1}^{p})(U, V) + ∑_{i=1}^{M−1} dB̃_{S_{i}} ∧ (Φ_{S_{i}}^{∗}ω_{1}^{p} − Φ_{S_{M}}^{∗} ω_{1}^{p})(U, V)

− ∑_{i<j} B̃_{S_{i}}(x)B̃_{S_{j}}(x) [Φ_{S_{i}}^{∗}ω_{1}^{p}(U) − Φ_{S_{j}}^{∗}ω_{1}^{p}(U), Φ_{S_{i}}^{∗}ω_{1}^{p}(V) − Φ_{S_{j}}^{∗}ω_{1}^{p}(V)]

where Φ_{S}^{∗}ω_{1}^{p} denotes the connection form of Φ_{S}^{∗}∇_{1}^{p}. Let us compute this connection form. Suppose that S is a chain satisfying (12.7.2). For 1 ≤ j ≤ s−1 let P_{S(j)} be the corresponding normalizing maximal parabolic subgroup and set Q = P_{S(1)} ∩ P_{S(2)} ∩ … ∩ P_{S(s−1)}. Then P ⊂ Q and P^{♭} = Q^{♭} = P_{1} which implies (as in §7.3) that Q ⊂ P ⊂ P_{1} have compatible decompositions,

P_{1} = U_{1}G_{1h}G_{1ℓ}
Q = U_{1}G_{1h}(U_{P_{1}Q}G_{Qℓ}) with U_{Q} = U_{1}U_{P_{1}Q}
P = U_{1}G_{1h}U_{P_{1}Q}(U_{QP} G_{P ℓ}) with U_{P} = U_{1}U_{P_{1}Q}U_{QP} .

Here, U_{QP} G_{P ℓ} is the parabolic subgroup of G_{Qℓ} determined by P ⊂ Q. We also note that

U_{P_{1}P} = U_{P_{1}Q}U_{QP}  (12.9.1)

is the unipotent radical of the parabolic subgroup ν_{1ℓ}(P) ⊂ G_{1ℓ} determined by P (where ν_{1ℓ} : P_{1} → G_{1ℓ} is the projection). Let 𝔑_{P_{1}P} denote its Lie algebra.

By Proposition 12.2, the connection form ω_{1}^{p}(π_{W} (x)) ∈ A^{1}(G_{1h}, End(V)) commutes with the adjoint action of λ_{1}(G_{1ℓ}) ⊂ GL(V). So we may apply Proposition 10.9 to determine Φ_{S}^{∗}(ω_{1}^{p}). Let g = u_{P} g_{1h}g_{P ℓ} ∈ P = U_{P} G_{1h}G_{P ℓ} and let

ġ = u̇_{1} + ġ_{1h} + u̇_{P_{1}Q} + u̇_{QP} + ġ_{P ℓ} ∈ Lie(U_{1}G_{1h}U_{P_{1}Q}U_{QP} G_{P ℓ}).

Apply Proposition 10.9 using u̇_{Q} = u̇_{1} + u̇_{P_{1}Q} ∈ Lie(U_{Q}) and ġ_{Qℓ} = u̇_{QP} + ġ_{P ℓ} ∈ 𝔤_{Qℓ} to find:

Φ_{S}^{∗}(ω_{1}^{p})(L_{g∗}(ġ)) = ω_{1}^{p}(L_{g_{1h}∗}ġ_{1h})) + λ′_{1}(u̇_{QP}) + λ′_{1}(ġ_{P ℓ}) .

Moreover, λ′_{1}(u̇_{QP}) ∈ λ′_{1}(𝔑_{P_{1}P}) ⊂ λ′_{1}(𝔤_{1ℓ}).

Now suppose that R is another chain in the sum (12.7.1) which makes a nonzero contribution Φ_{R}^{∗}∇_{1}^{p} = d+Φ_{R}^{∗}(ω_{1}^{p}) to the connection ∇^{p}, say, W = R(1) < R(2) < ⋯ < R(r−1) < X. Then Q′ = P_{R(1)} ∩ P_{R(2)} ∩ … ∩ P_{R(r−1)} and P also have compatible decompositions:

Q′ = U_{1}G_{1h}(U_{P_{1}Q′}G_{Q′ℓ}) with U_{Q′} = U_{1}U_{P_{1}Q′}
P = U_{1}G_{1h}U_{P_{1}Q′}(U_{Q′P} G_{P ℓ}) with U_{P} = U_{1}U_{P_{1}Q′}U_{Q′P} .

The same element ġ ∈ Lie(P) decomposes as

ġ = u̇_{1} + ġ_{1h} + u̇_{P_{1}Q′} + u̇_{Q′P} + ġ_{P ℓ}.

So the same argument gives Φ_{R}^{∗}(ω_{1}^{p})(L_{g∗}(ġ)) = ω_{1}^{p}(L_{g_{1h}∗}(ġ_{1h})) + λ′_{1}(u̇_{Q′P}) + λ′_{1}(ġ_{P ℓ}). We conclude that:

(Φ_{S}^{∗}ω_{1}^{p} − Φ_{R}^{∗}ω_{1}^{p})(L_{g∗}(ġ)) = λ′_{1}(u̇_{QP} − u̇_{Q′P}) ∈ λ′_{1}(𝔑_{P_{1}P}) ⊂ λ′_{1}(𝔤_{1ℓ}).  (12.9.2)

Consequently each term in the sum (12.9) (except for the first) lies in 𝔑_{P_{1}P} .

