SINGULARITIES AND CHERN-WEIL THEORY, I

The Local MacPherson Formula

Dedicated to Kunihiko Kodaira, our teacher and friend

F. Reese Harvey and H. Blaine Lawson, Jr.*

Abstract.

Let α : E → F be a smooth bundle map between vector bundles with connection on a manifold X, and let Φ(Ω) be a Chern-Weil characteristic form of either E or F. A notion of “geometric atomicity” for α is introduced. For any such map α we establish a canonical cohomology

(∗) Φ(Ω) − ∑_{k≥0} Res_{Φ,k} [Σ_{k}(α)] = dT

where Σ_{k}(α) = {x ∈ X : dim ker(α) = k}, Res_{Φ,k} is a

smooth residue form along Σ_{k}(α), and T is a canonical L_{loc}^{1}- form on X. When rank E = rank F, (*) can be written

Φ(Ω^{F}) − Φ(Ω^{E}) = ∑_{k>0} Res_{Φ,k} [Σ_{k}(α)] + dT.

Normal sections of Hom(E, F) (those by definition which are transversal to the universal singularity sets Σ_{k}) are always geometrically atomic, and for such maps equation (*) expresses a classical formula of R. MacPherson at the level of forms and currents. Every real analytic map α is geometrically atomic, no matter how misbehaved its singularities. For those where each Σ_{k}(α) has the expected dimension, analogous formulas are established. In all cases, each term in the sum in equation (*) is a d-closed current. Proofs entail a direct application of the methods of singular connections and of finite volume flows developed by the authors.

Geometrically atomic maps prove to be generic or “typical” in all structured situations such as: direct sum mappings, tensor product mappings, mappings given by Clifford multiplication, etc. In each case the methods yield new formulas. This will be done in Part II.

*Research of both authors partially supported by the NSF.

Table of Contents

0. Introduction

1. Characteristic Currents

2. The Universal Case

3. Morse-Stokes Kernels

4. Morse-Stokes Operators

5. Analysis of the Currents Γ_{0} and Γ_{∞} — The Decomposition 6. Analysis of the Currents Γ_{0} and Γ_{∞} — The Residues

7. The First Main Theorem

8. The Formula in the Universal Case

9. Existence for Normal Bundle Maps — Geometric Atomicity

10. The Local MacPherson Formula

11. Real Analytic Bundle Maps

12. Analysis of the Currents Res_{Φ,k}[Σ_{k}] and Residue Calculations.

13. Results for Real Vector Bundles

§0. Introduction. Some of the most useful theorems in topology are those which relate singularities of maps to topological invariants, such as Hopf’s Theorem on vector fields or the Lefschetz Fixed-point Theorem. One of the most general results of this type is the beautiful formula of R. MacPherson [Mac_{∗}] which relates the topology of the primary singularities of a normal smooth bundle map α : E → F to characteristic classes of E and F

In geometry the classical theory of Gauss-Chern-Weil relates topological invariants to local curvature data. Given two connections on a smooth bundle and a characteristic polynomial Φ, the theory produces a formula: Φ(Ω_{1}) − Φ(Ω_{2}) = dT, where Ω_{i} is the curvature of the i^{th} connection and T is a canonically defined smooth form. The gauge-invariant forms T are important in the study of the space of connections and they lead to well-known secondary invariants [CS], [ChS].

The aim here is to combine these results and derive MacPherson-type formulas locally on the manifold. Assume bundles E and F are equipped with metrics and connections, and let α : E → F be a smooth bundle map. We shall derive formulas which explicitly express each Chern-Weil form Φ(Ω) of E or F as a sum

(0.1)  Φ(Ω) = ∑_{k} Res_{Φ,k}[Σ_{k}(α)] + dT

where Σ_{k}(α) = {x : dim ker(α_{x}) = k}, Res_{Φ,k} is a smooth residue form defined along Σ_{k}(α), and T is a canonical transgression form with L_{loc}^{1}-coefficients. The sum on the right in (0.1) is a characteristic current. It is the Chern-Weil representative of the class Φ for a certain singular connection in the sense of [HL_{1}].

When rank E = rank F, equation (0.1) has the form

Φ(Ω^{F}) − Φ(Ω^{E}) = ∑_{k>0} Res_{Φ,k} [Σ_{k}(α)] + dT,

expressing the difference of the Φ-characteristic classes of E and F in terms of the singularities of α.

If α is a normal bundle map (cf. Definition 9.3) on a compact manifold, then passing to cohomology in (0.1) yields MacPherson’s formula.

However, there are many important types of bundle mappings which are far from normal, such as direct sum mappings

α_{1} ⊕ ⋯ ⊕ α_{ℓ} : E_{1} ⊕ ⋯ ⊕ E_{ℓ} ⟶ F_{1} ⊕ ⋯ ⊕ F_{ℓ},

tensor product mappings

α_{1} ⊗ ⋯ ⊗ α_{ℓ} : E_{1} ⊗ ⋯ ⊗ E_{ℓ} ⟶ F_{1} ⊗ ⋯ ⊗ F_{ℓ},

and mappings given by Clifford multiplication. We shall establish MacPhersontype formulas in all of these cases. In fact we shall present a method for deriving such formulas in any case of interest. The method is based on a “finite-volume” property of bundle maps called geometric atomicity – one of the key ideas of the paper. This property guarantees the existence of formulas for every characteristic polynomial Φ. It holds for normal bundle maps and for all real analytic bundle maps. Furthermore, it cuts robustly across the cases mentioned above. Within each special case the geometrically atomic maps are generic.

The concept of geometric atomicity strictly generalizes the notion of atomicity introduced in [HS], that is, any section α : \underline{R} → F which is atomic is geometrically atomic. Furthermore, there is an analytic criterion analogous to that in [HS], which implies geometric atomicity. This will be discussed in part II.

A basic feature of geometric atomicity is that it enables the construction of canonical homologies between universal singularity sets. (See §4.) The main ideas involved here carry over to dynamical systems and have yielded a new approach to Morse Theory [HL_{3}].

Geometric atomicity guarantees the existence of the limit of characteristic forms for the families of approximate push-forward connections constructed in [HL_{1}]. Here in Part I we examine the resulting formulas (0.1) in detail for normal maps and for real analytic maps whose singularity sets have the expected dimension. For each k, it is proved that

d ([Σ_{k}(α)]) = d (Res_{Φ,k}[Σ_{k}(α)]) = 0.

We explicitly compute the residue forms in many cases. We also show they are completely canonical in the following sense. Along Σ_{k}(α) there are orthogonal splittings:

E = ker α ⊕ Im α  and  F = coker α ⊕ Imα

with respect to which α = 0 ⊕ I. The given connections induce direct sum connections with respect to these splittings. This in turn induces a connection on the bundle Hom(ker α, coker α) which is equivalent to the normal bundle of Σ_{k}(α). The residue form Res_{Φ,k} is expressed directly, in the spirit of Chern-Weil, from these bundles and connections.

It is a philosophically significant point that all the formulas here drop out directly from the methods of singular connections introduced in [HL_{1}]. The idea is this. Given any bundle map α : E → F between bundles with connection, one can construct canonical families \overrightarrow{D}_{s}, 0 ≤ s < ∞, of smooth “push-forward” connections on F (and “pull-back” connections \overleftarrow{D}_{s} on E) which begin with the given connection at infinity and limit to a “singular push-forward connection” (or “pull-back connection”) at 0. Applying standard Chern-Weil theory to this family essentially yields the results. MacPherson’s special blow-ups, the canonical residue forms, and (therefore) the topological formula all fall out.

It is possible that versions of the these local formulas over Z/2 can be established using ideas and results in [HZ].

§1. Characteristic currents. Let E and F be smooth vector bundles of rank m and n respectively over a manifold X, and let

α : E → F

be a smooth vector bundle map. We suppose that E and F are provided with metrics and with connections D^{E} and D^{F} (which need not respect the metrics). From this data the authors have constructed in [HL_{1}] certain smooth 1-parameter families of connections \overleftarrow{D}_{t} on E and \overrightarrow{D}_{t} on F, for 0 < t ≤ ∞, which connect the background connections

\overleftarrow{D}_{∞} = D^{E}  and  \overrightarrow{D}_{∞} = D^{F}

at time t = ∞ to certain “singular” pullback and pushforward connections at time t = 0 on E and F respectively. These limiting connections are well defined only outside the singularities of the map α. However, for Ad-invariant polynomials Φ and Ψ on the Lie algebras of the structure groups of E and F, it is possible that the limits

(1.1)  Φ((\overleftarrow{D})) ≡ lim_{t→0} Φ(\overleftarrow{D}_{t})  and  Ψ((\overrightarrow{D})) ≡ lim_{t→0} Ψ(\overrightarrow{D}_{t})

exist in the space of generalized forms (i.e., currents) on X. Here Φ(\overleftarrow{D}_{t}) ≡ Φ(\overleftarrow{Ω}_{t}) denotes the smooth characteristic form obtained by applying Φ to the curvature 2-form \overleftarrow{Ω}_{t} of \overleftarrow{D}_{t} in the standard way. In [HL_{1,2}] it is shown that for certain classes of bundle mappings these limits, called characteristic currents, do exist and give rise to formulas of the sort

Φ(Ω^{E}) − Φ((\overleftarrow{D})) = dT  and  Ψ(Ω^{F}) − Ψ((\overrightarrow{D})) = dT′

where Ω^{E}, Ω^{F} denote the curvature 2-forms of E and F, and where T, T′ are forms with L_{loc}^{1}-coefficients on X. Such formulas give a direct relationship between the singularities of the bundle map a and characteristic forms of E and F. They generalize classical results of Poincaré and Hopf and lead to a wide variety of interesting geometric residue theorems. (See [HL_{2}].)

