GEOMETRIC RESIDUE THEOREMS Reese Harvey and H. Blaine Lawson, Jr.*

*Research of both authors was supported by the National Science Foundation

Contents

Introduction

1. Divisors and Atomicity
2. Degeneracy currents
3. Poincaré-Lelong families
4. Thom-Porteous families (complex case)
5. Thom-Porteous families (real case)
6. Characteristic forms and the degeneracies of k-frame fields
7. Singularities of projections
8. Singularities of maps
9. Milnor currents
10. CR-singularities
11. Invariants for pairs of complex structures
12. Invariants for plane fields and foliations
13. Higher self intersections of plane fields and invariants for pairs of foliations

Appendix A Orientations and tensor products Appendix B Integration over the fibres in G_{r}(E) Appendix C Explicit formulae Introduction.

Roughly speaking residue theorems in geometry are results which associate topological invariants to the singularities of geometric objects. The discovery and use of such theorems has a history dating back at least to Riemann. A classical example is Hopf’s theorem relating the singularities of vector fields to the Euler characteristic.

A somewhat different topic in modern geometry is Chern-Weil Theory. This associates to a smooth bundle with connection a canonical family of differential forms which represent characteristic classes of the bundle. The forms are written explicitly as universal polynomials in curvature. Furthermore, for two distinct connections ω, ω′ on a bundle, the difference of the characteristic forms can be written as a coboundary p(Ω) − p(Ω′) = dT where T = T (ω, ω′) is also canonically expressed in terms of the connections. These transgression forms T lead to important secondary invariants (cf. [CS], [ChS]).

Recently the authors developed a generalized Chern-Weil Theory for singular connections [HL_{2}] where characteristic forms are replaced by characteristic currents written in terms of curvature and the singularities of some given geometric object. In this paper we shall use our theory to systematically deduce a wide variety of geometric residue theorems. Our formulas refine the classical ones in several ways. To begin they are derived canonically at the level of differential forms and currents. For example, for a mapping α between bundles with connection there are formulas

p(Ω) − Σ(α) = dT

where: p(Ω) is a canonical characteristic form as above, Σ(α) is a current defined purely in terms of the singularities of α, and T is a canonical transgression form (with L_{loc}^{1}-coefficients). This enables us to define secondary invariants for certain connections and singularities.

Furthermore, our theory generates canonical smooth families

p(Ω) − p(Ω_{s}) = dT_{s} for 0 < s ≤ ∞

where T_{∞} = 0 and where one has convergence

T_{s} → T everywhere in L_{loc}^{1}

as s → 0. In particular the families of smooth characteristic forms p(Ω_{s}) converge to the singular current, i.e.,

p(Ω_{s}) → Σ(α)

as s → 0. In certain “approximation modes” p(Ω_{s}) will be supported in the stubular neighborhood of Σ(α). The virtue of the explicit nature of these formulas was seen in [HL_{2}] where the procedure gave simple explicit formulas for the Thom class of a bundle with connection. In fact it gave families interpolating between the pull-back of the Euler (or top Chern) form and the current given by zero-section of the bundle.

The emphasis in this paper will be as much on the general method as on the detailed structure of various formulas. The authors hope to provide the reader with the techniques for explicitly relating singularities of maps, sections of bundles etc. to characteristic forms in the manner above whenever the need arises. However, we shall also derive here a series of such explicit formulas in a broad range of fields. These will include: Thom-Porteous formulas at the level of forms and currents, formulas of Poincaré-Lelong type between Chern/Pontjagin forms and linear dependency currents of families of cross-sections of a bundle, residue theorems relating degeneracies of maps between manifolds and characteristic forms, residue theorems for singularities of CR-structures, new invariants for pairs of complex structures, invariants for pairs of plane fields, higher self-intersection formulas for tangent plane fields, higher order contact currents for pairs of foliations and relations to characteristic forms. In a subsequent paper we shall similarly establish various determinental formulas and, in particular, explicit Poincaré-Lelong equations for Shubert cells on Grassmann manifolds.

The authors want to thank Bill Fulton for introducing them to the methods of modern enumerative geometry so beautifully presented in his book [Fu]. They are also indebted to John Zweck for many useful comments on early versions of this manuscript.

A notational convention: Throughout this paper X will denote a manifold which is oriented unless it is stated otherwise.

§1. Divisors and Atomicity.

In this section we review briefly the theory of atomic sections and divisors introduced in [HS]. This material enhances the range and applicability of the subsequent results, but it is not necessary for understanding their proofs. The reader could skip this section and simply replace “atomicity” everywhere by “non-degenerate vanishing”.

Let f : U → R^{p} be a C^{∞}-map where U ⊆ R^{n} is an open set.

Definition 1.1. The map f is atomic if

f^{∗} (\frac{dy^{I}}{|y|^{|I|}} ) ∈ L_{loc}^{1}

for all I = (i_{1}, …, i_{p}) such that |I| = Σi_{k} < p.

Let ρ : R^{p} − {0} ⟶ S^{p−1} denote radial projection onto the unit sphere, and define

(1.2)  Θ = \frac{1}{c_{p}} ρ^{∗}(d vol_{S^{p−1}})

= \frac{1}{c_{p}} (∑_{k=1}^{p} y_{k} \frac{∂}{∂y_{k}} ) (\frac{dy_{1} ∧ ⋯ ∧ dy_{p}}{|y|^{p}} )

= \frac{1}{c_{p}} ∑_{k=1}^{p}(−1)^{k−1} \frac{y_{k}}{|y|^{p}} dy_{1} ∧ ⋯ ∧ d̂y_{k} ∧ ⋯ ∧ dy_{p}

where c_{p} = vol(S^{p−1}). The coefficients of this form are integrable in bounded neighborhoods of 0, and Θ satisfies the current equation

dΘ = [0] in R^{p}.

Note that if f : U → R^{p} is atomic, then f^{∗}Θ ∈ L_{loc}^{1} on U.

Definition 1.3. Let f : U → R^{p} be atomic. Then the divisor of f is the current of degree p (and dimension n − p) on U given by taking the exterior derivative of the potential d(f^{∗}Θ), i.e.,

Div(f) = d(f^{∗}Θ).

This current has the following properties.

d Div(f) = 0 (1.4)

(1.5) supp Div(f) ⊆ {x ∈ U : f(x) = 0} =^{def} Z(f) (1.6) If 0 is a regular value of f, then Div(f) = [Z(f)]

where [Z(f)] is the current given by integration over the manifold Z(f). The first two properties are obvious. The last is straightforward to verify. Note that the definition of [Z(f)] involves a choice of orientation on Z(f).

Theorem 1.7. ([HS]) Let f̃(x) = g(x)f(ϕ(x)) where ϕ : U → U is a diffeomorphism and g : U → GL_{p}(R) is a smooth map. Then f̃ is atomic if and only if f is atomic. Furthermore, if det(g) > 0 on U and ϕ is orientation preserving, then

ϕ_{∗} Div(f̃) = Div(f).

As an immediate corollary the concepts of atomicity and divisor extend to sections of a vector bundle.

Definition 1.8. Let E → X be a smooth vector bundle over an n-manifold X. A smooth section μ ∈ Γ(E) is said to be atomic if each point x ∈ X has a neighborhood with local coordinates and a local trivialization of E with respect to which μ is an atomic R^{p}-valued function.

If E and X are oriented, and if μ ∈ Γ(E) is atomic, then Div(μ) is a well defined current of degree p (and dimension n − p) on X, called the divisor of μ.

Remark 1.9. Note that Div(μ) is well-defined in the non-orientable case provided that the first Stiefel-Whitney classes satisfy

w_{1}(E) = w_{1}(X)

in H^{1}(X; Z_{2}). This condition guarantees that we can choose local trivializations of E over a coordinate covering so that the changes of trivialization g_{αβ} and the Jacobian matrices of the changes of local coordinates ϕ_{αβ} satisfy det(g_{αβ})·det(ϕ_{αβ}) > 0. (See [Z].)

In [HS] effective criteria are established which guarantee atomicity.

Theorem 1.10. ([HS]) Let f : U → R^{p} be real analytic. If dim Z(f) ≦ n − p, then f is atomic.

Theorem 1.11. ([HS]) Suppose f : U → R^{p} satisfies:

(1) There are constants c > 0, N > 0 such that

∥f(x)∥ ≧ c dist(x, Z(f))^{N},

(2) The Minkowski dimension of Z(f) is < n − p + 1. Then f is atomic.

It is also proved in [HS] that if f is atomic, its divisor is integrally flat. Hence one has the following “regularity”.

Theorem 1.12. ([HS]) Let f : U → R^{p} be atomic. If the mass of Div(f) is locally finite, then Div(f) is locally rectifiable.

§2. Degeneracy currents.

In this section we introduce the notion of the k^{th} degeneracy current of a bundle map. This is a current associated to the drop in rank of the map to rank ≦ k.

For the definitions we must fix some notation. Let E → X and F → X be smooth vector bundles over an oriented manifold X, where E and F are either both complex or both real, and let

m = rank E and n = rank F.

Fix an integer k with 0 ≦ k ≦ min{m, n} and set

r = m − k.

Let

(2.1)  π : G_{r}(E) ⟶ X

be the smooth bundle whose fibre at x ∈ X is the set of all r-dimensional linear subspaces of E_{x} (the fibre of E at x). Over G_{r}(E) there is a tautological vector bundle U of rank r whose fibre at P ∈ G_{r}(E) consists of all vectors v ∈ P. There is a natural bundle embedding

(2.2)  U ⊂ π^{∗}E

and if we introduce a metric in E, this gives a natural splitting

(2.3)  π^{∗}E ≅ U ⊕ U^{⊥}.

Suppose now that we are given a smooth bundle map

α : E ⟶ F.