12.10. Completion of the proof. Using Lemma 12.8, at the point x̃ we may write

Ω^{p}(U, V) = π^{∗}(Ω_{1}^{p})(U, V) + n

where n ∈ λ′_{1}(G_{1ℓ}) is nilpotent and in fact lies in 𝔑_{P_{1}P} . Moreover, by Proposition 12.2, the curvature Ω_{1}^{p} commutes with n. If f : End(V) → ℂ is any Ad-invariant polynomial, it follows from Lemma 6.4 that f(Ω^{p}(U, V)) = f(Ω_{1}^{p}(π_{∗}U,)π_{∗}V) hence also f(Ω_{X}^{p}(U, V)) = f(Ω_{W}^{p} (π_{XW ∗}U, π_{XW ∗}V)). So it follows from (6.3.1) that the corresponding characteristic form satisfies the π-fiber condition relative to W. This completes the proof of Theorem 11.6.

13. Toroidal compactification

13.1. Throughout this section we assume that G = G(ℝ) is the set of real points of a connected semisimple algebraic group G defined over ℚ, that D = G/K is a Hermitian symmetric space, Γ ⊂ G(ℚ) is a neat arithmetic group, X = Γ\G/K is the corresponding locally symmetric space with Baily-Borel Satake compactification \overline{X} = Γ\D^{∗}. Fix a representation λ : K → GL(V) on some complex vectorspace V and let E = G ×_{K} V be the corresponding homogeneous vectorbundle on D, and E′ = Γ\E the automorphic vectorbundle on X. Choose a system of control data on the Baily-Borel compactification \overline{X} and a partition of unity as in §3, and let ∇_{X}^{p} denote the resulting patched connection on E′ → X. For each i, the Chern form σ^{i}(∇_{X}^{p}) ∈ Ω_{π}^{•}(\overline{X}; ℂ) (cf §6.3) is a π-fiber differential form on \overline{X} so it determines a cohomology class c̄^{i}(E′) = [σ^{i}(∇_{X}^{p})] ∈ H^{2i}(\overline{X}; ℂ).

We also fix a nonsingular toroidal compactification \overline{X}_{Σ}. This corresponds to a Γ-compat-ible collection of simplicial polyhedral cone decompositions Σ_{F} of certain self adjoint homogeneous cones. These compactifications were constructed in [AMRT] and are reviewed in [Har3], [HZ1], [FC], [Na]. In [Mu1], D. Mumford shows that the automorphic vectorbundle E′ → X admits a canonical extension Ē′_{Σ} over the toroidal compactification \overline{X}_{Σ}. In [Har3] Theorem 4.2, M. Harris shows that Mumford’s canonical extension coincides with Deligne’s canonical extension [D] (for an appropriately chosen flat connection with unipotent monodromy).

The identity mapping X → X has a unique continuous extension, τ : \overline{X}_{Σ} → \overline{X} of X, and this is a resolution of singularities.

13.2. Theorem. The patched connection ∇_{X}^{p} on E′ → X extends to a smooth connection \overline{∇}_{Σ}^{p} on Ē′_{Σ} → \overline{X}_{Σ}. Moreover for each i,

τ^{∗}c̄^{i}(E′) = τ^{∗}([σ^{i}(∇_{X}^{p})]) = [σ^{i}(\overline{∇}_{Σ}^{p})] = c^{i}(Ē′_{Σ}) ∈ H^{2i}(\overline{X}_{Σ}; ℂ).  (13.2.1)

The proof will appear in Section 14. S. Zucker has pointed out that it follows from mixed Hodge theory that the image of c̄^{i}(E′) in Gr_{2i}^{W} H^{2i}(\overline{X}; ℂ) is uniquely determined by (13.2.1).

13.3. Proportionality theorem. Fix representations λ_{j} : K → GL(V_{j}) for j = 1, 2, …, r and fix nonnegative integers I = (i_{1}, i_{2}, … i_{r}) with i_{1} + i_{2} + … + i_{r} = n = dim(D). For j = 1, 2, …, r let E′_{j} = Γ\G ×_{K} V_{j} → X be the resulting automorphic vectorbundle on X and let Ě_{j} = G_{u} ×_{K} V_{j} be the corresponding vectorbundle on the compact dual symmetric space Ď = G_{u}/K (where G_{u} is a compact real form of G containing K). Define “generalized” Chern numbers

č^{I}(λ_{1}, λ_{2}, …, λ_{r}) = (c^{i_{1}}(Ě_{1}) ∪ c^{i_{2}}(Ě_{2}) ∪ … ∪ c^{i_{r}}(Ě_{r})) ∩ [Ď] ∈ ℤ  (13.3.1)

and

(c̄^{I}(λ_{1}, λ_{2}, …, λ_{r}) = c̄^{i_{1}}(∇_{1}^{p}) ∪ c̄^{i_{2}}(∇_{2}^{p}) ∪ … ∪ c^{i_{r}}(∇_{r}^{p})) ∩ [\overline{X}] ∈ ℂ  (13.3.2)

where ∇_{j}^{p} denotes the patched connection on E′_{j} → X and where [\overline{X}] ∈ H_{2n}(\overline{X}; ℂ) denotes the fundamental class of the Baily-Borel compactification. Let v(Γ) ∈ ℚ denote the constant which appears in the proportionality theorem of Hirzebruch [Hr1], [Mu1].