For example in the category of real oriented bundles, suppose that E = \underline{R} is the trivial line bundle and Ψ is the normalized Pfaffian, so that Ψ(Ω^{F}) = χ(Ω^{F}) is the Euler-Chern form of F. Then for cross-sections α : \underline{R} ⟶ F which are transversal to 0, one obtains

χ(Ω^{F}) − Div(α) = dT

where Div(α) is the oriented submanifold of zeros of α. This result extends to quite general cross-sections of F, referred to as atomic sections. (See [HS].)

More generally one can consider the singularity sets

(1.2)  Σ_{k}(α) ≡ {x ∈ X : dim (ker α) = k}

for general k. There are similar results relating these singularities to Shur polynomials in the Chern classes (or Pontrjagin classes) of E and F (See [HL_{2}]). One also gets local versions of the differentiable Riemann-Roch Theorem for embeddings.

In the general case one expects to find a formula of the sort

Ψ(Ω^{F}) = ∑_{k≥0} Res_{Ψ,k}[Σ_{k}] + dT

where Res_{Ψ,k} is a smooth form defined on Σ_{k} universally in terms of Ψ. When rank(E) = rank(F), it would have the form

Ψ(Ω^{F}) − Ψ(Ω^{E}) = ∑_{k>0} Res_{Ψ,k}[Σ_{k}] + dT

The point of this paper is to derive these general formulas and to establish their existence under fairly weak hypotheses on α. For normal maps we recover the formula of R. MacPherson [Mac_{1,2,3}] concerning characteristic classes and singularities of bundle maps. Our formula is “local” on X, in the spirit of modern versions of the Atiyah-Singer Index Theorem. It is an equation of forms and currents with an explicit transgression term T. The MacPherson formula is obtained by passing to cohomology. The class of bundle maps α for which our local formula holds is broad and includes arbitrary real analytic maps whose singularity sets have the expected dimension.

Note 1. To simplify exposition we shall assume that E and F are complex bundles. Modifications required for the real case will be discussed in the last section on real vector bundles.

Note 2. The results in [HL_{1}] allow a choice of approximation mode. Here we shall always work with the algebraic approximation mode, where \overrightarrow{D}_{t} has a particularly nice form. For example if m ≤ n

\overrightarrow{D}_{t} = (t^{2}D^{F} + αD^{E}α^{∗})(αα^{∗} + t^{2})^{−1}

§2. The universal case. A bundle morphism

(2.1)  \begin{aligned}E & & ⟶^{α} & & F \\ & ↘ & & ↙ \\ & & X \end{aligned}

as above can be considered to be a cross-section of the vector bundle

Hom(E, F) ⟶^{π} X.

Now over the total space of Hom(E, F) there is a tautological bundle morphism

(2.2)  \begin{aligned}π^{∗}E & & ⟶^{α α α} & & π^{∗}F \\ & ↘ & & ↙ \\ & & Hom(E, F) \end{aligned}

which at A ∈ Hom(E, F) is given by A itself. Everything is induced by pullback from this universal case. In particular α^{∗}(α α α) = α, and the set-up in (2.1), metrics and connections included, is the pullback of that in (2.2). Our methods proceed as follows. We first analyse the problems posed in §1 for the universal case. We then examine normal maps, which are transversal to the universal singularities, and show that the universal formula can essentially be pulled back to X. Finally, using the notion of geometric atomicity, we establish results for quite general maps α.

We now focus our attention on the universal case (2.2). We begin by observing that there is a natural compactification

(2.3)  Hom(E, F) ⊂ G

of Hom(E, F) given by

G ≡ G_{m}(E ⊕ F) ⟶^{π} X,

the Grassmann bundle of complex m-planes in E ⊕F. The embedding (2.3) assigns to a linear map A : E_{x} → F_{x} at x ∈ X its graph P_{A} in E_{x} ⊕ F_{x}. Over G there is a natural orthogonal decomposition

(2.4)  π^{∗}(E ⊕ F) = U ⊕ U^{⊥}

where U is the tautological m-plane bundle over G.

The multiplicative flow φ_{t} : E ⊕ F ⟶ E ⊕ F defined by φ_{t}(e, f) = (te, f) naturally induces a flow

φ_{t} : G ⟶ G for t ∈ C^{∗}

which restricts to the linear flow

(2.5)  φ_{t}(A) = \frac{1}{t}A

on Hom(E, F). The importance of this flow comes from the following fact proved in [HL_{1}; Section I.8].

Proposition 2.1. Let \overrightarrow{D}_{t} and \overleftarrow{D}_{t} be the families of connections and Φ, Ψ the Ad-invariant polynomials discussed in §1. Then for all t > 0,

Φ(\overleftarrow{D}_{t}) = φ_{t}^{∗} Φ(Ω^{U})  and  Ψ(\overrightarrow{D}_{t}) = φ_{t}^{∗} Ψ(Ω^{U^{⊥}}).

Note 2.2. In the case where rank(E) = 1 John Zweck [Z] uses 2.1 to calculate the characteristic currents associated to a section of P(E ⊕ F) over X.

§3. Morse-Stokes kernels. Proposition 2.1 brings us to study the limits of differential forms under the flow φ_{t} : G → G for 0 < t < ∞. (Examination of this question led to the new approach to Morse Theory in [HL_{3}].)

Denote by G^{⊕2} the fibre product of G with itself over X, and consider the standard embedding R ⊂ P^{1}(R) = R ∪ {∞} as an affine algebraic chart. We consider the submanifold

T =^{def} {(t, φ_{t}(P), P) ∈ P^{1}(R) × G^{⊕2} : 0 < t < ∞ and P ∈ G} ,

called the total graph of the flow, and orient T by some choice of orientation on G. (We are essentially working locally on X, so its orientability is not a question.) Let [T ] denote the current given by integration over T and define

(3.1)  T = pr_{∗}[T ]

where pr : P^{1}(R) × G^{⊕2} ⟶ G^{⊕2} is the projection. Closely related to this is the family

T_{s,s′} =^{def} {(t, φ_{t}(P), P) ∈ T : s < t < s′}

and its pushforward

(3.2)  T_{s,s′} = pr_{∗}[T_{s,s′}]

for 0 < s < s′ < ∞. Note that T_{s,s′} is a compact manifold with boundary

∂T_{s,s′} = {s′} × Γ_{s′} − {s} × Γ_{s}

where

(3.3)  Γ_{s} =^{def} {(φ_{s}(P), P) ∈ G^{⊕2} : P ∈ G}.

It follows that

(3.4)  dT_{s,s′} = Γ_{s′} − Γ_{s}

in G^{⊕2}. This brings us to our main observation.

Proposition 3.1. T is a submanifold of finite volume in P^{1}(R) × G^{⊕2} over each compact subset of X.

Proof. This follows from real analyticity. In local trivializations of E and F and local coordinates on X we have that Hom(E, F) ≅ R^{p} × Hom(C^{m}, C^{n}) and G ≅ R^{p} × G_{m}(C^{m+n}) where G_{m}(C^{m+n}) denotes the Grassmannian of complex m-planes in C^{m+n} and p = dim(X). Now from (2.5) we deduce that in this presentation \overline{T} has the form R^{p} × A where A is a semi-algebraic subset of P^{1}(R) × G_{m}(C^{m+n}) × G_{m}(C^{m+n}). Because it is semi-algebraic, A has finite volume. (See, for example [F].) It follows that T does also. □

Corollary 3.2. The limit

(3.5)  T = lim_{s→0}_{s′→∞}T_{s,s′}

exists in the mass topology on currents on G^{⊕2}.

Proof. By Proposition 3.1 the analogous limit of T_{s,s′} exists and equals T on P^{1}(R) × G^{⊕2}. Now apply the projection pr which decreases mass. □

Corollary 3.3. The limits

Γ_{0} = lim_{s→0} Γ_{s}  and  Γ_{∞} = lim_{s→∞} Γ_{s}

exist in integrally flat currents on G^{⊕2} and

(3.6)  dT = Γ_{∞} − Γ_{0}.

§4. Morse-Stokes operators. Each of the results of the previous section can be reinterpreted from the point of view of operators – operating on forms on G (cf. [HP], [HL_{3}]). As noted above, we want to understand the limit of the pull-back of differential forms under the flow φ_{s} on G. Consider

G^{⊕2} ⟶^{pr_{2}} G
pr^{1}↓
G

where pr_{1} and pr_{2} are the projections in the fibre product, and note that the submanifold Γ_{s} = [graphφ_{s}] determines the pullback operator φ_{s}^{∗} via the equation

(4.1)  φ_{s}^{∗}(ω) = (pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ Γ_{s}} .

This leads us to consider, for each smooth p-form ω on G, the smooth (p − 1)-form defined by the expression

(4.2)  T_{s,s′}(ω) ≡ (−1)^{degω}(pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ T_{s,s′}} .

Note that T_{s,s′} defines a continuous linear operator of degree -1

T_{s,s′} : E^{∗}(G) ⟶ E^{∗}(G).

with “kernel” T_{s,s′}, on the space of smooth forms on G. The current equation (3.4) gives rise to the following operator equation.

Proposition 4.1.

{d ∘ T_{s,s′} + T_{s,s′} ∘ d}(ω) = φ_{s′}^{∗}ω − φ_{s}^{∗}ω

for all differential forms ω ∈ E^{∗}(G).