Then this lifts to a mapping π^{∗}α : π^{∗}E → π^{∗}F over G_{r}(E), and composing with j gives a map

(2.4)  α̂ =^{def} π^{∗}α|_{U} : U ⟶ π^{∗}F.

Definition 2.5. The bundle map α is said to be k-atomic if α̂ is an atomic section of the bundle Hom(U, π^{∗}F) = U^{∗} ⊗ π^{∗}F over G_{r}(E).

Definition 2.6. For a bundle map α which is k-atomic, we define its k^{th} degeneracy current on X to be

𝔻_{k}(α) = π_{∗} Div(α̂)

where π_{∗} denotes the push-forward of currents by π : G_{r}(E) → X.

Note that when E and F are real bundles, Div(α̂), and therefore 𝔻_{k}(α), only make sense when w_{1}(U^{∗} ⊗ π^{∗}F) = w_{1}(G_{r}(E)) (cf. 1.9). This condition holds when m ≡ n ≡ k (mod 2) (See Appendix A, A.6 - A.10).

The codimension of Div(α̂) is rn (or 2rn in the complex case), and the fibre dimension of G_{r}(E) is rk (or 2rk respectively). Hence, we have

codim 𝔻_{k} = r(n − k) = (m − k)(n − k)

in the real case, and

codim 𝔻_{k} = 2(m − k)(n − k)

in the complex case.

Lemma 2.7. For any k-atomic section α, one has

supp 𝔻_{k}(α) ⊆ {x ∈ X : rank α_{x} ≦ k}

Proof. If x ∈ supp 𝔻_{k}(α), then there exist a subspace U ⊂ E_{x} of dimension r such that α_{x} |_{U}= 0. Hence, rank α_{x} ≦ m − r = k. □

Note that if rank α_{x} = k, then there is exactly one subspace of dimension r (namely ker α_{x}) on which α_{x} = 0. That is, above each point of X where rank α = k, there is exactly one point in the zero set Z(α̂) of α̂.

Proposition 2.8. Suppose α̂ vanishes non-degenerately. Then

RK_{k}(α) =^{def} {x ∈ X : rank α_{x} = k}

is a locally rectifiable set, and

𝔻_{k}(α) = [RK_{k}(α)]

i.e., 𝔻_{k}(α) is the current given by integration over this set.

Proof. By hypothesis we know that Z(α̂) is a smooth proper submanifold of G_{r}(E), and that Div(α̂) = [Z(α̂)]. Therefore, 𝔻_{k}(α) = π_{∗}[Z(α̂)], i.e., 𝔻_{k}(α) is the d-closed locally rectifiable current given by the push-forward of the manifold Z(α̂). This current has dimension N = dim X−(m−k)(n−k) (N = dim X−2(m−k)(n−k) in the complex case). The Federer-Sard Theorem [Fe] implies that the set of critical values of the map π |_{Z(α̂)}, from Z(α̂) to X, has Hausdorff N-dimensional measure zero. Hence π_{∗}[Z(α̂)] = p[R] where R is the set of regular values and p is an integer. It remains to show that

(2.9)  R ⊂ RK_{k}(α),

since, as noted above,

π : Z^{0}(α̂) = {u ∈ Z(α̂) : π(U) ∈ RK_{k}(α)} ⟶ RK_{k}(α)

is one to one. To see (2.9), we observe that if rank α_{x} = k − p, then

{U ⊂ E_{x} : rank U = r and U ⊂ ker α_{x}} = π^{−1}(x) ∩ Z(α̂)

is a submanifold of G_{r}(E_{x}) diffeomorphic to the Grassmannian of r-planes in (r+p)- space. Thus, the preimage under π |_{Z(α̂)} of each point x with rank α_{x} < k is a smooth submanifold of positive dimension. □

It is appropriate here to point out that bundle maps are generically k-atomic, in fact they generically satisfy the hypothesis of Proposition 2.8. Recall that a smooth cross-section of a bundle is said to vanish non-degenerately if its graph is transversal to the zero-section. The following Propsition is a minor modification of the standard Thom Transversality Theorem [GG].

Proposition 2.10. Suppose X is compact with (possibly empty) boundary. Then the set of smooth bundle maps α for which α̂ vanishes non-degenerately is open and dense in the C^{1}-topology. Consequently, for any manifold X the set of such α is residual (i.e., contains the intersection of a countable family of open dense subsets.)

Proof. Openness is clear. To prove density we fix a section α and a point x_{0} ∈ X. Choose trivializations of E and F in a neighborhood U of x_{0}. Then we have a family of sections of Hom(E, F) over U given by

α_{L} = α |_{U} +L

for L ∈ Hom(E_{x_{0}}, F_{x_{0}}). This gives a family of sections α̂_{L} of Hom(U, π^{∗}F) over Ũ =^{def} π^{−1}(U). We think of this as a map of manifolds

Ũ × V ⟶ Hom(U, π^{∗}F) |Ũ (2.11)

(u, L) ⟼ α̂_{L}(u)

where V = Hom(E_{x_{0}}, F_{x_{0}}). This map (2.11) is actually a submersion. To see this note first that π ∘ α̂_{L}(u) = u, and so the image of T_{u}Ũ × {0} ⊂ T_{(u,L)}(Ũ × V) is a transversal to the fibre of π at all points. However, the map {u} × V → Hom(U, π^{∗}F)_{u} is a surjective linear map at each u (which sends L to “L |_{U}”). This shows that (2.11) is a submersion.

Using a partition of unity and standard constructions one can globalize to a submersion

(2.12)  G_{r}(E) × W ⟶^{Ψ} Hom(U, π^{∗}F)

where W is a finite-dimensional vector space, where

Ψ_{w} =^{def} Ψ |_{G_{r}(E)×{w}} = α̂_{w}

for a section α_{w} ∈ Γ(Hom(E, F)), and where α_{0} = α. (Here W will be a direct sum of V ’s as above.) Since Ψ is a submersion, it is transversal to the zero-section of Hom(U, π^{∗}F). By a standard argument using Sard’s Theorem for families (cf. [HL_{1}]) we conclude that Ψ_{w} is transversal to the zero section of Hom(U, π^{∗}F) for almost all w ∈ W. □

In general k-atomicity is much weaker than requiring α̂ to vanish non-degenerately. One useful criterion for k-atomicity is the following.

Lemma 2.13. Let α be a real analytic bundle map. If the set Z(α̂) ⊂ G_{r}(E) has codimension ≧ (m − k)n, then α is k-atomic.

Proof. This is an immediate application of Theorem 1.10 above. □

Proposition 2.14. Suppose α is real analytic, and satisfies the condition that

codim RK_{k−p} ≧ (m − k)(n − k + p),
(codim RK_{k−p} ≧ 2(m − k)(n − k + p) in the complex case),

for all p, 0 ≤ p ≤ k. Then α is k-atomic.

Proof. If rank α_{x} = k−p, then dim ker α_{x} = r+p and π^{−1}(x)∩Z(α̂) is a Grassmannian of r-dimensional subspaces of (r + p)-space. This is a manifold of dimension rp (2rp in the complex case). It follows that

dim{π^{−1}(RK_{k−p}) ∩ Z(α̂)} = dim(RK_{k−p}) + rp

for each p. Let d = dim X. Then the condition of 2.13 will be satisfied if dim(RK_{k−p}) + rp ≦ d − (m − k)(n − k) for all p, i.e., if codim(RK_{k−p}) ≧ (m − k)(n − k) + rp = (m − k)(n − k + p) for all p (with appropriate changes in the complex case). □

We now address the problem of real bundles. If E and F are real, then there will be two cases of interest for our discussion. In neither case is E or F assumed to be orientable.

2.15. Real Bundles; Case I. Here we assume that m, n and k are all even integers. In this case Hom(U, π^{∗}F) is canonically oriented even when U and F are not. Furthermore the fibre of π, which is the Grassmannian G_{r}(R^{m}) of unoriented r-planes in R^{m} (r = m − k), is also oriented. See Corollary A.6 and Corollary A.8 in Appendix A for proofs.

2.16. Real Bundles; Case II. Here we assume that m, n and k are all odd integers. In this case neither Hom(U, π^{∗}F) nor G_{r}(E) are oriented, but they have the same first Stiefel-Whitney class. (See Corollary A.6 and Corollary A.8 in Appendix A.) Consequently, Div(α̂) is defined. The fibre G_{r}(R^{m}) of π is also not oriented, so one must be careful when integrating over the fibre. (See Proposition A.11 and it’s proof.)

In the remaining cases a direct analysis shows that either w_{1}(G_{r}(E)) ≠ w_{1}(Hom(U, π^{∗}F)), and so Div(α̂) is not defined as a current, or the fibre integral of the Euler form χ(Hom(U, π^{∗}F)) is zero, and so results of the type obtained in §5 are uninteresting.

§3. Poincaré-Lelong families.

In this section we briefly recall a basic result of [HL_{2}] which provides the first step in all of our subsequent constructions. Let V → X be a smooth complex vector bundle of rank m with a complex connection D and a metric < ·, · > (which is in general unrelated to the connection), and let μ be an atomic section of V.

From this data we introduce a family of connections \overrightarrow{D}_{s} on V which become singular as s ↘ 0 precisely along the zeros of μ. Taking the determinant of the curvature \overrightarrow{R}_{s} of \overrightarrow{D}_{s} gives a family of smooth 2m-forms which converge as s → 0 to Div(μ). Moreover, the transgression forms for this family also converge and provide a canonical and functorial coboundary between c_{m}(R^{V}) and Div(μ), where R^{V} = D^{2} is the curvature of the given connection.