13.4. Proposition. For any choice λ_{1}, λ_{2}, …, λ_{r} of representations and for any partition I = (i_{1}, i_{2}, …, i_{r}) of n = dim_{ℂ}(D) we have

c̄^{I}(λ_{1}, λ_{2}, …, λ_{r}) = v(Γ)č^{I}(λ_{1}, λ_{2}, …, λ_{r}).  (13.4.1)

13.5. Proof. Each of the vectorbundles E′_{j} has a canonical extension Ē′_{j,Σ} → \overline{X}_{Σ}. The same proof as in [Mu1] (which is the same proof as in [Hr1]) (cf. [Hr3]) shows that the Chern classes of these extended bundles satisfy the proportionality formula

(c^{i_{1}}(Ē′_{1,Σ}) ∪ c^{i_{2}}(Ē′_{2,Σ}) ∪ … ∪ c^{i_{r}}(Ē′_{r,Σ})) ∩ [\overline{X}_{Σ}] = v(Γ)č^{I}(λ_{1}, λ_{2}, …, λ_{r}).  (13.5.1)

The result now follows immediately from Theorem 13.2 .

14. Proof of Theorem 13.2

14.1. Fix ϵ ≤ ϵ_{0}. We claim that the partition of unity (3.3.1) ∑_{Z≤X} B_{Z}^{ϵ} = 1 on \overline{X} pulls back to a smooth partition of unity on \overline{X}_{Σ}. Fix a pair of strata Y, Z of \overline{X}. We must verify that τ^{∗}B_{Z}^{ϵ} is smooth near τ^{−1}(Y). It can be shown that the mapping τ : \overline{X}_{Σ} → \overline{X} is a complex analytic morphism between complex analytic varieties, as is the projection π_{Y} : T_{Y} (ϵ) → Y. It follows that the composition π_{Y} τ : τ^{−1}T_{Y} (ϵ) → Y is a complex analytic morphism between smooth complex varieties, so it is smooth. If the stratum Z is not comparable to Y or if Z > Y then B_{Z}^{ϵ} vanishes on T_{Y} (ϵ/2) hence τ^{∗}B_{Z}^{ϵ} vanishes on τ^{−1}(T_{Y} (ϵ/2)) so it is smooth. If Z ≤ Y then by (3.2.4) and (3.2.5), τ^{∗}B_{Z}^{ϵ}(x) = B_{Z}^{ϵ}π_{Y} τ(x) for all x ∈ τ^{−1}(T_{Y} (ϵ/2)) so τ^{∗}B_{Z}^{ϵ} is smooth in this open set.

14.2. We may assume that G is simple over ℚ. The “boundary” \overline{X}_{Σ} − X of the toroidal compactification has a distinguished covering by open sets U_{Y}, one for each stratum Y ⊂ \overline{X} of the Baily-Borel compactification, such that τ(U_{Y}) ⊂ \overline{X} is a neighborhood of Y, and for which the restriction Ē′_{Σ}|U_{Y} arises from an automorphy factor ([HZ1] §3.3).

14.3. Proposition. For any smooth connection ∇_{Y} on (E′_{Y} , Y) the parabolically induced connection Φ_{XY}^{∗} (∇_{Y}) (which is defined only on E′|(U_{Y} ∩ X) = Ē′_{Σ}|(U_{Y} ∩ X)) extends canonically to a smooth connection (which we denote by \overline{Φ}_{XY}^{∗} (∇_{Y})) on \overline{E}′_{Σ}|U_{Y} .

14.4. Proof of Theorem 13.2. We may assume that ϵ_{0} > 0 was chosen so small that T_{Y} (ϵ_{0}) ⊂ U_{Y} for each stratum Y < X of \overline{X}. By §14.1 the partition of unity which is used to construct the patched connection ∇_{X}^{p} extends to a smooth partition of unity on \overline{X}_{Σ}. Hence, using Proposition 14.3,

\overline{∇}_{Σ}^{p} = τ^{∗}B_{X}^{ϵ_{X}} + ∑_{Y <X} τ^{∗}B_{Y}^{ϵX} \overline{Φ}_{XY}^{∗} (∇_{Y}^{p} )

is a smooth connection on \overline{E}′_{Σ} → \overline{X}_{Σ} which coincides with the patched connection ∇_{X}^{p} on E′ → X. Therefore its Chern forms are smooth and everywhere defined and they restrict to the Chern forms of ∇_{X}^{p}. It follows from Lemma 4.4 that each Chern class of \overline{∇}_{Σ}^{p} is the pullback of the corresponding Chern class of ∇_{X}^{p}.

The remainder of §13 is devoted to the proof of Proposition 14.3 which is essentially proven in [HZ1] (3.3.9) (following [Har2]). We will now verify the details.

14.5. Let us recall the construction [AMRT] of the toroidal compactification. Fix a rational boundary component F with normalizing parabolic subgroup P = UG_{h}G_{ℓ} and let Y = Γ_{h}\F ⊂ \overline{X} denote the corresponding stratum in the Baily-Borel compactification of X = Γ\D. Let Z_{F} = Center(U) and 𝔷 = Lie(Z_{F}). The vectorspace 𝔷 is preserved under the adjoint action of G_{ℓ} and contains a unique open orbit C_{F} ⊂ 𝔷; it is a rationally defined self adjoint homogeneous cone. Its Satake compactification C_{F}^{∗} consists of C_{F} together with all its rational boundary components (in the Satake topology). The toroidal compactification is associated to a collection Σ = {Σ_{F} } of rational polyhedral cone decompositions of the various C_{F}^{∗} which are compatible under Γ.

Let Ď denote the compact dual symmetric space (so Ď = G(ℂ)/K(ℂ)P^{−} in the notation of Proposition 8.2; cf.[AMRT], [Sa]) and let β : D → Ď denote the Borel embedding. Set D_{F} = Z_{F} (ℂ).β(D). The domain D_{F} is homogeneous under P.Z_{F} (ℂ) and it admits “Siegel coordinates” D(F) ≅ Z_{F} (ℂ)×ℂ^{a} ×F in which the subset β(D) ⊂ D_{F} is defined by a certain well known inequality ([AMRT] p. 239, [Sa] §III (7.4)). Now consider the commutative diagram [HZ1] (1.2.5), reproduced in figure 5.