Proof. By (3.4) and (4.1),

dT_{s,s′}(ω) = (pr_{2})_{∗} {(−1)^{degω}d {(pr_{1}^{∗} ω) ∧ T_{s,s′}}}
= (pr_{2})_{∗} {(−1)^{degω}(pr_{1})^{∗}dω ∧ T_{s,s′} + (pr_{1}^{∗} ω) ∧ dT_{s,s′}}
= (pr_{2})_{∗} {(−1)^{degω}(pr_{1})^{∗}dω ∧ T_{s,s′} + (pr_{1}^{∗} ω) ∧ (Γ_{s′} − Γ_{s})}
= −T_{s,s′}(dω) + φ_{s′}^{∗}ω − φ_{s}^{∗}ω □

As in (4.2) above the current or “kernel” T can be used to define the operator

(4.3)  T(ω) ≡ (−1)^{degω}(pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ T } .

Theorem 4.2. For any smooth k-form ω on G, the limits

Γl_{0}(ω) = lim_{s→0} φ_{s}^{∗}ω  and  Γl_{∞}(ω) = lim_{s→∞} φ_{s}^{∗}ω

exist in the space of flat currents on G, and are given by the formulas (4.4) Γl_{0}(ω) = (−1)^{k}(pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ Γ_{0}} and Γl_{∞}(ω) = (−1)^{k}(pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ Γ_{∞}} ,

Furthermore, these limits satisfy the equation

(4.5)  dT(ω) + T(dω) = Γl_{∞}(ω) − Γl_{0}(ω).

Proof. By Corollary 3.2 and equation (4.2) we see that

T(ω) = lim_{s→0}_{s′→∞}T_{s,s′}(ω)

for all ω. The result now follows from Corollary 3.3, equation (4.1) and Proposition 4.1. □

§5. Analysis of the currents Γ_{0} and Γ_{∞} – the decomposition. To understand the operators Γl_{0} and Γl_{∞} we must analyse the currents Γ_{0} and Γ_{∞} that define them in (4.4). Since these currents are interchanged by time reversal t ↦ \frac{1}{t}, it will suffice to study Γ_{0}. Note that each of the graphs Γ_{t} ⊂ G^{⊕2} is independent of base parameters. That is, if G|_{U} ≅ U × G_{m} is a local trivialization of G, then over U, Γ_{t} has the form {(x, φ_{t}(P), P) : x ∈ U and P ∈ G_{m}}, which is invariant under changes of the trivialization of E and F because the flow commutes with such changes. We conclude that the limit is similarly independent of base parameters. Consequently we shall drop all mention of X and simply analyse the multiplicative flow φ_{t} on G ≡ G_{m}(C^{m} ⊕ C^{n}) induced by the map (z, w) ↦ (tz, w) on C^{m} ⊕ C^{n}.

To simplify the formulas we assume that m ≤ n. The results hold in all cases as the reader will easily see.

Our first observation is that the fixed point set of the flow in a disjoint union of submanifolds

F = ∐_{k≥0} F_{k}

where (5.1)

F_{k} = {P ∈ G_{m}(C^{m} ⊕ C^{n}) : dim(P ∩ C^{m}) = k and dim(P ∩ C^{n}) = m − k}
≅ G_{k}(C^{m}) × G_{m−k}(C^{n})

Consider the subsets

Σ_{k} ≡ {P : dim(P ∩ C^{m}) = k}  and  Υ_{k} ≡ {P : dim(P ∩ C^{n}) = m − k}

and note that

Σ_{k} ∩ Hom(C^{m}, C^{n}) = {A : dim(ker A) = k} and
\{0\} = Σ_{m} ⊂ \overline{Σ}_{m−1} ⊂ \overline{Σ}_{m−2} ⊂ \overline{Σ}_{m−3} ⊂ ⋯ ⊂ \overline{Σ}_{0} = G.

Furthermore, we observe that

Υ_{k} ∩ Hom(C^{m}, C^{n}) = ∅ if k < m and
G = \overline{Υ}_{m} ⊃ \overline{Υ}_{m−1} ⊃ \overline{Υ}_{m−2} ⊃ \overline{Υ}_{m−3} ⊃ … Υ_{0} = G_{m}(C^{n})

and furthermore

(5.2)  Hom(E, F) = G − \overline{Υ}_{m−1}.

Note 5.1. Σ_{k} and Υ_{k} are the stable and unstable manifolds of F_{k} for φ_{t}^{−1}, that is,

Σ_{k} = {P ∈ G : lim_{t→0} φ_{t}(P) ∈ F_{k}}  and  Υ_{k} = {P ∈ G : lim_{t→∞} φ_{t}(P) ∈ F_{k}}.

This follows immediately from the next Lemma whose proof is easy.

Lemma 5.2. For any P ∈ G,

lim_{t→0} φ_{t}(P) = (P ∩ C^{m}) ⊕ pr_{C^{n}}(P) =^{def} π_{1}(P)

lim_{t→∞} φ_{t}(P) = pr_{C^{m}}(P) ⊕ (P ∩ C^{n}) =^{def} π_{2}(P)

where pr_{C^{m}} and pr_{C^{n}} are the projections of C^{m} ⊕ C^{n} onto the factors.

The right hand side of the formulas in Lemma 5.2 give us projections

Υ_{k} ⟶^{π_{2}} G_{k}(C^{m}) × G_{m−k}(C^{n}) ⟵^{π_{1}} Σ_{k}
||
F_{k}

Proposition 5.3.

Γ_{∞} = ∑_{k=0}^{m} [Σ_{k} ×_{F_{k}} Υ_{k}]  and  Γ_{0} = ∑_{k=0}^{m} [Υ_{k} ×_{F_{k}} Σ_{k}]

where

Υ_{k} ×_{F_{k}} Σ_{k} = {(P, Q) ∈ Υ_{k} × Σ_{k} ⊂ G × G : π_{2}(P) = π_{1}(Q)}

and Σ_{k} ×_{F_{k}} Υ_{k} is defined similarly.

Proof. We shall only sketch the argument since a similar, more general assertion is proved in [HL_{3}]. From its definition (cf. (3.3) and Corollary 3.3) it is straightforward to show that suppΓ_{0} ⊆ Υ_{k} ×_{F_{k}} Σ_{k}. Now each of the submanifolds Υ_{k} ×_{F_{k}} Σ_{k} is a Zariski dense subset of an algebraic subvariety; in particular it has finite volume in G × G. The Federer Flat Support Lemma [F; 4.1.15] now implies that Γ_{0} = ∑_{k} n_{k}[Υ_{k} ×_{F_{k}} Σ_{k}]. Analysis of the limit at points of F_{k} shows that n_{k} = 1 (cf. (10.2)). □

Note that Υ_{k} ×_{F_{k}} Σ_{k} is a fibre product over F_{k} embedded diagonally in G × G. For x ∈ F_{k} the fibre π_{2}^{−1}(x, x) lies in G×{x} and the fibre π_{1}^{−1}(x, x) lies in {x}×G.

Combining Proposition 5.3 with (4.4) above gives the following.

Corollary 5.4. The operators Γl_{0} and Γl_{∞} can be written as

Γl_{0} = ∑_{k=0}^{m} P_{k}  and  Γl_{∞} = ∑_{k=0}^{m} P̃_{k}

where

P_{k}(ω) = (pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ [Υ_{k} ×_{F_{k}} Σ_{k}]} and P̃_{k}(ω) = (pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ [Σ_{k} ×_{F_{k}} Υ_{k}]}

for any smooth form ω on the Grassmann bundle G.

§6. Analysis of the currents Γ_{0} and Γ_{∞} – the residues. We now show that the operator P_{k} in Corollary 5.4 has the form P_{k}(ω) = Res_{k}(ω)[Σ_{k}] where Res_{k}(ω) is an explicitly computable residue form defined on Σ_{k} in terms of ω, and analogously P̃_{k}(ω) = Res′_{k}(ω)[Υ_{k}] where Res′_{k}(ω) is a smooth residue form on Υ_{k}.

We begin with the following observation.

Lemma 6.1. Each of the submanifolds Υ_{k} and Σ_{k} has finite volume in G. So also do the fibre products Υ_{k} ×_{F_{k}} Σ_{k}.

Proof. This is evident from the fact that their closures are algebraic subvarieties. □

Definition 6.2. Given a smooth differential form ω defined in a neighborhood of \overline{Υ}_{k}, and a smooth differential form ω′ defined in a neighborhood of \overline{Σ}_{k}, set

Res_{k}(ω) = (π_{1})^{∗} {(π_{2})_{∗}ω}  and  Res′_{k}(ω′) = (π_{2})^{∗} {(π_{1})_{∗}ω′}

using the projections

Υ_{k} ⟶^{π_{2}} F_{k} ⟵^{π_{1}} Σ_{k}.

We shall show that the maps π_{1} and π_{2} have the natural structure of algebraic vector bundles over F_{k}.

Proposition 6.3. Each operator P_{k} can be expressed by the formula

P_{k}(ω) = Res_{k}(ω)[Σ_{k}].

Furthermore, for any smooth form ω on G, Res_{k}(ω) is a smooth form on Σ_{k} which has finite L^{1}-norm (i.e., finite mass), so that Res_{k}(ω)[Σ_{k}] is a well defined current on G. Similarly, P̃_{k} can be written as

P̃_{k}(ω) = Res′_{k}(ω)[Υ_{k}].

where Res′_{k}(ω) is a smooth L^{1} form on Υ_{k}.