To begin the process we fix an approximation mode by choosing a function χ ∈ C^{∞}([0, ∞]) with χ(0) = 0, χ(∞) = 1, and χ′ ≧ 0. We then define the family of connections \overrightarrow{D}_{s} by setting

(3.1)  \overrightarrow{D}_{s}ν = Dν − χ_{s} \frac{< ν, μ >}{|μ|^{2}}Dμ

on sections ν ∈ Γ(V), where χ_{s} = χ(|μ|^{2}/s^{2}). This is a smooth family of smooth connections on F for all s > 0. Let

(3.2)  \overrightarrow{R}_{s} = (\overrightarrow{D}_{s})^{2}

denote the curvature of \overrightarrow{D}_{s}, and consider the family of forms

(3.3)  τ_{s} =^{def} det (\frac{i}{2π} \overrightarrow{R}_{s}) = c_{m}(\overrightarrow{R}_{s}).

Note that τ_{∞} = c_{m}(R^{V}) is the top Chern form of the given connection. For all s > 0 there is a transgression form

(3.4)  σ_{s} = (\frac{i}{2π} )^{m} ∫_{s}^{∞}det ( ˙\overrightarrow{D}_{t} ; \overrightarrow{R}_{t}) dt

where

det(A; B) = \frac{d}{dt} det(B + tA) |_{t=0} = tr(B̃A)

where B̃ is the transposed matrix of cofactors of B, and where ˙ \overrightarrow{D}_{t} = (d/dt)\overrightarrow{D}_{t}. These (2m − 1)-forms σ_{s} have the property that

(3.5)  dσ_{s} = c_{m}(R^{V}) − τ_{s}.

Our main result is the following.

Theorem 3.6. ([HL_{2}]) Let μ be an atomic section of the bundle V with connection D as above. Then the limit σ = lim_{s→0} σ_{s} exists in the space of forms on X with

L_{loc}^{1}(X)-coefficients. This limit is independent of choice of approximation mode and satisfies the current equation

(3.7)  c_{m}(R^{V}) − Div(μ) = dσ.

In particular, the family of L_{loc}^{1}-forms ρ_{s} =^{def} σ − σ_{s} satisfies the equation

(3.8)  τ_{s} − Div(μ) = dρ_{s}

and has the property that

(3.9)  lim_{s→0} ρ_{s} = 0 and lim_{s→∞} ρ_{s} = σ

in L_{loc}^{1}.

Equation (3.7) generalizes the classical Poincaré-Lelong formula for line bundles and for this reason we call τ_{s} the Poincaré-Lelong family. This family provides canonical smoothings of the divisor Div(μ). If χ has the property that χ(t) = 1 for all t ≧ 1, then τ_{s} has the additional property that

supp(τ_{s}) ⊆ {x ∈ X : ∥μ_{x}∥ ≦ s}

for all s > 0.

There is a companion result when V is a real oriented bundle of rank 2m with an orthogonal connection D, and an atomic section μ. We fix any χ as above and introduce a smooth family of orthogonal connections defined on a section ν ∈ Γ(V) by

(3.10)  \overrightarrow{D}_{s}ν = Dν − χ_{s} \frac{< ν, μ >}{|μ|^{2}}Dμ + χ_{s} \frac{< Dν, μ >}{|μ|^{2}}μ

where χ_{s} = χ(|μ|^{2}/s^{2}) as above. Let \overrightarrow{R}_{s} = (\overrightarrow{D}_{s})^{2}, and consider the family of closed 2m-forms

(3.11)  τ_{s}^{R} =^{def} Pf (−\frac{1}{2π} \overrightarrow{R}_{s})

where Pf(A) denotes the Pfaffian of a skew-symmetric matrix A. We also define transgression forms

(3.12)  σ_{s}^{R} = (\frac{−1}{2π})^{m} ∫_{s}^{∞}Pf( ˙\overrightarrow{D}_{t} ; \overrightarrow{R}_{t})dt

where

Pf(A ; B) = \frac{d}{dt} Pf(B + tA) |_{t=0} .

These forms satisfy the equation

dσ_{s}^{R} = χ(R^{V}) − τ_{s}^{R} for s > 0

where χ(R^{V}) = Pf (−\frac{1}{2π} R^{V} ) is the Chern-Euler form of the connection D.

Theorem 3.13. ([HL_{2}]) Let μ be an atomic section of the oriented bundle V with orthogonal connection D as above. Then the limit σ^{R} = lim_{s→0} σ_{s}^{R} exists in L_{loc}^{1}(X) and is independent of the choice of approximation mode. It satisfies the current equation

(3.14)  χ(R^{V}) − Div(μ) = dσ^{R}.

In particular, the L_{loc}^{1}-forms ρ_{s}^{R} = σ^{R} − σ_{s}^{R} satisfy

(3.15)  τ_{s}^{R} − Div(μ) = dρ_{s}^{R}

on X and have the property that

(3.16)  lim_{s→0} ρ_{s}^{R} = 0 and lim_{s→∞} ρ_{s}^{R} = σ^{R}

in L_{loc}^{1}(X).

We will call τ_{s}^{R} the Euler family associated to D, μ and χ.

The results above enhance the fundamental work of of [Ch] and [ChB], [ChB_{2}] where the potentials σ and σ_{s} were introduced in the special case of the tautalogical section over the total space of V. In this universal case our family τ_{s} (and τ_{s}^{R}) provide families of canonical Thom forms for s > 0 which converge to the zero section as s → 0. See Appendix C for simple explicit formulas for τ_{s}, σ_{s}, σ, τ_{s}^{R}, σ_{s}^{R}, and σ^{R} taken from [HL_{2}].

§4. Thom-Porteous families (complex case).

In this section we generalize the results of §3 to arbitrary degeneracy currents. Let us fix smooth complex vector bundles E → X and F → X equipped with connections D^{E} and D^{F} and with hermitian metrics (not necessarily related to the connections). We suppose that rank(E) = m and rank(F) = n and we assume that X is oriented.

In terms of the given connections we shall derive characteristic forms which are cohomologous to the degeneracy currents (cf. [T][P][M1][R]). To do this we must introduce the Shur polynomials. Suppose

ξ = 1 + ξ_{1} + ξ_{2} + ⋯

is a differential form on X where each ξ_{k} is homogeneous of degree 2k. Then for non-negative integers a and b we define the Shur polynomial in ξ by

(4.1)  Δ_{a}^{(b)}(ξ) = det((ξ_{a−i+j}))_{\begin{matrix}1≤i≤b \\ 1≤j≤b \end{matrix}}

i.e., Δ_{a}^{(b)}(ξ) is the homogeneous form given by the determinant of the b × b matrix whose (i, j)^{th} entry is ξ_{a−i+j}. These polynomials satisfy the fundamental identity (cf. [Fu, pg. 264])

(4.2)  Δ_{a}^{(b)}(ξ) = (−1)^{ab}Δ_{b}^{(a)}(ξ^{−1})

where ξ^{−1} is defined by the relation ξ · ξ^{−1} = 1, and also the identity

(4.3)  Δ_{a}^{(b)}(ξ) = (−1)^{ab}Δ_{a}^{(b)}(ξ̃)

where ξ̃ =^{def} 1 − ξ_{1} + ξ_{2} − ξ_{3} + · · ·.

Consider now the total Chern form of the connection D^{F} given by

c(R^{F}) = det (I + \frac{i}{2π} R^{F} ) = 1 + c_{1}(R^{F}) + ⋯ + c_{n}(R^{F})

where R^{F} = (D^{F})^{2}. The form c(R^{E}) is defined similarly, and there is the inverse form c(R^{E})^{−1} determined by c(R^{E})c(R^{E})^{−1} = 1. The Thom-Porteous form of type (a, b) is then defined to be the form

Δ_{a}^{(b)}{c(R^{F})c(R^{E})^{−1}}.

We consider now a smooth bundle map

α : E ⟶ F

and we fix an approximation mode χ as in §3. Let k be an integer with 0 ≤ k < min{m, n} and set N = 2(m − k)(n − k). Then we have the following.

Theorem 4.4. Suppose that α is k-atomic. Then there exist a canonical smooth family of smooth N-forms TP_{s} and a smooth family of L_{loc}^{1} (N − 1)-forms S_{s} on X, for 0 < s ≦ ∞, such that

(4.5)  TP_{s} − 𝔻_{k}(α) = dS_{s}

for all s, and

lim_{s→0} S_{s} = 0

in L_{loc}^{1}. Furthermore, the transgression form S = S_{∞} is independent of the choice of approximation mode χ and satisfies the current equation

(4.6)  Δ_{n−k}^{(m−k)}{c(R^{F})c(R^{E})^{−1}} − 𝔻_{k}(α) = dS.

Proof. Consider the (atomic) section α̂ of the bundle H =^{def} Hom(U, π^{∗}F) introduced in §2. The metrics and connections on E and F induce a natural metric and connection on H. Thus, using χ, we can define a Poincaré-Lelong family of smooth 2(m − k)n-forms τ_{s} and canonical L_{loc}^{1}-forms ρ_{s} associated to α̂. They satisfy the equation

(4.7)  τ_{s} − Div(α̂) = dρ_{s}

for all 0 < s ≦ ∞. Applying π_{∗} to (4.7) gives the equation

(4.8)  TP_{s} − 𝔻_{k}(α) = dS_{s}

where TP_{s} =^{def} π_{∗}τ_{s} (called the Thom-Porteous family) and S_{s} =^{def} π_{∗}ρ_{s} are the smooth and L_{loc}^{1} forms respectively obtained by integration over the fibre of the smooth bundle π : G_{r}(E) → X. From 3.6 we know that σ = ρ_{∞} satisfies the equation

(4.9)  c_{M}(R^{H}) − Div(α̂) = dσ

on G_{r}(E), where M = (m − k)n. Therefore, applying π_{∗} to (4.9) gives the formula

(4.10)  π_{∗}c_{M}(R^{H}) − 𝔻_{k}(α) = dS

on X, where S = π_{∗}ρ_{∞} = π_{∗}σ.