D  ⟶ Γ′_{F} \D ⊂ D_{F,Σ} ⟶_{φ_{F,Σ}} \overline{X}_{Σ}
∩  ∩  ∩
D_{F}  ⟶_{Γ′_{F}} M′_{F} ⊂ M′_{F,Σ}
θ_{2}↓  ^{π_{2}}↓^{T_{F}}  ^{π_{2,Σ}}↓^{T_{F,Σ}}
D_{F} /Z_{F} (ℂ) ⟶ A_{F}  A_{F}
θ_{1}↓  ^{π_{1}}↓
F  ⟶^{Γ_{h}} Y

Figure 5. Toroidal compactification

Here, Γ′_{F} = Γ ∩ (G_{h}U) and M′_{F} = Γ′_{F} \D_{F}. The algebraic torus T_{F} = Z_{F} (ℂ)/(Γ ∩ Z_{F}) acts on M′_{F} with quotient A_{F}, which is in turn an abelian scheme over Y. The choice Σ_{F} of polyhedral cone decomposition determines a torus embedding T_{F} ↪ T_{F,Σ} and a partial compactification M′_{F,Σ} = M′_{F} ×_{T_{F}} T_{F,Σ} of M′_{F} . Let D_{F,Σ} denote the interior of the closure of M′_{F} = Γ′_{F} \D in M′_{F,Σ}. The quotient mapping Γ′_{F} \D → X extends to a local isomorphism φ_{F,Σ} : D_{F,Σ} → \overline{X}_{Σ} (cf. [AMRT] p. 250). In other words, φ_{F,Σ} is an open analytic mapping with discrete fibers which, near the boundary, induces an embedding D_{F,Σ}/(Γ′_{F} \Γ_{P}) ↪ \overline{X}_{Σ} whose image is the neighborhood U_{Y} referred to in §14.2. The mappings φ_{F,Σ} for the various strata Y ⊂ \overline{X} cover the boundary of \overline{X}_{Σ}. The composition θ_{1}θ_{2}|D : D → F coincides with the canonical projection π.

14.6. Each of the spaces in Figure 5 comes equipped with a vectorbundle and most of the mappings in this diagram are covered by vectorbundle isomorphisms. We will make the following notational convention: If E_{1} → M_{1} and E_{2} → M_{2} are (smooth) vectorbundles, α : M_{1} → M_{2} is a (smooth) mapping, and Φ : E_{1} → E_{2} is a vectorbundle mapping which induces an isomorphism E_{1} ≅ α^{∗}(E_{2}), then we will write Φ : (E_{1}, M_{1}) ∼ (E_{2}, M_{2}) and refer to Φ as being a vectorbundle isomorphism which covers α.

As in [Mu1], complexify λ : K → GL(V) and extend it trivially over P^{−} to obtain a representation λ̃ : K(ℂ)P^{−} → GL(V). The homogeneous vectorbundle E = G ×_{K} V has a canonical extension

Ě = G(ℂ) ×_{K(ℂ)P^{−}} V

over the compact dual symmetric space Ď. Its restriction E_{F} to D_{F} is a P Z_{F} (ℂ)-homogen-eous bundle, and it passes to a vectorbundle E′_{F} → M′_{F} upon dividing by Γ′_{F} . The restriction E′_{F} |(Γ′_{F} \D) coincides with the vectorbundle obtained from the homogeneous vectorbundle E = G ×_{K} V upon dividing by Γ′_{F}. We will denote this restriction also by (E′_{F} , Γ′_{F} \D).

Define Ẽ = P.Z_{F} (ℂ) ×_{K_{P} .Z_{F} (ℂ)} V. This vectorbundle on D_{F} /Z_{F} (ℂ) is homogeneous under P.Z_{F} (ℂ) and it passes to a vectorbundle E_{F}^{A} → A_{F} upon dividing by Γ′_{F} . As in [HZ1] (3.2.1) and (3.3.5), there is a canonical vectorbundle isomorphism

ψ : (E′_{F} , M′_{F}) ∼ (E_{F}^{A}, A_{F})  (14.6.1)

which covers π_{2}. In fact, the isomorphism ψ is obtained from the canonical isomorphism of P.Z_{F} (ℂ)-homogeneous vectorbundles,

Ψ : (E_{F} , D_{F}) ∼ (Ẽ, D_{F} /Z_{F} (ℂ))  (14.6.2)

which covers θ_{2} and which is given by the quotient mapping

E_{F} = P.Z_{F} (ℂ) ×_{K_{P}} V → Ẽ = P.Z_{F} (ℂ) ×_{K_{P} .Z_{F} (ℂ)} V.

Let Ē′_{Σ} denote Mumford’s canonical extension of the vectorbundle E′ = Γ\E → X to the toroidal compactification \overline{X}_{Σ}, and let Ē′_{F,Σ} = φ_{F,Σ}^{∗}(E′_{Σ}) be its pullback to D_{F,Σ}. Then we have a further canonical identification E′ = Ē′_{F,Σ}|(Γ′_{F} \D) = E′_{F} |(Γ′_{F} \D). We also have vectorbundles E_{h} = G_{h} ×_{K_{h}} V on F and its quotient E′_{Y} → Y = Γ_{h}\F.