Proof. To begin we note that there is a commutative diagram

(6.1)  \begin{matrix}Υ_{k} ×_{F_{k}} Σ_{k} & ⟶^{p_{2}} & Σ_{k} \\ ^{p_{1}}↓ & & ↓^{π_{1}} \\ Υ_{k} & ⟶_{π_{2}} & F_{k} \end{matrix}

where p_{1} and p_{2} are induced from the two projections G × G onto G, i.e., there is a commutative diagram

Υ_{k} ⟵^{p_{1}} Υ_{k} ×_{F_{k}} Σ_{k} ⟶^{p_{2}} Σ_{k}
↓  ↓  ↓
G ⟵^{pr_{1}} G × G ⟶^{pr_{2}} G

Each of the maps π_{1} and π_{2} in (6.1) has the natural structure of an algebraic vector bundle over F_{k}. We show this explicitly as follows. Let

ξ ⟶ G_{k}(C^{m})  and  η ⟶ G_{m−k}(C^{n})

be the tautological bundles of rank k and m − k respectively, and extend them by pullback to F_{k} = G_{k}(C^{m}) × G_{m−k}(C^{n}). Then there are commutative diagrams

Hom(ξ, η^{⊥})  ⟶^{j_{2}} Υ_{k}  Hom(η, ξ^{⊥})  ⟶^{j_{1}} Σ_{k}
π′_{2}↓  ↓^{π_{2}}  ^{π′_{1}}↓  ↓^{π_{1}}
G_{k}(C^{m}) × G_{m−k}(C^{n}) ⟶^{≅} F_{k}  G_{k}(C^{m}) × G_{m−k}(C^{n}) ⟶^{≅} F_{k}

where π′_{1} and π′_{2} are bundle projections and where at (ξ, η) ∈ G_{k} × G_{m−k}

j_{2}(a) = η ⊕ graph(a)  and  j_{1}(b) = ξ ⊕ graph(b).

The maps j_{1} and j_{2} are biholomorphisms and give Υ_{k} and Σ_{k} the structure of vector bundles as claimed. Consider the Grassmann compactificatons

Hom(ξ, η^{⊥})  ⊂  G(ξ ⊕ η^{⊥})  Hom(η, ξ^{⊥}) ⊂  G(η ⊕ ξ^{⊥})
π′_{2}  ↘  ↙ π̃_{2}  π′_{1}  ↘  ↙  π̃_{1}
F_{k}  F_{k}

where G(ξ ⊕ η^{⊥}) is the bundle of k-planes in ξ ⊕ η^{⊥} and G(η ⊕ ξ^{⊥}) is the bundle of (m − k)-planes in η ⊕ ξ^{⊥}. The maps j_{i} extend to surjective algebraic maps

G(ξ ⊕ η^{⊥}) ⟶^{j̄_{2}} \overline{Υ}_{k} G(η ⊕ ξ^{⊥}) ⟶^{j̄_{1}} \overline{Σ}_{k}

given on the fibres above (ξ, η) ∈ G_{k} × G_{m−k} by

(6.2)  j̄_{2}(ℓ_{2}) = η ⊕ ℓ_{2}  and j̄_{1}(ℓ_{1}) = ξ ⊕ ℓ_{1}

Note 6.4. The normal bundle to Σ_{k} is equivalent to the pullback of the vector bundle Υ_{k} via the map π_{2}. Similarly the normal bundle to Υ_{k} is the π_{1}-pullback of Σ_{k}.

We now observe that by the commutativity of (6.1) we have

(pr_{2})_{∗} {(pr_{1}^{∗} ω) ∧ [Υ_{k} ×_{F_{k}} Σ_{k}]} = (p_{2})_{∗} {(p_{1}^{∗}ω)}
= (π_{1})^{∗} {(π_{2})_{∗}ω}
= Res_{k}(ω)[Σ_{k}]

This proves the formula asserted in 6.3. The integrability of Res_{k}(ω) on Σ_{k} is equivalent to the fact that the current Res_{k}(ω)[Σ_{k}] has finite mass. This finiteness of mass is a consequence of Lemma 6.1, which implies that (pr_{1}^{∗} ω) ∧ [Υ_{k} ×_{F_{k}} Σ_{k}] has finite mass, and the fact that pushforward of currents is mass non-increasing.

This completes the proof of 6.3 for P_{k}. The argument for P̃_{k} is completely analogous. □

The proof above used the “Grassmann desingularization” of \overline{Υ}_{k} and \overline{Σ}_{k} by the maps \overline{j}_{2} and \overline{j}_{1}. This gives us another way to look at the residues which will be useful to us when we consider characteristic forms in §9 and onward.

Proposition 6.5. The form Res_{k}(ω) can be expressed as

Res_{k}(ω) = (π_{1})^{∗} {(π̃_{2})_{∗}ω̃}

where π̃_{2} : G(ξ ⊕ η^{⊥}) ⟶ F_{k} is the Grassmann compactification above, and where ω̃ = \overline{j}_{2}^{∗}ω. The analogous statements hold for Res′_{k}(ω′).

Proof. We have seen that (π_{2})_{∗}(ω) = (π′_{2})_{∗}(j_{2}^{∗}ω). Since the fibres of π′_{2} are Zariski dense, and in particular of full measure, in the fibres of π̃_{2}, we see that integration of a smooth form on G(ξ ⊕ η^{⊥}) over the fibres of π̃_{2} and over the fibres of π′_{2} are equal. □

§7. The first main theorem. Combining 4.2, 5.4 and 6.3 immediately yields our first main result.

Theorem 7.1. Let G = G_{m}(E ⊕ F) be the Grassmann bundle of m-planes in the smooth vector bundle E ⊕ F → X, and let φ_{t}, 0 < t < ∞ be the multiplicative flow on G engendered by (e, f) ↦ (te, f) in E ⊕ F. Then there are continuous linear operators Γl_{0}, Γl_{∞}, T : E^{∗}(G) ⟶ E′^{∗}(G) from smooth differential forms to generalized differential forms (in fact, flat currents) on G with the following properties. For all ω ∈ E^{∗}(G),

Γl_{0}(ω) = lim_{t→0} φ_{t}^{∗}ω = ∑_{k=0}^{m} Res_{k}(ω)[Σ_{k}]  and  Γl_{∞}(ω) = lim_{t→∞} φ_{t}^{∗}ω = ∑_{k=0}^{m} Res′_{k}(ω)[Υ_{k}]

where

Res_{k}(ω) = π_{1}^{∗}{(π_{2})_{∗}ω}  and  Res′_{k}(ω) = π_{2}^{∗}{(π_{1})_{∗}ω}.

Furthermore, T, defined by (4.3), is an operator of degree -1 which satisfies the equation

(7.1)  d ∘ T + T ∘ d = Γl_{∞} − Γl_{0}

§8. The formula in the universal case. Let U ⟶ G be the tautological bundle and let Φ and Ψ be Ad-invariant polynomials on the Lie algebras of the structure groups of E and F respectively. Because of Proposition 2.1 we want to apply the Theorem above to the situation where

ω = Φ(Ω^{U})  or  ω = Ψ(Ω^{U^{⊥}}).

The main point is to compute the residues. For simplicity we will treat the first case. According to Proposition 6.3 Res_{k}(ω) on Σ_{k} is computed by restricting ω to Υ_{k}, integrating over the projection π_{2} and then pulling back to Σ_{k} via π_{1}. Note that by the naturality of the Chern construction

ω|_{Υ_{k}} = Φ (Ω^{U|_{Υ_{k}}} ) .

From the proof of Proposition 6.3 we see that on Υ_{k} the bundle U splits as

U|_{Υ_{k}} ≅ η ⊕ U_{k}

where U_{k} is the restriction to Hom(ξ, η^{⊥}) ⊂ G(ξ⊕η^{⊥}) of the tautological k-plane bundle

U_{k} ⟶ G(ξ ⊕ η^{⊥}).

Consequently we have from 7.1 that

(8.0)  Res_{k}(ω) = π_{1}^{∗}(π_{2})_{∗}Φ (Ω^{η⊕U_{k}})

Restricting to the coordinate chart Hom(E, F) we get

(8.1)  Res_{k}(ω) = (π_{2})_{∗}Φ (Ω^{Imα α α⊕U_{k}})

where U_{k} is the tautological bundle over the Grassmann compactification G_{k}(kerα α α ⊕ cokerα α α) of the normal bundle Hom(kerα α α, cokerα α α) to Σ_{k}.

On the other hand by (5.2) one sees directly that on the chart Hom(E, F)

(8.2)  Res′_{k}(ω) = 0  for k < m

and

(8.3)  Res′_{m}(ω) = π_{1}^{∗} (ω|_{Σ_{m}}) = Φ(Ω^{E})

Theorem 8.1. Let E → X and F → X be smooth complex vector bundles with rank(E) ≤ rank(F), and let Φ be an invariant polynomial on the Lie algebra of the structure group of E. Then for any choice of connections on E and F there exists an L_{loc}^{1}-form T on Hom(E, F) so that

(8.4)  Φ(Ω^{E}) = ∑_{k=0}^{m} Res_{Φ,k}[Σ_{k}] + dT

Res_{Φ,k} = (π_{2})_{∗}Φ (Ω^{Im α α α⊕U_{k}})

where α α α : π^{∗}E → π^{∗}F denotes the tautological bundle map on Hom(E, F) and where (π_{2})_{∗} denotes integration over the fibres of the Grassmann compactification G_{k} = G_{k}(ker α α α ⊕ coker α α α) of the normal bundle N_{Σ_{k}} ≅ Hom(ker α α α, coker α α α), and where U_{k} is the tautological k-plane bundle over G_{k}.