It remains to compute π_{∗}c_{M}(R^{H}). To begin we recall from Appendix A that the Shur polynomials are exactly what is needed to compute the top Chern form of a tensor product connection such as that on H = U^{∗} ⊗ π^{∗}F. The formula is

(4.11)  c_{M}(R^{H}) = Δ_{n}^{(m−k)}{π^{∗}c(R^{F}) · c(R^{U})^{−1}}.

We now recall the splitting

(4.12)  π^{∗}E = U ⊕ U^{⊥}.

There are two connections on π^{∗}E: the induced one π^{∗}D^{E}, and the direct sum connection D^{U} ⊕ D^{U^{⊥}} induced from π^{∗}D^{E} by projection onto the factors in (4.12). Taking the canonical convex family of connections joining these two and using the standard transgression formula [HL_{2}, I.1.19] gives an equation of smooth forms

c(R^{U})c(R^{U^{⊥}}) = π^{∗}c(R^{E}) + dη

on G_{r}(E). This can be rewritten as

c(R^{U})^{−1} = π^{∗}c(R^{E})^{−1} · c(R^{U^{⊥}}) + dη′

where η′ = c(R^{U})^{−1}π^{∗}c(R^{E})^{−1}η. Plugging this into (4.9) gives the formula

(4.13)  c_{M}(R^{H}) = Δ_{n}^{(m−k)} {π^{∗}(c(R^{F})c(R^{E})^{−1})c(R^{U^{⊥}})} + dη′′

for a smooth form η′′ on G_{r}(E).

We now observe that c(R^{U^{⊥}}) is of the form

c(R^{U^{⊥}}) = 1 + c_{1}(R^{U^{⊥}}) + ⋯ + c_{k}(R^{U^{⊥}}).

Furthermore, the fibre dimension of π is 2(m−k)k, and so π_{∗} has the property that

π_{∗}{(π^{∗}ϕ)ψ} = ϕ(π_{∗}ψ_{0})

for smooth forms ϕ on X and ψ on G_{r}(E), where ψ_{0} is the homogeneous component of ψ of degree 2(m − k)k. Consequently, applying π_{∗} to (4.13) gives

(4.14)  π_{∗}c_{M}(R^{H}) = Δ_{n−k}^{(m−k)}{c(R^{F})c(R^{E})^{−1}}π_{∗}(c_{k}(R^{U^{⊥}})^{(m−k)}) + dS′

where S′ = π_{∗}η′′ is a smooth form on X. To see this observe that the only (m − k)- fold product of Chern classes of U^{⊥} which has degree 2(m−k)k is c_{k}(U)^{m−k}. Hence all other terms in the determinant can be dropped when integrating over the fibre. Now π_{∗}c_{k}(R^{U^{⊥}})^{m−k} is a closed 0-form, i.e., a constant. This constant is determined by the topological class which is computed in Proposition B.1 of Appendix B. We prove there that

π_{∗}(c_{k}(U^{⊥})^{m−k}) = 1.

Hence, we have that

π_{∗}c_{M}(R^{H}) = Δ_{n−k}^{(m−k)}{c(R^{F})c(R^{E})^{−1}} + dS′.

Remark 4.15 The adjoint problem. Let α : E ⟶ F be as above and consider the adjoint mapping

α^{∗} : E^{∗} ⟶ F^{∗}

Replacing S_{s} by S_{s} − (\frac{s}{1+s}) S′ and TP_{s} by TP_{s} − (\frac{s}{1+s}) dS′ in equation (4.8) now gives the result. □ of the dual bundles. If α^{∗} is k-atomic, the degeneracy current 𝔻_{k}(α^{∗}) is defined and Theorem 4.4 applies. In this case equation (4.6) becomes

(4.16)  Δ_{m−k}^{(n−k)}{c(R^{E^{∗}})c(R^{F^{∗}})^{−1}} − 𝔻_{k}(α^{∗}) = dS.

We note from the Shur relations (4.2) and (4.3) and the fact that c(E^{∗}) = c̃(E) = 1 − c_{1}(E) + c_{2}(E) − …, that

Δ_{m−k}^{(n−k)}{c(R^{E^{∗}})c(R^{F^{∗}})^{−1}} = (−1)^{d}Δ_{m−k}^{(n−k)}{c(R^{E})c(R^{F})^{−1}} = Δ_{n−k}^{(m−k)}{c(R^{F})c(R^{E})^{−1}}

where d = (m − k)(n − k). Hence the left hand sides of (4.6) and (4.16) coincide. Furthermore one can show that

supp 𝔻_{k}(α^{∗}) = supp 𝔻_{k}(α),

and that for generic maps

(4.17)  𝔻_{k}(α^{∗}) = 𝔻_{k}(α).

One conjectures that (4.17) holds for general k-atomic bundle maps.

Remark 4.18. Holomorphic Case. In the special case where X, E, F and α are all holomorphic and where the D^{E} and D^{F} are canonical hermitian connections, equation (4.7) and therefore also its push-forward (4.8) can be written as ∂ ¯∂-equations, that is, the right hand sides can be replaced by a ∂ ¯∂T where T is a current of bidegree q, q for appropriate q. Consequently in this case Theorem 4.4 has ∂ ¯∂-refinement.

§5. Thom-Porteous families (real case).

In this section we shall establish the analogue of Theorem 4.2 for morphisms α : E → F of real vector bundles. To state the result in its proper generality we need some preliminary discussion.

Throughout this section X will be an oriented manifold and E → X and F → X will be smooth, real vector bundles furnished with orthogonal connections D^{E} and D^{F} respectively. We assume rank E = m and rank F = n and we fix an integer k with 0 ≤ k < min{m, n}. We do not assume that E and F are orientable. There are two cases of interest.

Case I. The integers m = 2m_{0}, n = 2n_{0}, and k = 2k_{0} are all even.

Case II. The integers m = 2m_{0} + 1, n = 2n_{0} + 1 and k = 2k_{0} + 1 are all odd.

To state the theorem we need to consider the Shur polynomials of Pontrjagin forms. Suppose

η = 1 + η_{1} + η_{2} + ⋯

is a differential form on X where each η_{k} is homogeneous of degree 4k. Then for a, b ≧ 0 we define

(5.1)  Δ̃_{a}^{(b)}(η) = det((η_{a−i+j}))_{\begin{matrix}1≤i≤b \\ 1≤j≤b \end{matrix}}.

Associated to the connection D^{F} is the total Pontrjagin form

p(R^{F}) = 1 + p_{1}(R^{F}) + ⋯ + p_{n_{0}}(R^{F}),

where p(A) = det (I + \frac{1}{4π^{2}} A^{2}). The form p(R^{E}) is given similarly, and we define the Thom-Porteous form of type (a, b) in the Pontrjagin classes to be the homogeneous form

Δ̃_{a}^{(b)}{p(R^{F})p(R^{E})^{−1}}.

We now consider a smooth bundle map

α : E ⟶ F,

and we fix an approximation mode χ as in §3.

Theorem 5.2. Let E, F and k be as above (either Case I or Case II), and suppose that α is k-atomic. Let N = (m − k)(n − k) = 4(m_{0} − k_{0})(n_{0} − k_{0}). Then there exists a canonical smooth family of smooth N-forms TP_{s}, s > 0, and a smooth family of L_{loc}^{1} (N − 1)-forms S_{s} on X, for 0 < s ≦ ∞ such that

(5.3)  TP_{s} − 𝔻_{k}(α) = dS_{s}

for all s, and

lim_{s→0} S_{s} = 0

in L_{loc}^{1}. Furthermore, the transgression form S = S_{∞} is independent of approximation mode and satisfies the equation

(5.4)  Δ̃_{n_{0}−k_{0}}^{(m_{0}−k_{0})}{p(R^{F})p(R^{E})^{−1}} − 𝔻_{k}(α) = dS

of currents on X.

Proof. Let π : G_{r}(E) → X be the fibre bundle whose fibre at x is the Grassmannian of unoriented r-planes in E_{x}, where

r = m − k = 2(m_{0} − k_{0}) = 2r_{0}.

Let U → G_{r}(E) be the tautological r-plane bundle, and let α̂ be the section of

H = Hom(U, π^{∗}F) ≅ U^{∗} ⊗ F

defined as in (2.4). By assumption α̂ is atomic. At this point our discussion breaks into the two cases above.

We begin with Case I. Here both the bundle Hom(U, π^{∗}F) and the maninfold G_{r}(E) are oriented. Using the approximation mode we can define the Euler family of smooth forms τ_{s}^{R} and the L_{loc}^{1}-forms ρ_{s}^{R} associated to α̂ as in Theorem 3.11. They satisfy the equation

(5.5)  τ_{s}^{R} − Div(α̂) = dρ_{s}^{R}

on G_{r}(E) for all 0 < s ≦ ∞. Applying π_{∗} to (5.5) gives the equation

(5.6)  TP_{s} − 𝔻_{k}(α) = dS_{s}

where TP_{s} =^{def} π_{∗}τ_{s}^{R} (called the Thom-Porteous family) and S_{s} =^{def} π_{∗}ρ_{s}^{R} are smooth and L_{loc}^{1} forms respectively on X. From 3.11 we know that σ^{R} = ρ_{∞}^{R} satisfies the equation

(5.7)  χ(R^{H}) − Div(α̂) = dσ^{R}

on G_{r}(E) where H = Hom(U, π^{∗}F), and where χ(R^{H}) = Pf (−\frac{1}{2π} R^{H}) is the Chern-Euler class of H. Applying π_{∗} in (5.7) gives the equation

(5.8)  π_{∗}χ(R^{H}) − 𝔻_{k}(α) = dS

on X, where S = π_{∗}σ.