According to [HZ1] (3.3.9) (which in turn relies on [Har2]), the canonical isomorphism 14.6.1 extends to a vectorbundle isomorphism

ψ_{Σ} : (Ē′_{F,Σ}, D_{F,Σ}) ∼ (E_{F}^{A}, A_{F})  (14.6.3)

which covers π_{2,Σ}. This is the key point in the argument: the isomorphism ψ_{Σ} identifies Mumford’s canonical extension (which is defined using a growth condition on a singular connection) with a vectorbundle, π_{2,Σ}^{∗}(E_{F}^{A}) which is defined topologically, and which is trivial on each torus embedding π_{2,Σ}^{−1}(a) ≅ T_{F,Σ}. We will use this isomorphism to extend the parabolically induced connection over the toroidal compactification, because such a parabolically induced connection is also pulled up from E_{F}^{A}.

As in (10.1.1), define an action of P.Z_{F} (ℂ) on the vectorbundle E_{h} → F by

ug_{h}g_{ℓ}z.[g′_{h}, v] = [g_{h}g′_{h}, λ_{1}(g_{ℓ})v]  (14.6.4)

(where u ∈ U_{P}, g_{h}, g′_{h} ∈ G_{h}, g_{ℓ} ∈ G_{ℓ}, z ∈ Z_{f}(ℂ), and v ∈ V). Define a mapping

Φ̃_{F} : P Z_{F} (ℂ) ×_{K_{P} Z_{F} (ℂ)} V → G_{h} ×_{K_{h}} V  (14.6.5)

by Φ̃_{F} ([ug_{h}g_{ℓ}z, v]) = [g_{h}, λ_{1}(g_{ℓ})v]. Then Φ̃_{F} is well defined, it is P.Z_{F} (ℂ)-invariant, and it gives a P.Z_{F} (ℂ)-equivariant isomorphism of vectorbundles,

Φ_{F} : (Ẽ, D_{F} /Z_{F} (ℂ)) ∼ (E_{h}, F)  (14.6.6)

which covers θ_{1}. Moreover the composition Φ_{F} Ψ|(E, D) is precisely the isomorphism Φ : (E, D) ≅ π^{∗}(E_{h}, F) of (10.1.2). In summary, this array of vectorbundles appears in figure 6.

(E, D) − −−→^{mod Γ′_{F}}(E′_{F} , Γ′_{F} \D) − −−→^{extend} (Ē′_{F,Σ}, D_{F,Σ})− −−→^{extend}(Ē′_{Σ}, \overline{X}_{Σ})

\begin{matrix}extend & ↓ \\ (E_{F} , D_{F}) \\ Ψ & ↓ & ∼ \end{matrix} − −−→\begin{matrix}extend & ↓ \\ (E′_{F} , M′_{F}) \\ ψ & ↓ & ∼ \end{matrix}  ^{ψ_{Σ}}↓^{∼}

(Ẽ, D_{F} /Z_{F} (ℂ))− −−→ (E_{F}^{A}, A_{F})  (E_{F}^{A}, A_{F})
Φ↓^{∼}  ↓
(E_{h}, F) − −−→^{mod Γ_{h}} (E′_{Y} , Y)

Figure 6. Vectorbundles on the toroidal compactification

14.7. Each of the vectorbundles in Figure 6 comes equipped with a connection. Let ∇_{Y} be a given connection on E′_{Y} → Y and let ∇_{h} be its pullback to E_{h} → F. Define ˜∇ = Φ_{F}^{∗} (∇_{h}) to be the pullback of ∇_{h} under the isomorphism (14.6.6). Then Ψ^{∗}( ˜∇) is an extension of the parabolically induced connection Φ^{∗}(∇_{h}) on (E, D). Both connections are invariant under Γ′_{F} . Let ∇′_{F} denote the resulting connection on the quotient (E′_{F} , M′_{F}) (where again we use the same symbol to denote this connection as well as its restriction to (E′_{F} , Γ′_{F} \D)). We need to show that this connection ∇′_{F} on (E′_{F} , Γ′_{F} \D) has a smooth extension to a connection ∇′_{F,Σ} on (Ē′_{F,Σ}, D_{F,Σ}) which is invariant under Γ′_{F} \Γ_{P} .

The connection ˜∇ passes to a connection ∇_{F}^{A} on (E_{F}^{A}, A_{F}) such that ∇′_{F} = ψ^{∗}(∇_{F}^{A}). Therefore the connection

∇′_{F,Σ} = ψ_{Σ}^{∗}(∇_{F}^{A})

on (Ē′_{F,Σ}, M′_{F,Σ}) is a smooth extension of ∇′_{F} . The Γ′_{F} \Γ_{P} -invariance of ∇′_{F,Σ} follows from the P.Z_{F} (ℂ)-invariance of ˜∇. This completes the proof of Proposition 14.3.

15. Chern classes and constructible functions

15.1. A constructible function F : W → ℤ on a complete (complex) algebraic variety W is one which is constant on the strata of some algebraic (Whitney) stratification of W. The Euler characteristic of such a constructible function F is the sum

χ(W; F) = ∑_{α} χ(W_{α})F (W_{α})

over strata W_{α} ⊂ W along which the function F is constant. If f : W → W′ is an (proper) algebraic mapping, then the pushforward of the constructible function F is the constructible function

f_{∗}(F)(w′) = χ(f^{−1}(w′); F)  (15.1.1)

(for any w′ ∈ W′). According to [Mac], for each constructible function F : W → ℤ it is possible to associate a unique Chern class c_{∗}(W; F) ∈ H_{∗}(W; ℤ) which depends linearly on F, such that f_{∗}c_{∗}(W; F) = c_{∗}(W′; f_{∗}F) (whenever f : W → W′ is a proper morphism), and such that c_{∗}(W; 1_{W}) = c^{∗}(W) ∩ [W] if W is nonsingular. (Here, [W] ∈ H_{2 dim(W)}(W; ℤ) denotes the fundamental class of W.) The MacPherson-Schwartz Chern class of W is the Chern class of the constructible function 1_{W} .