If rank(E) = rank(F), then formula (8.1) becomes

Φ(Ω^{E}) − Φ(Ω^{F}) = ∑_{k=1}^{m} Res_{Φ,k}[Σ_{k}] + dT

If Ψ is an invariant polynomial on the Lie algebra of the structure group of F, then there exists an L_{loc}^{1}-form T′ on Hom(E, F) so that

(8.5)  Ψ(Ω^{F}) = ∑_{k=0}^{m} Res_{Ψ,k}[Σ_{k}] + dT′

Res_{Ψ,k} = (π_{2})_{∗}Ψ (Ω^{(ker α α α)^{⊥}⊕U_{k}^{⊥}})

Note 8.2. The case where rank(E) > rank(F) follows by applying Theorem 8.1 to the adjoint of α α α.

Note 8.3. The bundle Hom(coker α α α, ker α α α) is the dual of the normal bundle Hom(ker α α α, coker α α α) of Σ_{k}.

Proof. If ω = Φ(Ω^{U}), then dω = 0. We apply Theorem 7.1 to ω and apply (8.1) to calculate the term Γl_{0}(ω). We then restrict to Hom(E, F) ⊂ G and apply (8.2) and (8.3) to calculate Γl_{∞}(ω). This proves the first part of the theorem. The calculations for Ψ(Ω^{U^{⊥}}) are completely analogous. □

Remark 8.4. The bundle Im(α α α) is a pull-back to G_{k} of a bundle defined on Σ_{k} via the fibration G_{k} → Σ_{k}. The tautological bundle U_{k} carries a natural connection which along the fibres of G_{k} → Σ_{k} is the standard connection. In §12 we shall see that the connection yielding the curvature form in formula (8.2) for the residue can be assumed to be the direct sum of the pull-back connection on Im(α α α) with the projected connection on U_{k}. This has particularly nice consequences when Φ is a multiplicative series of characteristic polynomials.

§9. Existence for normal bundle maps — geometric atomicity. Let E and F be smooth complex vector bundles over a manifold X of dimension ν, and suppose that

α : E ⟶ F

is a smooth bundle map. Given Ad-invariant polynomials Φ and Ψ as above, one can ask when the limits (1.1) exist. We shall now answer this question in some generality, and also establish the local MacPherson formula for α.

The following concept is crucial here. To begin we recall that a Borel measurable subset A of a locally compact topological space Z is said to have locally finite μmeasure if each point z ∈ Z has a compact neighborhood U such that μ(A∩U) < ∞.

Definition 9.1. The section α is called geometrically atomic if the subset

(9.1)  T_{α} ≡ \{(\frac{1}{t}α_{x}, α_{x}) ∈ G^{⊕2} : x ∈ X and 0 < t < ∞\}

has locally finite (ν + 1)-dimensional measure in G^{⊕2}.

Note. Above the open set X − Zero(α) where α ≠ 0, T_{α} is a submanifold. In fact, it is a line bundle over this set. The remaining points of T_{α} consist of the zeros of α and therefore have locally finite ν-dimensional measure. Hence they can be ignored, and the condition in 9.1 can be replaced by requiring that the remaining submanifold have locally finite volume in G^{⊕2}. (Thus, the zero-section α = 0 is always geometrically atomic.)

Note. The condition in Definition 9.1 is equivalent to the requirement that for each compact K ⊂ X, the subset

T_{K,α} ≡ \{(\frac{1}{t}α_{x}, α_{x}) ∈ G^{⊕2} : x ∈ K and 0 < t < ∞\}

has finite (ν + 1)-dimensional measure in G^{⊕2}.

The generality of Definition 9.1 is clear from the following result.

Proposition 9.2. If α is real analytic, then it is geometrically atomic.

Proof. The closure of the submanifold

(9.2) T_{α} = \{(t, \frac{1}{t}α_{x}, α_{x}) ∈ R × G^{⊕2} : x ∈ X and 0 < t < ∞\} ⊂ P^{1}(R) × G^{⊕2}

is an analytic subvariety of dimension (ν + 1) in P^{1}(R) × G^{⊕2} and hence has locally finite (ν + 1)-measure. It follows that its image T_{α} = pr_{∗} T_{α}, where pr : P^{1}(R) × G^{⊕2} → G^{⊕2} is the projection, also has locally finite (ν + 1)-measure. □

Note The singularities of a real analytic map can be monstrous. In particular, the sets Σ_{k}(α), defined in (1.2), need not have the expected dimension.

Definition 9.3. A bundle map α : X ⟶ Hom(E, F) is called normal if it is transversal to the submanifolds Σ_{k} for all k.

Proposition 9.4. Any normal bundle map is geometrically atomic.

The proof is postponed to section 10. Our first main result is the following.

Theorem 9.5. If α is geometrically atomic, then the limits

(9.3)  Φ((\overleftarrow{D})) ≡ lim_{t→0} Φ(\overleftarrow{D}_{t})  and  Ψ((\overrightarrow{D})) ≡ lim_{t→0} Ψ(\overrightarrow{D}_{t})

exist on X for all Ad-invariant polynomials Φ and Ψ on the Lie algebras of the structure groups of E and F. Furthermore, for all Φ, Ψ there exist L_{loc}^{1}-forms T_{Φ}, T_{Ψ} on X such that

(9.4)  Φ(Ω^{E}) − Φ((\overleftarrow{D})) = dT_{Φ}  and  Ψ(Ω^{F}) − Ψ((\overrightarrow{D})) = dT_{Ψ}

where Ω^{E}, Ω^{F} denote the curvature 2-forms E and F respectively.

Proof. Fix s′ > s > 0 and consider the subset

T_{α,s,s′} ≡ \{(\frac{1}{t}α_{x}, α_{x}) ∈ G^{⊕2} : x ∈ X and s < t < s′\} .

(Note that T_{α,s,s′} = pr_{∗} T_{α,s,s′} where T_{α,s,s′} is the compact submanifold with boundary defined as in 9.2 with s ≤ t ≤ s′.) As in §3 the assumption of locally finite volume implies that

lim_{s→0}_{s′→∞}T_{α,s,s′} = T_{α}

in locally integral currents on G^{⊕2}, and that

(9.5)  lim_{s→0}_{s′→∞}dT_{α,s,s′} = lim_{s′→∞} Γ_{α,s′} − lim_{s→0} Γ_{α,s} ≡ Γ_{α,∞} − Γ_{α,0}

where

Γ_{α,s} = \{(\frac{1}{s}α_{x}, α_{x}) ∈ G^{⊕2} : x ∈ X\}

We now reinterpret these equations as operator equations and apply them to the forms Φ(Ω^{U}) and Ψ(Ω^{U^{⊥}}) as in §4. Specifically, to each integral current S of dimension n + ℓ on G^{⊕2} we associate the operator S : E^{∗}(G) → 𝔈^{∗}(X) of degree −ℓ from forms on G to currents on X by setting

(9.6)  S(ω) = p_{∗} {(pr_{1}^{∗} ω) ∧ S}

where p : G → X is the bundle projection. Note that Γl_{α,s}(ω) = α^{∗}φ_{s}^{∗}(ω) where φ_{s} is the flow on G defined in (2.5). In particular, if ω = Φ(Ω^{U}), then by Proposition 2.1 and the universality of the construction of \overleftarrow{D}_{s} we have that

(9.7)  Γl_{α,s}(ω) = α^{∗}φ_{s}^{∗}Φ(Ω^{U}) = α^{∗}Φ(\overleftarrow{Ω Ω}_{s}) = Φ(\overleftarrow{Ω}_{s}).

where \overleftarrow{Ω Ω}_{s} is the curvature of the universal pushforward connection on Hom(E, F) ⊂ G. Consequently (9.5) and (9.7) imply that

lim_{s→∞} Φ(\overleftarrow{Ω}_{s}) − lim_{s→0} Φ(\overleftarrow{Ω}_{s}) = dT_{Φ}

where T_{Φ} = T_{α}(Φ(Ω^{U})). By the continuity of \overleftarrow{D}_{s} at infinity we have lim_{s→∞} Φ(\overleftarrow{Ω}_{s}) = Φ(Ω^{E}). This proves the result for Φ. The result for Ψ is similar. □

A determination of the limits in (9.3) for all Φ and Ψ will follow from understanding the limiting current Γ_{α,0}. When α is normal we shall see that this current is modeled on the universal case.

§10. The local MacPherson formula. In this section we analyse the current Γ_{α,0}. Our discussion is local on X, so we shall assume that E and F are trivialized bundles. Our section α is then just a map from X to Hom(C^{m}, C^{n}) ⊂ G_{m}(C^{m+n}) = G.

To begin the analysis we give a simple presentation of the flow φ_{s} in a neighborhood of F_{k} ≅ G_{k}(E) × G_{m−k}(F) in G. We return to the notation of §6 and consider the vector bundle

H_{k}≡ Hom(ξ, η^{⊥}) ⊕ Hom(η, ξ^{⊥})
↓^{π′_{2}⊕π′_{1}}
F_{k}.

There is a map j : H_{k} → G defined by

(10.1)  j(a, b) = gr(a) ⊕ gr(b)

where gr(a) denotes the graph of a. This map gives a diffeomorphism from a neighborhood of the zero-section to a neighborhood of F_{k} in G. We introduce fibre metrics and identify such a neighborhood of F_{k} with

U = {(a, b) ∈ H_{k} : |a| ≤ 1 and |b| ≤ 1}

In this presentation the flow φ_{s} has the form

φ_{s}(a, b) = (\frac{1}{s}a, sb) .