It remains to compute π_{∗}χ(R^{H}). To begin we recall from Theorem A.17 in Appendix A the formula

(5.9)  χ(R^{H}) = Δ̃_{n_{0}}^{(m_{0}−k_{0})}{π^{∗}p(R^{F}) · p(R^{U})^{−1}}.

Transgressing between the pullback connection and its projection onto the splitting

π^{∗}E = U ⊕ U^{⊥}

gives the equation

(5.10)  p(R^{U})p(R^{U^{⊥}}) = π^{∗}p(R^{E}) + dη

which can be rewritten as

(5.11)  p(R^{U})^{−1} = π^{∗}p(R^{E})^{−1}p(R^{U^{⊥}}) + dη′.

Plugging this into (5.9) gives the formula

(5.12)  χ(R^{H}) = Δ̃_{n_{0}}^{(m_{0}−k_{0})} {π^{∗}(p(R^{F})p(R^{E})^{−1})p(R^{U^{⊥}})} + dη′′

of smooth forms on G_{r}(E). Arguing exactly as in the proof of Theorem 4.2, we now see that

(5.13) π_{∗}χ(R^{H}) = Δ̃_{n_{0}−k_{0}}^{(m_{0}−k_{0})} {p(R^{F})p(R^{E})^{−1}} π_{∗}(p_{k_{0}}(R^{U^{⊥}})^{m_{0}−k_{0}}) + dS′

where S′ = π_{∗}η′′ is a smooth form on X.

The term π_{∗}p_{k_{0}}(R^{U^{⊥}})^{m_{0}−k_{0}} is a d-closed 0-form, i.e., a constant. In Proposition B.5 of Appendix B it is proved that

(5.14)  π_{∗}p_{k_{0}}(U^{⊥})^{m_{0}−k_{0}} = 1,

and so this constant is 1. Replacing S_{s} by S_{s} − \frac{s}{1+s}S′ and TP_{s} by TP_{s} − \frac{s}{1+s}dS′ in equation (5.6) now completes the proof for Case I.

The argument for Case II is highly analogous. The main difference comes from the fact that while X is oriented, G_{r}(E) is not because the fibre of π is not orientable. Consequently, when passing through the proof, one must keep in mind the following points. Let O denote the orientation bundle for the manifold G_{r}(E). Tensoring the exterior powers of the cotangent bundle of G_{r}(E) by the real line bundle O yields bundles whose sections are called twisted differential forms.

A. Currents of dimension k on the d dimensional manifold G_{r}(E) include twisted forms of degree d − k which can be allowed to have L_{loc}^{1} coefficients. B. Twisted forms with L_{loc}^{1} coefficients can be integrated over the fibre and

this corresponds to current push forward π_{∗}. C. χ(R^{U}) is a twisted form on G_{r}(E).

The main calculational difference in the argument is that equation (5.9) must be replaced by

(5.15)  χ(R^{H}) = (−1)^{(n_{0}+1)r_{0}}χ(R^{U})Δ̃_{n_{0}}^{(m_{0}−k_{0})}{π^{∗}p(R^{F}) · p(R^{U})^{−1}}

which is proved in Theorem A.20 in Appendix A, and (5.14) is replaced by

(5.16)  π_{∗}{χ(U)p_{m_{0}}(U^{⊥})^{m_{0}−k_{0}}} = 1,

which is proved in Proposition B.11 of Appendix B. □

The formulas which appear in Theorem 5.2 were computed at the cohomology level by R. MacPherson in his Harvard University Thesis in 1970 (cf. [M1][M2]). of the dual bundles. If α^{∗} is k-atomic, the degeneracy current 𝔻_{k}(α^{∗}) is defined and Theorem 5.2 applies. In this case equation (5.4) becomes

(5.18)  Δ̃_{m_{0}−k_{0}}^{(n_{0}−k_{0})}{p(R^{E})p(R^{F})^{−1}} − 𝔻_{k}(α^{∗}) = dS.

From (4.2) we see that

Δ̃_{m_{0}−k_{0}}^{(n_{0}−k_{0})}{p(R^{E})p(R^{F})^{−1}} = (−1)^{d}Δ̃_{n_{0}−k_{0}}^{(m_{0}−k_{0})}{p(R^{F})p(R^{E})^{−1}}

where d = (m_{0} − k_{0})(n_{0} − k_{0}). It can be shown that supp 𝔻_{k}(α^{∗}) = supp 𝔻_{k}(α) and that for generic maps

(5.19)  𝔻_{k}(α^{∗}) = (−1)^{d}𝔻_{k}(α).

One conjectures that (5.19) holds for general k-atomic bundle maps.

Remark 5.20. A version of Theorem 5.2 holds for any pair of bundles E and F whenever k = 0, provided that E^{∗} ⊗ F = Hom(E, F) is oriented or, more generally, that w_{1}(E^{∗} ⊗ F) = w_{1}(X). In this version the characteristic form in equation (5.4) is just the Euler form of χ(E^{∗} ⊗ F) and 𝔻_{k}(α) is just the divisor of α considered as a section of Hom(E, F). This result is merely an instance of the general results discussed in §3.

§6. Characteristic forms and the degeneracies of k-frame fields.

Among the classical theorems in topology are those which relate the characteristic classes of a bundle to the singularities of fields of k-frames in the bundle. We shall derive such formulas, at the level of forms and currents, as direct consequences of the general theory. The results generalize the classical Chern formula c_{n}(F) = Div(α) + dT and our families (cf. §2) to all Chern and Pontryagin classes.

Let F → X be a smooth complex vector bundle of rank n with connection D^{F} over an oriented manifold X, and consider a set of k + 1 cross-sections α_{0}, …, α_{k} ∈ Γ(F) where k < n. This is equivalent to a bundle map

α : \underline{C}^{k+1} ⟶ F

from the trivial bundle \underline{C}^{k+1} given by setting α_{x}(t_{0}, …, t_{k}) = Σt_{i}α_{i}(x) for x ∈ X. We shall say that this frame field (α_{0}, …, α_{k}) has a good dependency locus if the map α is k-atomic. In this case we can define the linear dependency current

(6.1)  𝕃𝔻(α_{0}, …, α_{k}) =^{def} 𝔻_{k}(α).

Theorem 6.2. Let F → X be a complex vector bundle with connection over an oriented manifold, and suppose that α_{0}, …, α_{k} ∈ Γ(F) are k + 1 smooth sections with a good dependency locus. Then there exists an L_{loc}^{1}-form S on X such that

(6.3)  c_{n−k}(R^{F}) = 𝕃𝔻(α_{0}, …, α_{k}) + dS

where n = rank(F). Furthermore, there exist smooth families of smooth forms TP_{s} and L_{loc}^{1}-forms S_{s}, 0 < s ≦ ∞, with TP_{∞} = c_{n−k}(R^{F}) and S_{∞} = S, such that

(6.4)  TP_{s} = 𝕃𝔻(α_{0}, …, α_{k}) + dS_{s}

and lim_{s→0} S_{s} = 0 in L_{loc}^{1}.

Proof. We endow the trivialized bundle \underline{C}^{k+1} over X with the canonical metric and flat connection. We introduce on F a smooth hermitian metric (not necessarily related to D^{F}), and we choose an approximation mode χ. In terms of this data, Theorem 4.4 provides the families TP_{s} and S_{s} for the current 𝔻_{k}(α). Formula (4.6) translates directly into (6.3) above. □

Note. In algebraic geometry these linear dependency classes are used to define the Chern classes in the Chow ring of a smooth variety (cf. [Fu]).

Theorem 6.2 gives a direct proof of the basic fact that if α_{0}, …, α_{k} are linearly dependent on a small set, i.e., one of dimension < n − k, and if they vanish algebraically as in Theorem 1.11 (1), then [c_{n−k}] = 0 in H^{2(n−k)}(X).

Theorem 6.2 can be generalized to higher dependencies. Fix integers ℓ > 0 and k with 0 ≤ k < rank(F), and consider k + ℓ smooth cross-sections α_{1}, …, α_{k+ℓ} ∈ Γ(F). They give rise to bundle map

α : \underline{C}^{k+ℓ} ⟶ F

as above. We say that the frame field α = (α_{1}, …, α_{k+ℓ}) has a good ℓ-dependency locus if α is k-atomic. In this case we define the ℓ-dependency current

(6.5)  𝕃𝔻_{ℓ}(α) = 𝔻_{k}(α)

which measures in a rigorous way the set of points x where at least ℓ of the vectors α_{1}(x), …, α_{k+ℓ}(x) become linearly dependent on the remaining ones.

Theorem 6.6. Let F be as above and suppose α = (α_{1}, …, α_{k+ℓ}) are k+ℓ sections of F with a good ℓ-dependency locus. Then there is an L_{loc}^{1}-form S on X such that

(6.7)  det_{ℓ×ℓ}((c_{n−k−i+j}(R^{F}))) = 𝕃𝔻_{ℓ}(α) + dS.

Furthermore there exist families TP_{s} and S_{s}, 0 < s ≦ ∞, with properties analogous to those in Theorem 6.2.

Proof. One applies Theorem 4.4 as in the proof above. □

Of course one retrieves Theorem 6.2 from 6.6 as the special case where ℓ = 1. At the other extreme we can take k = 0. Note that 𝕃𝔻_{ℓ}(α_{1}, …, α_{ℓ}) measures the simultaneous vanishing of ℓ generic sections of F. Here we get the predictable formula

(6.8)  c_{n}(R^{F})^{ℓ} = 𝕃𝔻_{ℓ}(α_{1}, …, α_{ℓ}) + dS.