15.2. Now let \overline{Z} be a nonsingular complete complex algebraic variety and let D = D_{1} ∪ D_{2} ∪ … ∪ D_{m} be a union of smooth divisors with normal crossings in \overline{Z}. Set Z = \overline{Z} − D. The tangent bundle T_{Z} of Z has a “logarithmic” extension to \overline{Z},

T\overline{Z}(− log D) = Hom(Ω1 ; \overline{Z}(log D), O\overline{Z})

which is called the “log-tangent bundle” of (\overline{Z}, D). It is the vectorbundle whose sheaf of sections near any k-fold multi-intersection {z_{1} = z_{2} = … = z_{k} = 0} of the divisors is generated by z_{1} \frac{∂}{∂z_{1}} , z_{2} \frac{∂}{∂z_{2}} , …, z_{k} \frac{∂}{∂z_{k}} , z_{k+1}, …, z_{n} (where n = dim(Z)). The following result was discovered independently by P. Aluffi [Al].

15.3. Proposition. The Chern class of the log tangent bundle is equal to the Chern class of the constructible function which is 1 on Z = \overline{Z} − D, that is,

c^{∗}(T\overline{Z}(− log D)) ∩ [\overline{Z}] = c_{∗}(1_{Z}).

15.4. Proof. For any subset I ⊂ {1, 2, …, m} let D_{I} = ⋂_{i∈I} D_{i}, let

D^{I} = D_{I} ∩ ⋃_{j∉I} D_{j}

denote the “trace” of the divisor D in D_{I}, and let D_{I}^{o} = D_{I} − D^{I} denote its complement. The restriction of the log tangent bundle of (\overline{Z}, D) to any intersection D_{I} is (topologically) isomorphic to the direct sum of vectorbundles

T\overline{Z}(− log D)|D_{I} ≅ T_{D_{I}}(− log(D^{I})) ⊕ |I|1  (15.4.1)

(the last symbol denoting |I| copies of the trivial bundle). (This follows from the short exact sequence of locally free sheaves on D_{j},

0 ⟶ Ω_{D_{j}}^{1}(log D^{\{j\}}) ⟶ Ω1 ; \overline{Z}(log D)|D_{j} ⟶ O_{D_{j}} ⟶ 0

by dualizing and induction.) We will prove Proposition 15.3 by induction on the number m of divisors, with the case m = 0 being trivial. For any constructible function F on \overline{Z}, denote by c(F) ∈ H^{∗}(\overline{Z}) the Poincaré dual of the (homology) Chern class of F. Each divisor D_{j} carries a fundamental homology class whose Poincaré dual we denote by [D_{j}] ∈ H^{2}(\overline{Z}). The Chern class of the line bundle O(D_{j}) is 1 + [D_{j}]. Let c̃ denote the Chern class of the bundle T\overline{Z}(− log D). If I ⊂ \{1, 2, …, m\} and if i : D_{I} → \overline{Z} denotes the inclusion then

c̃ · [D_{I}] = i_{∗}(c(T\overline{Z}(− log D))|D_{I}) = i_{∗}c(T_{D_{I}}(− log(D^{I})) = i_{∗}c(1_{D_{I}^{o}})

by (15.4.1) and induction. Using [Tu] Proposition 1.2 we see,

c(\overline{Z}) = c̃ · ∏_{i}(1 + [D_{i}]) = c̃ + c̃ · ∑_{I}[D_{I}]

= c̃ + ∑_{I} c(1_{D_{I}^{o}}) = c̃ + c(1_{D})

since each point in D occurs in exactly one multi-intersection of divisors.

15.5. Theorem. Let X = Γ\G/K be a Hermitian locally symmetric space as in §9, with Baily-Borel compactification \overline{X}. Let c̄^{i}(\overline{X}) ∈ H^{2i}(\overline{X}; ℂ) denote the cohomology Chern class of the tangent bundle, constructed in Theorem 11.6. Then its homology image

c̄^{∗}(\overline{X}) ∩ [\overline{X}] = c_{∗}(1_{X}) ∈ H_{∗}(\overline{X}; ℤ)

lies in integral homology and coincides with the (MacPherson) Chern class of the constructible function which is 1 on X and is 0 on \overline{X} − X.

15.6. Proof. Let τ : \overline{X}_{Σ} → \overline{X} denote a smooth toroidal resolution of singularities, having chosen the system of polyhedral cone decompositions Σ so that the exceptional divisor D is a union of smooth divisors with normal crossings. Let T_{\overline{X}_{Σ}}(− log D) denote the log tangent bundle of (\overline{X}_{Σ}, D). As in [Mu1] Prop. 3.4, this bundle is isomorphic to Mumford’s canonical extension \overline{T}_{X,Σ} of the tangent bundle. Therefore

c̄^{∗}(\overline{X}) ∩ [\overline{X}] = τ_{∗}(τ^{∗}c̄^{∗}(\overline{X}) ∩ [\overline{X}_{Σ}]))
= τ_{∗}(c^{∗}(\overline{T}_{X,Σ}) ∩ [\overline{X}_{Σ}])
= τ_{∗}c_{∗}(1_{X}) = c_{∗}(τ_{∗}(1_{X})) = c_{∗}(1_{X})

by Theorem 13.2, Proposition 15.3 and (15.1.1).