Now in this neighborhood our set T can be written as

\begin{aligned}T ∩ (U × U) & = \{(\frac{1}{s}a, sb, a, b) : |a| ≤ 1, |b| ≤ 1, |\frac{1}{s}a| ≤ 1, |sb| ≤ 1, and 0 < s ≤ 1\} \\ & = \{(a, sb, sa, b) : |a| ≤ 1, |b| ≤ 1, and 0 < s ≤ 1\} \end{aligned}

The boundary of this set is clearly given by

(10.2)  ∂{T ∩ (U × U)} = {(a, 0, 0, b) : |a| ≤ 1, |b| ≤ 1} ≅ Υ_{k} ×_{F_{k}} Σ_{k}.

This is in fact a manifold with boundary in U × U − F_{k}. We can resolve the singularity at F_{k} by considering

{(s, a, sb, sa, b) : |a| ≤ 1, |b| ≤ 1, and 0 ≤ s ≤ 1} ⊂ R × G × G.

This is a manifold with boundary whose projection is the set above.

Suppose now that α is a normal bundle map and fix x_{0} ∈ X with α_{x_{0}} ∈ Σ_{k}. Let z_{0} = π_{1}(α_{x_{0}}) ∈ F_{k} and fix a neighborhood V of z_{0} in F_{k} with local trivializations

ξ|_{V} ≅ V × C^{k}  and  η|_{V} ≅ V × C^{m−k}.

Thus over V we have

(10.3)  Hom(ξ, η^{⊥}) ⊕ Hom(η, ξ^{⊥})|_{V} ≅ V × Hom_{k,n−m+k} × Hom_{m−k,m−k}.

where Hom_{r,s} ≡ Hom(C^{r}, C^{s}). In this picture α_{x_{0}} ≅ (z_{0}, 0, b_{0}) for some point b_{0} ∈ Hom_{m−k,m−k}, which we may assume (by homothety) to satisfy |b_{0}| < 1. Restricting to triples (v, a, b) with |a| ≤ 1 and |b| ≤ 1 parameterizes a neighborhood in G containing z_{0} and α_{x_{0}}. In this neighborhood

Σ_{k} ≅ V × {0} × Hom_{m−k,m−k}.

Now the transversality of α to Σ_{k} implies the following.

Lemma 10.1. Suppose α is normal and x_{0} ∈ Σ_{k}(α). Then there exist local coordinates on a neighborhood U of x_{0} in X of the form

(y, a) ∈ R^{N} × Hom_{k,n−m+k}

where N = ν − 2k(n − m + k), such that in the coordinates (10.3) above

(10.4)  α(y, a) = (v(y), a, b(y)).

Note that in U, Σ_{k}(α) is the submanifold corresponding to a = 0. Furthermore, Lemma 10.1 shows that in U

(10.5)  Σ_{ℓ}(α) = R^{N} × {a ∈ Hom_{k,n−m+k} : dim ker(a) = ℓ}

for all ℓ ≤ k. We conclude the following.

Corollary 10.2. If α is normal , then each Σ_{ℓ}(α) has locally finite volume in X.

Proof of Proposition 9.4 Fix x_{0} ∈ Σ_{k}(α) and choose coordinates on G and X as above. Note that the map j defined in (10.1) extends smoothly to the compactification G(ξ ⊕ η^{⊥}) ⊕ G(η ⊕ ξ^{⊥}). In particular, via (10.3) this gives a map

V × G_{k}(C^{k} ⊕ C^{n−m+k}) × Hom_{m−k,m−k} ⟶ G

which we compose with our coordinate representation of α above. We then consider the map

(0, 1] × U ⟶ G × G

given by

(s, y, a) ↦ (v(y), \frac{1}{s}a, sb(y); v(y), a, b(y)).

The volume element induced by this map is dominated by the volume element induced by the product mapping

(s, y, a) ↦ (v(y), \frac{1}{s}a, b(y); v(y), a, b(y)).

Now the map (s, a) ↦ (\frac{1}{s}a, a) is algebraic and its image is a submanifold of finite volume in G_{k}(C^{n−m+2k}) × G_{k}(C^{n−m+2k}). This proves that T_{α,K} has finite volume for compact subsets K ⊂ U. □

We now consider the local MacPherson formula for a normal bundle map α. We have seen that for each k, Σ_{k}(α) is a smooth submanifold of locally finite volume and of (real) codimension 2k(n − m + k) in X. Along each Σ_{k}(α) it is clear that ker α ⊂ E and Im α ⊂ F are smooth vector bundles. Furthermore, the bundle Hom(ker α, coker α) is naturally equivalent to the normal bundle of Σ_{k}(α) in X. (See [Mac_{1}] for example.) Now fix x_{0} ∈ Σ_{k}(α) and choose coordinates as in Lemma 10.1. Then from (10.5) we see that for each ℓ ≤ k we have a splitting in U:

Σ_{ℓ}(α) = R^{N} × Σ_{ℓ}

where

Σ_{ℓ} ⊂ Hom_{k,n−m+k} ⊂ G_{k}(C^{n−m+2k})

is the universal degeneracy locus (where dim ker a = ℓ). Our section of G^{⊕2} can now be written

(φ_{s}α(y, a), α(y, a)) = (v(y), \frac{1}{s}a, sb(y); v(y), a, b(y)).

From here it is straightforward to see that in these coordinates on G^{⊕2}|_{U}

(10.6)  Γ_{α,0} = R^{N} × Γ_{0}

where Γ_{0} is the current in G_{k}(C^{n−m+2k}) × G_{k}(C^{n−m+2k}) defined universally in §3 as the limit of the sets Γ_{s} = Cl\{(\frac{1}{s}a, a) : a ∈ Hom_{k,n−m+k}\} as s → 0. Thus the analysis of §§5-6 applies directly and we conclude the following.

Theorem 10.3. Let E → X and F → X be smooth complex vector bundles with rank(E) ≤ rank(F), and let α : E → F be a normal bundle map. Suppose Φ is an invariant polynomial on the Lie algebra of the structure group of E. Then for any choice of connections on E and F there exists an L_{loc}^{1}-form T on X so that

(10.7)  Φ(Ω^{E}) = ∑_{k=0}^{m} Res_{Φ,k}[Σ_{k}(α)] + dT

Res_{Φ,k} = (π_{2})_{∗}Φ (Ω^{Im α⊕U_{k}})

where (π_{2})_{∗} denotes integration over the fibres of the Grassmann compactification G_{k} = G_{k}(ker α ⊕ coker α) of the normal bundle N_{Σ_{k}} ≅ Hom(ker α, coker α) to Σ_{k}(α), and where U_{k} → G_{k} is the tautological k-plane bundle.

If rank(E) = rank(F), then formula (10.7) becomes

Φ(Ω^{E}) − Φ(Ω^{F}) = ∑_{k=1}^{m} Res_{Φ,k}[Σ_{k}(α)] + dT

If Ψ is an invariant polynomial on the Lie algebra of the structure group of F, then there exists an L_{loc}^{1}-form T′ on X so that

(10.8)  Ψ(Ω^{F}) = ∑_{k=0}^{m} Res_{Ψ,k}[Σ_{k}(α)] + dT′

Res_{Ψ,k} = (π_{2})_{∗}Ψ (Ω^{(ker α)^{⊥}⊕U_{k}^{⊥}})

§11. Real analytic bundle maps. In this section we derive a Local MacPherson Formula for real analytic bundle maps under the assumption that the degeneracy loci have the expected dimension. We begin with the following general result.

Lemma 11.1. Let α : E → F be a geometrically atomic bundle map over a smooth manifold X, and let Γ_{α,0} be the current from (9.5). Then

(11.1)  supp Γ_{α,0} ⊂ ⋃_{k} Υ_{k} ×_{F_{k}} Σ_{k}(α)

Proof. Fix x ∈ Σ_{k}(α) and choose local trivializations of E and F in a neighborhood U of x so that G^{⊕2}|_{U} ≅ U ×G_{m}×G_{m} where G_{m} = G_{m}(C^{m+n}). Suppose (x, P′, P) ∈ supp Γ_{α,0}. Then there exist sequences x_{j} → x and t_{j} → 0 such that

γ_{j} ≡ gr(α_{x_{j}}) ⟶ P  and  γ′_{j} ≡ gr(\frac{1}{t_{j}} α_{x_{j}}) = φ_{t_{j}}γ_{j} ⟶ P′

where φ_{t} is the flow from §2. It is now an elementary argument (as in [HL_{3}, Lemma 2.10]) to see that P must be joined to P′ by a piecewise flow line in G_{m} which passes through F_{k}. In particular, P′ ∈ Υ_{k} and π_{2}(P′) = π_{1}(P), where π_{1} and π_{2} are the projections from §6. Thus (P′, P) ∈ Υ_{k} ×_{F_{k}} Σ_{k}(α) as claimed. □

Suppose now that α : E → F is a real analytic bundle map between complex vector bundles over a ν-dimensional manifold X, and that

rank(E) = m ≤ n = rank(F).

Then for each k the degeneracy locus Σ_{k}(α) is an analytic subset of X of some dimension, say ν_{α,k}. Therefore, Σ_{k}(α) has locally finite ν_{α,k}-measure, and integration over the regular points of Σ_{k}(α) defines an integral current [Σ_{k}(α)] of dimension ν_{α,k}. Recall that the “expected” dimension of Σ_{k}(α) in X is ν_{k} ≡ ν −2k(n−m+k).