There are corresponding theorems in the real case. Let F → X be a smooth real vector bundle of rank n with orthogonal connection D^{F}. Consider smooth cross-sections α_{1}, …, α_{k+ℓ} ∈ Γ(F) where 0 ≦ k < n and ℓ is even. We say that α = (α_{1}, …, α_{k+ℓ}) has a good ℓ-dependency locus if the bundle map

α : \underline{R}^{k+ℓ} ⟶ F

given by α_{x}(t_{1}, …, t_{k+ℓ}) = Σt_{i}α_{i}(x) is k-atomic. In this case we define the ℓdependency current

𝕃𝔻_{ℓ}(α) = 𝔻_{k}(α).

Since ℓ = 2ℓ_{0} is even, there are two cases to consider:

Case I. n = 2n_{0} and k = 2k_{0} Case II. n = 2n_{0} + 1 and k = 2k_{0} + 1

for integers n_{0}, k_{0} ≧ 0. In Case II we assume F to be orientable.

Theorem 6.9. Let F and α be as above (either Case I or Case II) and suppose that α has a good ℓ-dependency locus. Then there is an L_{loc}^{1}-form S on X such that

(6.10)  det_{ℓ_{0}×ℓ_{0}}((p_{n_{0}−k_{0}−i+j}(R^{F}))) = 𝕃𝔻_{ℓ}(α) + dS.

Furthermore there exist families TP_{s} and S_{s}, 0 < s ≤ ∞, with properties analogous to those in Theorem 6.2.

Proof. Introduce the canonical flat orthogonal connection on the trivialized bundle \underline{R}^{k+ℓ} and apply Theorem 5.2. □

Setting ℓ_{0} = 1 gives the following analogue of the formula in Theorem 6.2:

(6.11)  p_{n_{0}−k_{0}}(R^{F}) = 𝕃𝔻_{2}(α_{1}, …, α_{k+2}) + dS

where k = 2k_{0} or 2k_{0}+1 depending on which of the Cases I or II we are considering. If we set k_{0} = 0, we obtain

(6.12)  p_{n_{0}}(R^{F})^{ℓ_{0}} = 𝕃𝔻_{2ℓ_{0}}(α_{1}, …, α_{2ℓ_{0}}) + dS

where 𝕃𝔻_{2ℓ_{0}}(α_{1}, …, α_{2ℓ_{0}}) is measuring the simultaneous vanishing of the 2ℓ_{0} sections.

Theorem 6.9 has some interesting applications in the next section.

§7. Singularities of projections.

Let X be a smooth oriented m-manifold with an immersion

j : X ↬ R^{N}

into euclidean N-space. Fix an integer n < N and consider the set of linear maps from R^{N} to R^{n}. Each P ∈ Hom(R^{N}, R^{n}) restricts to give a smooth mapping

P̂ = P ∘ j : X ⟶ R^{n}.

We are interested in studying the singularities of these projections. In particular for fixed k < min{m, n} and generic P we want to understand the locus where the differential

dP̂ : T X ⟶ \underline{R}^{n}

has rank ≦ k.

The projection P is called k-atomic on X if the bundle map dP̂ is k-atomic, and under this hypotheses one can define the k^{th} degeneracy current of the projection P on X to be

𝔻_{k}(P) =^{def} 𝔻_{k}(dP̂).

Theorem 7.1. Let j : X ↬ R^{N} be a C^{∞} immersion of a smooth oriented mmanifold into euclidean space. Fix integers k and n with k < min{m, n} and with k ≡ m ≡ n (mod 2). Then for almost all P ∈ Hom(R^{N}, R^{n}) the projection P is katomic on X and the following holds. There is a canonical L_{loc}^{1}-form S = S(P, k, n) such that

(7.2)  det_{ℓ_{0}×ℓ_{0}}((p_{m_{0}−k_{0}−i+j}(X))) = 𝔻_{k}(P) + dS

where m_{0} = [m/2], n_{0} = [n/2], k_{0} = [k/2], ℓ_{0} = n_{0} − k_{0}, and where

p(R^{X}) = 1 + p_{1}(R^{X}) + p_{2}(R^{X}) + ⋯

is the total Pontrjagin form of X for its induced riemannian connection. Furthermore, there are smooth families of smooth forms TP_{s} and L_{loc}^{1}-forms S_{s}, 0 < s ≦ ∞, with TP_{∞} = Δ̃_{m_{0}−k_{0}}^{(n_{0}−k_{0})}{p(R^{X})} and S_{∞} = S, such that

(7.3)  TP_{s} = 𝔻_{k}(P) + dS_{s}

and lim_{s→0} S_{s} = 0 in L_{loc}^{1}.

In particular, if n_{0} = k_{0} + 1 then there is a canonical cohomology

(7.4)  p_{m_{0}−k_{0}}(R^{X}) = 𝔻_{k}(P) + dS

between the (m_{0} − k_{0})^{th} Pontrjagin form and the current which measures where the differential of the projection P̂ : X → R^{k+2} drops rank by 2.

Proof. To prove that P is k-atomic for almost all P we mimick the proof of Proposition 2.10. Set W = Hom(R^{N}, R^{n}) and let π : G_{r}(\underline{R}^{n}) → X be the bundle projection and \underline{U} → G_{r}(\underline{R}^{n}) the tautological r-plane bundle, where r = n − k. Then there is a C^{∞} map

(7.5)  W × G_{r}(\underline{R}^{n}) = W × G_{r}(R^{n}) × X ⟶^{Ψ} Hom(\underline{U}, π^{∗}T X)

(into the total space of the bundle Hom(\underline{U}, π^{∗}T X) over G_{r}(\underline{R}^{n})) given by

Ψ(P, U, x) = P̃^{∗} |_{U} where P̃ = P |_{T_{x}X} .

We claim that Ψ is transversal to the zero section of this bundle. To see this suppose that Ψ(P, U, x) = 0 and let Q_{0} ∈ Hom(U, T_{x}X) be given. Extend Q_{0} to Q ∈ Hom(R^{n}, R^{N}) by defining Q to be zero on U^{⊥}. Consider the curve

P_{t} = P + tQ^{∗} in W.

Then

\frac{d}{dt}Ψ(P_{t}, U, x) |_{t=0} = (Q^{∗} |_{T_{x}X})^{∗} |_{U} = Q |_{U} = Q_{0}.

Hence dΨ, restricted to W ≅ T_{p}W ⊂ T_{(P,U,x)} (W × G_{r}(R^{n}) × X), maps onto the fibre of Hom(\underline{U}, π^{∗}T X) and is therefore transversal to the zero-section as claimed.

It now follows from Sard’s Theorem for Families ([HL_{1}]) that for almost all P ∈ W the restriction of Ψ to \{P \} × G_{r}(\underline{R}^{m}) is transversal to the zero section of Hom(\underline{U}, π^{∗}T X). All such P are k-atomic. This proves the first assertion of the theorem. The remaining assertions are straightforward consequences of Theorem 6.9 applied to dP̂ : T X ⟶ \underline{R}^{n} where T X carries the riemannian connection for the metric induced by j and where \underline{R}^{n} carries the canonical flat orthogonal connection. □

§8. Singularities of maps.

We shall now considerably generalize the results of the last section. Let X and Y be smooth riemannian manifolds of dimensions m and n respectively (where Y need not be orientable) and consider a smooth mapping

f : X ⟶ Y.

Fix an integer k < min{m, n} and assume that m ≡ n ≡ k (mod 2). Then the map f is said to be k-atomic if the differential

df : T X ⟶ f^{∗}T Y

is k-atomic. Arguing as in sections 2 and 7, one can show that generic smooth maps have this property. Whenever f is k- atomic, we can define its k^{th} degeneracy current 𝔻_{k}(f) =^{def} 𝔻_{k}(df).

Theorem 8.1. Let f : X → Y, m, n and k be as above, and suppose that f is k-atomic. Set m_{0} = [m/2], n_{0} = [n/2] and k_{0} = [k/2]. Then there is a canonical L_{loc}^{1}-form S on X such that

(8.2)  Δ̃_{n_{0}−k_{0}}^{(m_{0}−k_{0})} {f^{∗}p(R^{Y})/p(R^{X})} = 𝔻_{k}(f) + dS

where p(R^{X}) = 1 + p_{1}(R^{X}) + p_{2}(R^{X}) + ⋯ and p(R^{Y}) = 1 + p_{1}(R^{Y}) + p_{2}(R^{Y}) + ⋯ are the total Pontrjagin forms in the riemannian curvatures of X and Y. Furthermore there are smooth families of smooth forms TP_{s} and L_{loc}^{1}-forms S_{s}, 0 < s ≦ ∞, with TP_{∞} = Δ_{n_{0}−k_{0}}^{(m_{0}−k_{0})}{f^{∗}p(R^{Y})/p(R^{X})} and S_{∞} = S, such that

TP_{s} = 𝔻_{k}(f) + dS_{s}

and lim_{s→0} S_{s} = 0 in L_{loc}^{1}.

Proof. This is a straightforward application of Theorem 5.2. □

An interesting special case occurs when dim X = dim Y = 4 and k = 2. Here (8.2) has the form

(8.3)  f^{∗}p_{1}(Y) − p_{1}(X) = Σn_{i}x_{i} + dS

where 𝔻_{2}(f) = Σn_{i}x_{i} is a discrete sum of points with integer coefficients. Let N_{f} = Σn_{j} denote the total 2-degeneracy number of f.

This yields a 4-dimensional analogue of the classical Riemann-Hurwitz Theorem [M1], [M2], [R].

Corollary 8.4. Suppose that f : X → Y is a smooth map between compact oriented 4-manifolds with isolated points of 2-degeneracy. Then

M_{f}p_{Y} − p_{X} = N_{f}

where p_{X} and p_{Y} are the first Pontrjagin numbers of the manifolds X and Y respectively, M_{f} is the degree of the map f, and N_{f} is the total 2-degeneracy number of f.