15.7. Corollary. The MacPherson-Schwartz Chern class of the Baily-Borel compactification \overline{X} is given by the sum over strata Y ⊂ \overline{X},

c_{∗}(1\overline{X}) = c_{∗}( ∑_{Y ⊂\overline{X}} 1_{Y}) = ∑_{Y ⊂\overline{X}} i_{∗}c̄^{∗}(\overline{Y}) ∩ [\overline{Y}]

where i : \overline{Y} ↪ \overline{X} is the inclusion of the closure of Y (which is also the Baily-Borel compactification of Y) into \overline{X}.

16. Cohomology of the Baily-Borel Compactification

16.1. Let K be a compact Lie group and let EK → BK be the universal principal K-bundle. For any representation λ : K → GL(V) on a complex vectorspace V, let E_{λ} = EK ×_{K} V be the associated vectorbundle. The Chern classes c^{i}(E) ∈ H^{2i}(BK; ℂ) of all such vectorbundles generate a subalgebra which we denote H_{Chern}^{∗}(BK; ℂ). Two cases are of particular interest: if K = U(n) then BK = lim_{k→∞} G_{n}(ℂ^{n+k}) is the infinite Grassmann manifold and H^{∗}(BK; ℂ) = H_{Chern}^{∗}(BK; ℂ). In fact, the standard representation λ : U(n) → GL_{n}(ℂ) gives rise to a single vectorbundle E_{λ} → BK such that the algebra H^{∗}(BK; ℂ) is canonically isomorphic to the polynomial algebra in the Chern classes c^{1}(E_{λ}), c^{2}(E_{λ}), …, c^{n}(E_{λ}). If K = SO(n) then BK = lim_{k→∞} G_{n}^{o}(ℝ^{n+k}) is the infinite Grassmann manifold of real oriented n-planes. Let λ̃ : SO(n) → GL_{n}(ℝ) be the standard representation with resulting vectorbundle Eλ̃ → BK, and let λ : SO(n) → GL_{n}(ℂ) denote the composition of λ̃ with the inclusion GL_{n}(ℝ) ⊂ GL_{n}(ℂ). The associated vectorbundle E_{λ} = Eλ̃(ℂ) is the complexification of Eλ̃. If n is odd, then H^{∗}(BK; ℂ) is canonically isomorphic to the polynomial algebra generated by the Pontrjagin classes p^{i}(Eλ̃) = c^{2i}(E_{λ}) ∈ H^{4i}(BK; ℂ) for i = 1, 2, …, n. Hence H^{∗}(BK; ℂ) = H_{Chern}^{∗}(BK; ℂ). If n is even then the algebra H^{∗}(BK; ℂ) has an additional generator, the Euler class e = e(Eλ̃) ∈ H^{n}(BK; ℂ). (It satisfies e^{2} = p^{n/2}.) If n = 2 then e is the first Chern class of the line bundle corresponding to the representation SO(2) ≅ U(1) ⊂ GL_{1}(ℂ).

16.2. Now suppose that K = K_{1} × K_{2} × … × K_{r} is a product of unitary groups, odd orthogonal groups, and copies of SO(2). According to the preceding paragraph, there are representations λ_{1}, …, λ_{r} of K on certain complex vectorspaces V_{1}, V_{2}, …, V_{r} so that the Chern classes of the resulting “universal” complex vectorbundles E_{i} = EK ×_{K} V_{i} → BK generate the polynomial algebra H^{∗}(BK; ℂ) = H_{Chern}^{∗}(BK; ℂ).

16.3. Suppose that G is a semisimple algebraic group defined over ℚ, and that G(ℝ)^{0} acts as the identity component of the group of automorphisms of a Hermitian symmetric space D = G/K. Recall ([He] X §6, [Bo2]) that the irreducible components of D come from the following list:

Type Symmetric Space  Compact Dual
AIII U(p, q)/U(p) × U(q)  U(p + q)/U(p) × U(q)
DIII SO^{∗}(2n)/U(n)  SO(2n)/U(n)
BDI SO(p, 2)/SO(p) × SO(2) SO(p + 2)/SO(p) × SO(2)
CI Sp(n, ℝ)/U(n)  Sp(n)/U(n)
EIII E_{6}^{3}/Spin(10) × SO(2)  E_{6}/Spin(10) × SO(2)
EVII E_{7}^{3}/E_{6} × SO(2)  E_{7}/E_{6} × SO(2)

Let X = Γ\G/K, with Γ ⊂ G(ℚ) a neat arithmetic group, and let \overline{X} denote the Baily-Borel compactification of X. Let Ď = G_{u}/K be the compact dual symmetric space, where G_{u} ⊂ G(ℂ) is a compact real form containing K. The principal bundles Γ\G → X and G_{u} → Ď are classified by mappings Φ : X → BK and Ψ : Ď → BK (respectively) which are uniquely determined up to homotopy. A theorem of Borel [Bo2] states that (in this Hermitian case) the resulting homomorphism Ψ^{∗} : H^{∗}(BK; ℂ) → H^{∗}(Ď; ℂ) is surjective.

Suppose the irreducible factors of D = G/K are of type AIII, DIII, CI, or BDI for p odd or p = 2. The construction of π-fiber Chern forms in Section 11 determines a homomorphism Φ̃^{∗} : H^{∗}(BK; ℂ) → H^{∗}(\overline{X}; ℂ) by setting Φ̃^{∗}(c^{i}(E_{j})) = c̄^{i}(E′_{j}) (where E_{j} → BK is the universal vectorbundle corresponding to the representation λ_{j} of §16.2 and E′_{j} → X is the corresponding automorphic vectorbundle). Let us denote the image of Φ̃^{∗} by H_{Chern}^{∗}(\overline{X}; ℂ).