Theorem 11.2. Let α : E → F be as above and suppose that

(11.2)  dim Σ_{k}(α) ≤ ν_{k}

for each k. Then there exist integer-valued functions n_{k} on Σ_{k}(α), constant on each irreducible component, such that for all Φ, Ψ as in 10.3 and all connections on E and F, there exist L_{loc}^{1}-forms T, T′ on X such that

Φ(Ω^{E}) = ∑_{k=0}^{m} n_{k} Res_{Φ,k}[Σ_{k}(α)] + dT  and

Ψ(Ω^{F}) = ∑_{k=0}^{m} n_{k} Res_{Ψ,k}[Σ_{k}(α)] + dT′

where Res_{Φ,k} and Res_{Ψ,k} are smooth forms defined on the regular set of Σ_{k}(α) exactly as in Theorem 10.3.

Proof. By Theorem 9.5 and its proof (in particular the discussion from (9.5) to (9.7)) we need only to compute

(11.3)  Γl_{α,0}(ω) = p_{∗}{(pr_{1}^{∗} ω) ∧ Γ_{α,0}}

where ω = Φ(Ω^{U}) or Ψ(Ω^{U^{⊥}}). Now it follows from assumption (11.2) that

dim{Υ_{k} ×_{F_{k}} Σ_{k}(α)} ≤ ν = dim Γ_{α,0}

for all k. Hence, from (11.1), the fact that dΓ_{α,0} = 0, and the Federer Flat Support Lemma [F;4.1.15] it follows that

Γ_{α,0} = ∑_{k} n_{k}[Υ_{k} ×_{F_{k}} Σ_{k}(α)].

where n_{k} : Σ_{k}(α) → Z is locally constant on the regular set and 0 on any component of dimension < ν_{k}. Computing (11.3) at regular points of Σ_{k}(α) gives the residue forms as in §8. □

Note. The function n_{k} represents the order of k-degeneracy of α.

§12. Analysis of the currents Res_{Φ,k}[Σ_{k}] and residue calculations. In this section we shall study the singular currents which appear in our formulas. They have a surprizingly regular structure and the residues are explicitly computable in many cases. Our first result is that for regular bundle maps, each of the terms Res_{Φ,k}[Σ_{k}] occuring in the main formula is a d-closed current of finite mass. We begin with the following.

Proposition 12.1. Let α : E → F be as in Theorem 10.3 or Theorem 11.2. Then for each k, integration over the regular points of Σ_{k}(α) defines a locally rectifiable current [Σ_{k}(α)] in X with

(12.1)  d[Σ_{k}(α)] = 0

Proof. Corollary 10.2 and the discussion prior to 11.2 show that Σ_{k}(α) has locally finite volume in X and therefore defines a locally rectifiable current. Note that

supp {d[Σ_{k}(α)]} ⊆ ⋃_{ℓ>k} Σ_{ℓ}(α),

and that codim_{R} Σ_{ℓ} = 2ℓ(n−m+ℓ). Since d[Σ_{k}(α)] is a flat current of codimension 2k(n − m + k) − 1, it follows from [F, 4.1.15] that d[Σ_{k}(α)] = 0. □

Proposition 12.2. Let α : E → F, Φ and Ψ be as in 10.3 or 11.2. Then for each k, the currents

R_{k} ≡ Res_{Φ,k}[Σ_{k}(α)]  and  R′_{k} ≡ Res_{Ψ,k}[Σ_{k}(α)]

have locally finite mass in X and satisfy

dR_{k} = dR′_{k} = 0.

Proof. We begin with the universal case. In Propositions 6.3 and 6.5 it is proved that the currents R_{k} and R′_{k} have locally finite mass. We recall that this is done as follows. Consider the desingularization

\overline{j}_{1} : G(η ⊕ ξ^{⊥}) ⟶ \overline{Σ}_{k}

of the closure of Σ_{k} given by the Grassmann compactification of Σ_{k} ≅ Hom(η, ξ^{⊥}) (cf. (6.2)). The projection π_{1} : Σ_{k} → F_{k} extends to a smooth map

π̃_{1} : G(η ⊕ ξ^{⊥}) ⟶ F_{k}.

We pull the bundle Υ_{k} ≅ Hom(ξ, η^{⊥}) back via \overline{j}_{1} and take its Grassmann compactification G(ξ ⊕ η^{⊥}). Let U_{k} → G(ξ ⊕ η^{⊥}) be the tautological bundle. Then we have the identity U = U_{k} ⊕ η (cf. §8), and

Res_{Φ,k} = (\overline{j}_{1})_{∗}Φ (Ω^{U_{k}⊕η}) .

Since Φ (Ω^{U_{k}⊕η}) is a smooth form on the manifold G(η ⊕ ξ^{⊥}), its push-forward by \overline{j}_{1} has finite mass. Furthermore, since Φ (Ω^{U_{k}⊕η}) is d-closed on G(η ⊕ ξ^{⊥}), its push-forward is d-closed on X. A similar argument applies for R′_{k}. This gives the result in the universal case.

The normal case is proved in parallel fashion by using the regular singular structure of Σ_{k}(α) established in §10. Namely, from Lemma 10.1 we see that Σ_{k}(α) has the same singular structure as Σ_{k} in the universal case, and the arguments above apply straightforwardly. In the analytic case one replaces the desingularization \overline{j}_{1} by resolution of singularities. □

We now address the question of the residue forms themselves. From forumla (8.0) we see that it would be particularly nice if the connection on U|_{Υ_{k}}= η ⊕ U_{k} were a direct-sum connection D^{η} ⊕ D^{U_{k}}. Explicit calculation shows that this is not the case. However, one could hope that for an invariant form Φ, there is an equality

(12.2)  π_{2}_{∗}Φ (Ω^{η⊕U_{k}}) = π_{2}_{∗}Φ (Ω^{η} ⊕ Ω^{U_{k}})

where

Ω^{η} ⊕ Ω^{U_{k}} = ( \begin{matrix}Ω^{η} & 0 \\ 0 & Ω^{U_{k}} \end{matrix} )

is the curvature of the direct-sum connection obtained from the given connection by taking orthogonal projection of the covariant derivative D^{η⊕U_{k}} back onto the factors η and U_{k}. This is often the case (cf. [HL_{1}], [Z]). However, the work of John Zweck [Z, Thm. 4.17] on meromorphic sections of vector bundles shows that (12.2) does not always hold. Nevertheless, we do have the following.

Lemma 12.3. The general residue form Res_{Φ,k} on Σ_{k} appearing in Theorems 10.3 and 11.2 can be written as

(12.3)  Res_{Φ,k} = π_{2}_{∗}Φ (Ω^{Im α} ⊕ Ω^{U_{k}}) + dS_{k}

where S_{k} is a smooth form written universally in terms of the curvature and connection of E and F.

Proof. Consider the linear family D^{t} = (1 − t)D^{U} + t(D^{Im α} ⊕ D^{U_{k}}) joining the given connection on U = Im α ⊕ U_{k} and the direct-sum connection, and set S_{k} be the standard Chern transgression form (cf. [HL_{1}]). □

Remark 12.4. The universal expression (12.3) can be regarded in another way. To derive it, it suffices to consider the universal case. In fact via [NS] it suffices to consider the case where X = G_{m}(C^{M}) × G_{n}(C^{N}) and E and F are the pullbacks of the tautological bundles E → G_{m}(C^{M}) and F → G_{n}(C^{N}) respectively. Here π_{2}_{∗}Φ (Ω^{η⊕U_{k}}) is a U_{M} × U_{N}-invariant form on the universal k^{th} fixed-point set G_{k}(E) × G_{N}(F). Now F_{k} = G_{k}(E) × G_{N}(F) is a product of two-stage flag manifolds, and the invariant forms in a given cohomology class are not unique. However, any two cohomologous invariant forms differ by the exterior derivative of an invariant form. To derive equation (12.3) explicitly in any given case, it suffices to do it on this particular manifold X.

Proposition 12.5. Let α : E → F, Φ and Ψ be as in 10.3 and 11.2. For each k let

Res_{Φ,k} = R̃es_{Φ,k} + dS_{k}

be the canonical decomposition of the residue form given in (12.3). Then each of the currents

R̃_{k} ≡ R̃es_{Φ,k}[Σ_{k}(α)],  S_{k}[Σ_{k}(α)],  and  (dS_{k})[Σ_{k}(α)]

has locally finite mass in X, and furthermore the following equation holds on X:

(12.4)  d (S_{k}[Σ_{k}(α)]) = d(S_{k})[Σ_{k}(α)].

The parallel results hold for Res_{Ψ,k}.

Proof. The proof that these currents have finite mass follows exactly the lines of the proof of 12.2. Assertion (12.4) then follows from (12.1). □

Corollary 12.6. The cycles R_{k} ≡ Res_{Φ,k}[Σ_{k}(α)] and R̃_{k} ≡ R̃es_{Φ,k}[Σ_{k}(α)] are cohomologous in the complex of locally flat currents on X. In particular they represent the same class in H^{∗}(X; R). (The analogous assertion holds for Res_{Ψ,k}[Σ_{k}(α)].)

Combining the above gives the following.