In his thesis [S] Robert Stingley gives reinterpretations of N_{f} and methods of computing it in terms of the local geometry of the map f.

§9. Milnor currents.

There is a complete analogue of the formulas of §8 for the complex case. Such results are not new; they can be deduced from work by Fulton, Bismut, Gillet and Soulé. However we add them here for interest, and remark that the holomorphic assumptions here can be considerably relaxed to statements about smooth almost complex manifolds.

Let X and Y be complex manifolds of dimensions m and n respectively, and consider a holomorphic map

f : X ⟶ Y.

Introduce complex connections and hermitian metrics on X and Y. Fix k < min{m, n}. We say that f is k-atomic if

codim_{C}{x ∈ X : rank(df_{x}) ≦ k} ≧ (m − k)(n − k).

By 1.10 and 2.7 this hypothesis implies the existence of the k^{th} degeneracy cur-

rent 𝔻_{k}(f) =^{def} 𝔻_{k}(df).

Theorem 9.1. Let f : X → Y be a holomorphic k-atomic map as above. Then there exists a canonical L_{loc}^{1}-form S on X such that

(9.2)  Δ_{n−k}^{(m−k)}{f^{∗}c(Y)/c(X)} = 𝔻_{k}(f) + dS

where d = (m − k)(n − k) and where c(X) = det(1 + (i/2π)R^{X}) and c(Y) = det(1 + (i/2π)R^{Y}) are the total Chern forms of X and Y in their given connections. Furthermore, there are smooth families TP_{s}, τ_{s}, 0 < s ≦ ∞ with properties analogous to those in 8.1.

Proof. This is a direct consequence of Theorem 4.4 and (4.2). □

Remark. When X and Y are given the canonical hermitian connections associated to the metrics, the term dR in formula (9.2) can be replaced by ∂ ¯∂T as noted in Remark 4.18.

An interesting case occurs when X and Y are compact, dim Y = n = 1, and k = 0. Then f is a map to a complex curve, and it will be 0-atomic iff it has isolated singular points, say x_{1}, …, x_{ℓ} ∈ X. We define the Milnor current to be

𝕄(f) =^{def} 𝔻_{k}(f) = ∑_{i=1}^{ℓ} m_{i}[x_{i}]

where the integers m_{i} are the local Milnor numbers

m_{i} = dim \{O_{x_{i}}/ ⟨\frac{∂f}{∂t_{1}} , …, \frac{∂f}{∂t_{m}} ⟩\} .

Equation (9.2) has the form

(9.3)  (−1)^{m}{c_{m}(R^{X}) − c_{m−1}(R^{X})f^{∗}c_{1}(R^{Y})} = 𝕄(f) + dS.

When n = 1 this equation integrates to give the classical Riemann-Hurwitz Formula.

Another interesting case arises when X ⊂ P^{N} is a projective m-manifold and f : X → P^{m} is given by linear projection P^{N} − P^{N−m−1} → P^{m}, where X ∩ P^{N−m−1} = ∅ (Noether normalization). Let ω be the Kähler form of the metric induced on X. Then (9.2) implies formulas of the following type. If f is (m − k)- atomic for k^{2} ≦ m, then

(9.4)  Δ_{k}^{(k)}{c(X)(1 + ω)^{−(m+1)}} = (−1)^{k^{2}}𝔻_{m−k}(f) + dS_{k}

for an L_{loc}^{1}-form S_{k}. Every such f is (m − 1)-atomic, and 𝔻_{m−1}(f) = 𝔹(f) is the branching divisor of f. This gives the formula:

(9.5)  (m + 1)ω − c_{1}(X) = 𝔹(f) − dS.

For example if X is a curve of degree d and genus g, then (9.5) implies that 2(d + g − 1) = |𝔹| = the total order of branching of f.

There are of course many many such formulas coming from methods of enumerative geometry.

§10. CR-singularities.

The methods introduced above yield some interesting results in CR geometry. Consider an immersion

f : X ↬ Z

of a real manifold X into a complex manifold Z, where

m = dim_{R}(X) ≥ n = dim_{C}(Z).

Then the differential df : T X → f^{∗}T Z extends to a complex bundle map

(10.1)  df_{C} : T X ⊗_{R} C ⟶ f^{∗}T Z.

Definition 10.2. Assume that the bundle map df_{C} is k-atomic where 0 ≤ k < n, and let r = n − k. Then we define the r^{th} complex tangency current of f to be

ℂr_{r}(f) = 𝔻_{k}(df_{C}).

Roughly speaking this current corresponds to the locus of points x where f_{∗}T_{x}X contains a complex subspace having r “excess” dimensions, i.e., more complex tangency (by r) than expected. Specifically we have:

Lemma 10.3. The support of ℂr_{r}(f) satisfies

supp ℂr_{r}(f) ⫅ {x ∈ X : dim_{C}(T_{x}X ∩ JT_{x}X) ≧ m − n + r}

where J denotes the almost complex structure of Z (and where for notational convenience we have identified T_{x}X with f_{∗}T_{x}X).

Proof. By (2.7) we have supp ℂr_{r}(f) ⊆ {x ∈ X : rank(df_{C}) ≦ k}. Note that at x ∈ X,

ker(df_{C}) = {V + iW : V, W ∈ T_{x}X and V + JW = 0}
= {V + iJV : V, JV ∈ T_{x}X}
= [(T_{x}X ∩ JT_{x}X) ⊗ C]^{0,1} .

Since rank(df_{C}) = m − dim_{C} ker(df_{C}) we have rank(df_{C}) ≦ k iff dim_{C}(T_{x}X ∩ JT_{x}X) ≧ m − k = m − n + r. □

We now suppose that X carries a riemannian metric and that Z carries a hermitian metric and a complex connection. Define c(Z) as in (9.1) and set

p̃(X) = 1 − p_{1}(R^{X}) + p_{2}(R^{X}) − ⋯ + (−1)^{[m/4]}p_{[m/4]}(R^{X})

and

c̃(Z) = 1 − c_{1}(R^{Z}) + c_{2}(R^{Z}) − ⋯ + (−1)^{n}c_{n}(R^{Z})

where p_{i}(R^{X}) and c_{i}(R^{Z}) are the i^{th} Pontrjagin and Chern forms of X and Z respectively.

Theorem 10.4. Let f : X → Z be an immersion of a real m-manifold into a complex n-manifold with n ≦ m as above, and assume that df_{C} is (n − r)-atomic where 0 < r ≦ n. Then there is a canonical L_{loc}^{1} form S on X such that

(10.5)  Δ_{m−n+r}^{(r)}{p̃(X)/f^{∗}c̃(Z)} = Δ_{r}^{(m−n+r)}{f^{∗}c(Z)/p̃(X)}
= ℂr_{r}(f) + dR.

Furthermore there are smooth families TP_{s}, S_{s} 0 < s ≦ ∞ as in Theorems 6.2, 8.1, 9.1 etc.

Note 10.6. The term on the left is computed by writing p̃(X)/f^{∗}c̃(Z) = 1 + ξ_{1} + ξ_{2} + ⋯, where ξ_{i} is a 2i-form, and then applying formula (4.1).

Note 10.7. In Theorem 10.4 it suffices that Z be an almost complex manifold. Furthermore f need not be an immersion; but in this case Lemma 10.3 does not apply.

Proof of Theorem 10.4. Apply Theorem 4.4 to the bundle map df_{C} and note that

c(R^{T^{∗}X⊗C}) = c(R^{T X⊗C}) = p̃(X).

This establishes the second equality in (10.5). The first equality is a direct consequence of the Shur relations (4.2) and (4.3). □

Consider for example the case where Z = C^{n} and r = 1. This gives the following.

Corollary 10.8. Let f : X ↬ C^{k+1} be an immersion of a smooth m-manifold with the property that df_{C} is k- atomic. Suppose m − k = 2ℓ > 0. Then there is an L_{loc}^{1}-form S on X with

ℂr_{1}(f) = (−1)^{ℓ}p_{ℓ}(X) + dS.

Example 10.9. Suppose X^{4} is a smooth 4-manifold and

f : X^{4} ↬ C^{3}

an immersion as above. Then we have that

p_{1}(X) = Σn_{i}x_{i} + dS

where x_{1}, …, x_{N} are the points of complex tangency of the immersion and where n_{1}, …, n_{N} are integers computed from the local CR geometry of f.

Example 10.10. Consider the case where m = n. Let

f : X^{n} ↬ Z^{n}

be an immersion of a real n-manifold into a complex n-manifold. If df_{C} is (n − r)- atomic, we have

ℂr_{r}(f) = Δ_{r}^{(r)}{p̃(X)/c̃(Z)} + dS

(where c̃(Z) is assumed to be pulled back via f to X). When r = 1 we have

ℂr_{1}(f) = c_{1}(Z) + dS.

When r = 2, a calculation yields

ℂr_{2}(f) = Δ_{2}^{(2)}c(Z) − p_{1}(Z)p_{1}(X) + p_{1}(X)^{2} + dS.

Example 10.11. Consider an immersion

f : S^{m} ↬ Z

and give S^{m} the standard Riemannian connection for which p(S^{m}) ≡ 1. Then if df_{C} is (n − r)-atomic we have

ℂr_{r}(f) = Δ_{r}^{(m−n+r)}{c(Z)} + dS.

One case of interest is where m = n = 2r^{2}. Then we have

ℂr_{r}(f) = | c_{r}(Z) c_{r+1}(Z) ⋯ c_{2r}(Z) ; c_{r−1}(Z) c_{r}(Z) ; ⋮ ; c_{1}(Z) c_{r}(Z) | + dS.