16.4. Theorem. Suppose X = Γ\G/K is a Hermitian locally symmetric space such that the irreducible factors of D = G/K are of type AIII, DIII, CI, or BDI for p odd or p = 2. Then the mappings Φ̃^{∗} and Ψ^{∗} determine a surjection

h : H_{Chern}^{∗}(\overline{X}; ℂ) → H^{∗}(Ď; ℂ)  (16.4.1)

from this subalgebra of the cohomology of the Baily-Borel compactification, to the cohomology of the compact dual symmetric space. Moreover, for each “universal” vectorbundle E_{j} → BK we have

h(c̄^{i}(E′_{j})) = c^{i}(Ě_{j})

where E′_{j} → X and Ě_{j} → Ď are the associated automorphic and homogeneous vectorbundles, respectively.

16.5. Proof. Define the mapping h : H_{Chern}^{∗}(\overline{X}; ℂ) → H^{∗}(Ď; ℂ) by hΦ̃^{∗}(c) = Ψ^{∗}(c) for any c ∈ H^{∗}(BK; ℂ). If this is well defined, it is surjective by Borel’s theorem. To show it is well defined, let us suppose that Φ̃^{∗}(c) = 0. We must show that Ψ^{∗}(c) = 0, so we assume the contrary.

Let x = Ψ^{∗}(c) ∈ H^{i}(Ď; ℂ). By Poincaré duality, there exists a complementary class y ∈ H^{2n−i}(Ď; ℂ) so that (x ∪ y) ∩ [Ď] ≠ 0 (where n = dim_{ℂ}(Ď)). Then y has a lift, d ∈ H^{2n−i}(BK; ℂ) with Ψ^{∗}(d) = y. Let us write K = K_{1}K_{2} … K_{r} for the decomposition of K into irreducible factors. By §16.2 the polynomial algebra H^{∗}(BK; ℂ) is generated by the Chern classes of the universal vectorbundles E_{1}, E_{2}, …, E_{r} corresponding to representations λ_{i} : K_{i} → GL(V_{i}). Hence, both c and d are polynomials in the Chern classes of the vectorbundles E_{1}, E_{2}, …, E_{r}. Hence (Φ̃^{∗}(c) ∪ Φ̃^{∗}(d)) ∩ [\overline{X}] ∈ ℂ is a sum of “generalized” Chern numbers which, by the Proposition 13.4, coincides with the corresponding sum of “generalized” Chern numbers for the compact dual symmetric space, v(Γ)(x ∪ y) ∩ [Ď] ≠ 0. This implies that Φ̃^{∗}(c) ≠ 0 which is a contradiction.

16.6. Remarks. We do not know whether the surjection (16.4.1) has a canonical splitting. However, the intersection cohomology IH^{∗}(\overline{X}; ℂ) contains, in a canonical way, a copy of the cohomology H^{∗}(Ď; ℂ) of the compact dual symmetric space. By the Zucker conjecture ([Lo] and [SS]), the intersection cohomology may be identified with the L^{2} cohomology of X which, in turn may be identified with the relative Lie algebra cohomology H^{∗}(𝔤, K; L^{2}(Γ\G)). But L^{2}(Γ\G) contains a copy of the trivial representation 1 (the constant functions), whose cohomology H^{∗}(𝔤, K; 1) ≅ H^{∗}(Ď; ℂ) is the cohomology of the compact dual symmetric space. We sketch a proof that the following diagram commutes.

H_{Chern}^{2k}(\overline{X}; ℂ) ⟶_{j} IH^{2k}(\overline{X}; ℂ)
h↓  ↑^{i}
H^{2k}(Ď; ℂ)  H^{2k}(Ď; ℂ)

If E′ → X and Ě → Ď are vectorbundles arising from the same representation λ of K then the class j(c̄^{k}(E′)) is represented by the differential form σ^{k}(∇_{X}^{p}) which is π-fiber, hence bounded, hence L^{2}. The class i(c^{k}(Ě)) is represented by the differential form σ^{k}(∇_{X}^{Nom}) which is “invariant” (meaning that its pullback to D is invariant), hence L^{2}. The intersection cohomology of \overline{X} embeds into the ordinary cohomology of any toroidal resolution \overline{X}_{Σ}. But when these two differential forms are considered on \overline{X}_{Σ}, they both represent the same cohomology class, c^{k}(\overline{E}_{Σ}) (using Theorem 13.2 and [Mu1]). Alternatively, one may deform the connection ∇_{X}^{p} to ∇_{X}^{Nom}, obtaining a differential form Ψ ∈ A^{2k−1}(X) such that dΨ = σ^{k}(∇_{X}^{p}) − σ^{k}(∇_{X}^{Nom}), and check that Ψ is L^{2}.

In the case BDI, the compact dual is Ď = SO(p + 2)/SO(p) × SO(2). Suppose p is even. Its cohomology H^{∗}(Ď; ℂ) has a basis {1, c_{1}, c_{1}^{2}, …, c_{1}^{p−1}, e} where c_{1} is the Chern class of the complexification of the line bundle arising from the standard representation of SO(2) and where e is the Euler class of the vectorbundle arising from the standard representation of SO(p), [BoH] §16.5. All these classes lift canonically to IH^{∗}(\overline{X}; ℂ) and c_{1}^{j} lifts further to the (ordinary) cohomology of the Baily-Borel compactification. However (except in the case p = 2) we do not know whether e also lifts further to H^{∗}(\overline{X}; ℂ).

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