Theorem 12.7. Let α : E → F, Φ and Ψ be as in Theorem 10.3 or Theorem 11.2. Then the following equation holds on X:

Φ(Ω^{E}) = ∑_{k} R̃es_{Φ,k}[Σ_{k}(α)] + dT̃

where

R̃es_{Φ,k} = ∫_{π_{k}} Φ (Ω^{π_{k}^{∗}Im α} ⊕ Ω^{U_{k}})

and where π_{k} : G_{k}(kerα ⊕ cokerα) ⟶ Σ_{k}(α) is the Grassmann compactification of the normal bundle to Σ_{k}(α), Im α ⊂ F carries the induced connection, Ω^{π_{k}^{∗}Im α} ⊕ Ω^{U_{k}} denotes the curvature of the direct sum connection, and T̃ is a flat current on X. In particular, if Φ is a multiplicative series, then

R̃es_{Φ,k} = {∫_{π_{k}} Φ(Ω^{U_{k}})} Φ(Ω^{Im α}).

The analogous result holds for Ψ(Ω^{F}).

It is interesting to examine some basic examples. For convenience we shall drop the tilde from our notation.

Example 12.8. Let

Φ(Ω) = c(Ω) =^{def} det (I + \frac{i}{2π} Ω)

be the total Chern class and suppose that m = n. Then

Res_{c,1} = c(Ω^{Im α})
Res_{c,k} = 0 for all k > 1.

Example 12.9. Let

Φ(Ω) = ch(Ω) =^{def} exp\{\frac{i}{2π} Ω)\}

be the Chern character and suppose that m = n. Then

Res_{ch,1} = −\frac{ch(Ω^{ker α}) − ch(Ω^{coker α})}{c_{1}(Ω^{ker α}) − c_{1}(Ω^{coker α})}

Res_{ch,k} = 0 for all k > 1.

Example 12.10. Let

Φ(Ω) = c^{⊥}(Ω) =^{def} det \{(I + \frac{i}{2π} Ω)^{−1}\}

be the total dual Chern class and suppose that m = n. Then a calculation shows that for any ℓ ≥ 1

Res_{(c^{⊥})^{ℓ},1} = −c^{⊥}(Ω^{Im α})^{ℓ} \{\frac{c^{⊥}(Ω^{ker α})^{ℓ} − c^{⊥}(Ω^{coker α})^{ℓ}}{c_{1}(Ω^{ker α}) − c_{(}Ω^{coker α})} \}

Res_{(c^{⊥})^{ℓ},ℓ} = c^{⊥}(Ω^{Im α})^{ℓ}c^{⊥}(Ω^{ker α})^{ℓ}c^{⊥}(Ω^{coker α})^{ℓ}

Res_{(c^{⊥})^{ℓ},k} = 0 for all k > ℓ.

From this example we get the following pretty formulas for a normal bundle map α : E → F over a manifold X where rank(E) = rank(F). Fix any integer ℓ ≥ 1. Then there exists a flat current S_{ℓ} on X such that

c^{⊥}(Ω^{E})^{ℓ} − c^{⊥}(Ω^{F})^{ℓ} = −c^{⊥}(Ω^{Im α})^{ℓ} \{\frac{c^{⊥}(Ω^{ker α})^{ℓ} − c^{⊥}(Ω^{coker α})^{ℓ}}{c_{1}(Ω^{ker α}) − c_{(}Ω^{coker α})} \} [Σ_{1}]
+ …

+ c^{⊥}(Ω^{Im α})^{ℓ}c^{⊥}(Ω^{ker α})^{ℓ}c^{⊥}(Ω^{coker α})^{ℓ}[Σ_{ℓ}] + dS_{ℓ}

§13. Results for real vector bundles. Up to this point the bundles E and F have been assumed to be complex. We now re-examine our results under the assumption that E and F are real vector bundles. One verifies directly that in this case the fundamental constructions presented in §§2—12 carry through with virtually no change provided that the fibre diagonal Δ_{G} ⊂ G^{⊕2} and its isotopic deformations Γ_{s}, (cf. (3.3)) define currents on G^{⊕2}, and provided that the projection pr_{2} induces a map (pr_{2})_{∗}on currents (cf. (4.1) and (4.2)).

We recall that currents of dimension p on a manifold Y are defined as the topological dual space of the space Ẽ^{p}(Y) of compactly supported, smooth p-forms twisted by the orientation bundle Or_{Y} of Y (cf. [deR], [S]). Any p-dimensional submanifold with oriented normal bundle defines such a current. So also does any smooth form of degree n − p on Y, and in fact every current can be considered to be an (n − p)-form on Y with generalized coefficients.

Note that a smooth mapping f : Y → Y′ between smooth manifolds induces a continuous linear map f_{∗} on currents if and only if f^{∗}Or_{Y′} = Or_{Y}. When f is a submersion, this condition is guaranteed if the fibres of f are orientable.

Now the normal bundle N to the fibre diagonal G ≅ Δ_{G} ⊂ G^{⊕2} is isomorphic to the bundle of tangent vectors to the fibres of the fibration π : G → X. That is, there is a bundle equivalence

(13.1)  N ≅ Hom(U, U^{⊥})

We recall an important elementary fact. (See [HL_{2}, A.13], for example.)

Lemma 13.1. Let V, V′ be real vector bundles over a space Y. If rank(V) − rank(V′) is even, then Hom(V, V′) is orientable.

Recall that rank(U) = rank(E) and rank(U^{⊥}) = rank(F), and note that the fibres of the projections pr_{i} : G^{⊕2} → G are diffeomorphic to the Grassmannian G_{m}(R^{m+n}) whose tangent bundle is Hom(U, U^{⊥}). Observe also that for a normal bundle map α : E → F the normal bundle to each Σ_{k}(α) is isomorphic to Hom(ker α, coker α). Thus from (13.1) and the Lemma one deduces

Lemma 13.2. Suppose that rank(F) − rank(E) is even. Then the submanifolds Γ_{s} ⊂ G^{⊕2} define currents on G^{⊕2}, and the projections pr_{i}, i = 1, 2, induce continuous maps on currents. Furthermore, if α : E → F is a normal bundle map, then each submanifold Σ_{k}(α) defines a current [Σ_{k}(α)] on X.

Note that these results are independent of all considerations of orientability for E and F on X.

This brings us to the main result of this section.

Theorem 13.3. Let E → X and F → X be smooth real vector bundles where rank(F) − rank(E) is even. Then the analogues of all results in §§2—12 hold for these bundles. Furthermore, if E and F are given orthogonal connections, then in formulas (6.3), (8.4-5), (10.7-8) and in Theorems 11.2 and 12.6, one has that

(13.2)  Res_{k} = 0  for all k odd.

Proof. Once one knows that the submanifolds Γ_{s} define currents in G^{⊕2}, that pr_{2} induces a continuous map on currents, and that the submanifolds Σ_{k} and Υ_{k} define currents in X, the discussion given in §§2—12 carries through without change in the real case. This gives the first part of the theorem.

To prove (13.2) we use Theorem 12.6. Observe first that every O_{m}-invariant polynomial Φ is a polynomial in the Pontrjagin forms and that for the total Pontrjagin form we have

p \binom{}{Ω^{U}} = p (Ω^{U_{k}}) π^{∗}p (Ω^{Im α}) .

Hence any polynomial in the Pontrjagin classes p_{j} \binom{}{Ω^{U}} can be expressed as a

polynomial in the Pontrjagin classes p_{j} (Ω^{U_{k}}) with coefficients which are pull-backs over π of forms on Σ_{k}(α). Thus to prove (13.2) it will suffice to prove the following.

Lemma 13.4. Let V → Y be a smooth riemannian vector bundle of rank M with orthogonal connection, and let G = G_{ℓ}(V) → Y be the Grassmann bundle of unoriented ℓ-planes in V. Let π : U → G be the tautological ℓ-plane bundle, and write π^{∗}V = U ⊕ U^{⊥}. Give U the connection obtained by projection of the pull-back connection on π^{∗}V. Then if rank(V) is odd, one has

∫_{π} Φ (Ω^{U}) = 0

for all O_{M}-invariant polynomials Φ on the Lie algebra 𝔒_{M}.

Proof. To begin we observe that the result holds in the special case where V = V_{0} is the tautological bundle over the real Grassmannian G_{M}(R^{N}) ≡ Y_{0}. This follows because the closed form

∫_{π} Φ (Ω^{U_{0}})

(where U_{0} is the tautological bundle over Y_{0}) is O_{N}-invariant and hence harmonic, but also of odd degree and therefore zero since H^{odd}(Y_{0}; R) = 0.

The general case follows from the special one because by [NS] there exists an embedding j : Y ↪ Y_{0} such that j^{∗}V_{0} ≅ V as bundles with connection. It follows from the naturality of the constructions that there exists a bundle map j̃: G_{ℓ}(V) → G_{ℓ}(V_{0}) covering j such that j̃^{∗}U_{0} ≅ U as bundles with connection. Thus there is a commutative diagram

U \underrightarrow{˜j̃} U_{0}
↓  ↓
G_{ℓ}(V) \underrightarrow{j̃} G_{ℓ}(V_{0})
π↓  ↓^{π_{0}}
Y ⟶^{j} Y_{0}

where j̃ and ˜ j̃ are bundle maps. One concludes that

0 = j^{∗}(π_{0})_{∗}Φ (Ω^{U_{0}}) = (π)_{∗} (j̃^{∗}Φ (Ω^{U_{0}})) = π_{∗} (Φ (Ω^{U})) .

Note 13.5 . When F (or E) is orientable, these results extend to invariant polynomials on the Lie Algebra of SO_{n} (or SO_{m} respectively).

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