There is a counterpart to all of the discussion above for the case where m ≦ n.

Theorem 10.12. Let f : X ↬ Z be an immersion of a real m-manifold into a complex n-manifold where m ≦ n. Assume that the map df_{C} of (10.1) is (m − r)- atomic. For given connections and metrics on X and Z as above, there is a canonical L_{loc}^{1}-form S such that

(10.13)  ℂr_{r}(f) =^{def} 𝔻_{m−r}(df_{C}) = Δ_{(n−m+r)}^{(r)}{c(Z)/p̃(X)} − dS

with approximating families TP_{s} and S_{s} as in (10.4).

The proof of Lemma 10.3 shows that in this case

supp ℂr_{r}(f) ⊆ {x ∈ X : dim_{C}(T_{x}X ∩ JT_{x}X) ≧ r}.

Remark 10.14. Note that if f : X ↬ Z is a totally real immersion, then Theorem 10.12 implies that

(10.15)  Δ_{n−m+r}^{(r)}{c(Z)/p̃(X)} = 0 in H^{∗}(X)

for all r > 0. However in this case T Z |_{X}= (T X ⊕ ℂ) ⊕ ν where ν is a complex bundle of dimension n − m. Hence, on X we have

c(Z) = c(T X ⊕ ℂ)c(ν) = p̃(X)c(ν)

and so

c(Z)/p̃(X) = c(ν) = 1 + c_{1}(ν) + ⋯ + c_{n−m}(ν)

and (10.15) follows trivially.

On the other hand if f is Lagrangian (in the sense that f_{∗}T_{x}X)⊥J(f_{∗}T_{x}X) for all x), then for a natural choice of connections on X and Z one has Δ_{n−m+r}^{(r)}{c(Z)/p̃(X)} ≡ 0 on X, and via (10.15) a secondary invariant [T] ∈ H^{odd}(X; R) is defined.

Some of the results in this section are related to work of Lai [Lai], Webster [W1,2,3], and Wolfson [Wo].

§11. Invariants for pairs of complex structures.

In this section we shall introduce characteristic invariants which measure the relative singularities of a pair of complex structures. Consider a real C^{∞} vector bundle E → X of rank 2n, and suppose that J_{1} and J_{2} are smooth almost complex structures on E. These structures induce decompositions

(11.1)  E ⊗_{R} C = E_{1}^{1,0} ⊕ E_{1}^{0,1} = E_{2}^{1,0} ⊕ E_{2}^{0,1}

into the ±i eigenspaces of J_{1} and J_{2} respectively. (E_{k}^{1,0} is the +i eigenspace.) Consider the bundle map α given by the composition

E_{1}^{1,0} ↪ E ⊗_{R} C ↠ E_{2}^{0,1}

Lemma 11.2. At each x ∈ X, there is an isomorphism

ker α_{x} ≅ {V ∈ E_{x} : J_{1}V = J_{2}V }
= the maximal subspace of E_{x} which is simultaneously
J_{1} and J_{2} complex.

Proof. Note that E_{1}^{1,0} = {V − iJ_{1}V : V ∈ E_{x}}. Now for V ∈ E_{x}, we have

v =^{def} V − iJ_{1}V ∈ ker α ⟺ π_{2}^{1,0}(v) = 0
⟺ v + iJ_{2}v = V − iJ_{1}V + iJ_{2}(V − iJ_{1}V) = 0
⟺ J_{2}V = J_{1}V. □

Definition 11.3. Fix r, 1 ≤ r ≤ n. The structures J_{1}, J_{2} will be called rtransversal if the bundle map α is (n − r)-atomic. Under this hypothesis we define the r^{th} coincidence current of the pair J_{1}, J_{2} to be

ℚ_{r}(J_{1}, J_{2}) = 𝔻_{n−r}(α).

From 11.2 and 2.7 we have that

supp ℚ_{r}(J_{1}, J_{2}) ⫅ {x ∈ X : J_{1} = J_{2} on a subspace W ⊂ E_{x} with dim W ≧ r}.

Theorem 11.4. Suppose J_{1} and J_{2} are r- transversal. Then given complex connections on E_{1} ≡ (E, J_{1}) and E_{2} ≡ (E, J_{2}), there exists an L_{loc}^{1}-form S such that

ℚ_{r}(J_{1}, J_{2}) = Δ_{r}^{(r)}c(E_{2} − E_{1}) + dS

where c(E_{2} − E_{1}) = c(R^{E_{2}})c(R^{E_{1}})^{−1}. Furthermore, there are smooth families TP_{s} and S_{s}, 0 < s ≦ ∞ as in 6.2, 8.1, 9.1.

Proof. Apply Theorem 4.4. □

Example 11.5. Fix E_{1} = X × C^{n} = (X × R^{2n}, J_{0}). The product bundle with the canonical flat connection, and consider E_{2} = (X × R^{2n}, J) where J is any other almost complex structure on this trivialized R^{2n}-bundle. For those J which are r-transversal to the given flat structure we have

Q_{r}(J) =^{def} ℚ_{r}(J_{0}, J) = Δ_{r}^{(r)}c(R^{J}) + dS_{r}

where c(R^{J}) are the Chern forms of a J-complex connection. For example,

Q_{1}(J) = c_{1}(R^{J}) + dR_{1}
Q_{2}(J) = det ( \begin{matrix}c_{2} & c_{3} \\ c_{1} & c_{2} \end{matrix} ) + dR_{2}
= c_{2}^{2} − c_{1}c_{3} + dT_{2}
Q_{3}(J) = det ( \begin{matrix}c_{3} & c_{4} & c_{5} \\ c_{2} & c_{3} & c_{4} \\ c_{1} & c_{2} & c_{3} \end{matrix} ) + dR_{3}.

Note. There are cohomology relations

c(E)c(\overline{E}) = 1 and c_{n}(E) = 0

on any complex bundle which is trivial as a real bundle. Nevertheless, E may still be non-trivial as a complex bundle. Consider for example the complex line bundle L → S^{1} × S^{1} of Chern class 2, and set E = L ⊕ C. Then

c_{1}(E) = c_{1}(L) = 2

and so E is non-trivial. However E is trivial as an R^{4}-bundle. To see this note first that the classifying map S^{1} × S^{1} ⟶ BU_{2} ⟶ BSO_{4} can be lifted to a map S^{1} × S^{1} ⟶ BSpin_{4} since the only obstruction to this lifting is w_{2}(E) = c_{1}(E) (mod 2) = 0. However, BSpin_{4} is 3-connected, so this lift, and therefore also the map to BSO_{4}, are contractible.

Example 11.6. (Diffeomorphisms) Let f : X → X be a diffeomorphism of an almost complex manifold with almost complex structure J. For generic f, the structures J and f^{∗}J will be r-transversal for all r, and so given a J-compatible connection ∇ on X, we can take the f^{∗}J-compatible connection f^{∗}∇ and obtain L_{loc}^{1}-forms S_{r} with

ℚ_{r,J}(f) =^{def} ℚ_{r}(J, f^{∗}J) = Δ_{r}^{(r)}{f^{∗}c(R^{J})/c(R^{J})} + dS_{r}.

§12. Invariants for plane fields and foliations.

This discussion given in §10 for mappings can be applied to plane fields, and in particular to foliations. Even more generally suppose F → X is a smooth complex vector bundle of complex dimension n, and consider a real sub-bundle

j : P ↪ F

of (real) dimension m. Then there is a natural extension of j to a complex bundle map

(12.1)  j_{C} : P ⊗_{R} C ⟶ F.

Fix an integer r, with 0 < r ≦ min{m, n}, and set k = min{m, n} − r. We say that P has good complex r-tangencies if j_{C} is k-atomic. Under this hypotheses the complex tangency current

(12.2)  ℂr_{r}(P) =^{def} 𝔻_{k}(j_{C})

is defined. In analogy with 10.3 and (10.12) we have

supp ℂr_{r}(P) ⊆ {x ∈ X : dim_{C}(P_{x} ∩ J_{x}P_{x}) ≧ r + max{0, m − n}}.

This is the subset of X where the dimension of the maximal complex subspace of P is greater by at least r than the “expected” or “generic” dimension.

Theorem 12.3. Let j : P ↪ F be a real m-dimensional subbundle with good complex r-tangencies in a complex n-dimensional bundle F. Let P be equipped with a real connection and metric, and let F be given a complex connection and hermitian metric. (No relation among the four is assumed.)

Then there is a canonically defined L_{loc}^{1}-form S on X such that

(12.4)  ℂr_{r}(P) = \{ \begin{matrix}Δ_{m−n+r}^{(r)}\{p̃(P)/c̃(F)\} + dS, when m ≧ n \\ Δ_{n−m+r}^{(r)}\{c(F)/p̃(P)\} + dS, when m ≦ n \end{matrix}

where c(F), c̃(F) and p̃(P) are defined as in §10. Furthermore there are approximating families Σ_{s}, S_{s}, 0 < s ≦ ∞, as in previous theorems.

Note. The equations in (12.4) can be rewritten by using the elementary Shur relations (4.2) and (4.3). In particular, with k = min{m, n} − r, we have

Δ_{n−k}^{(m−k)}{c(F)/p̃(P)} = Δ_{m−k}^{(n−k)}{p̃(P)/c̃(F)}.

Example 12.5. (Plane fields and foliations). A case of geometric interest occurs when

j : P ↪ T Z

is a real m-plane field on an almost complex manifold Z. This arises for example when P is the tangent plane field T F of an m-dimensional foliation F of Z. In this case the currents ℂr_{r}(F) ≡ ℂr_{r}(T F) correspond to the excess complex tangencies of the foliation. The formulas in (12.4) give cohomological obstructions to finding an isotopy of F to a foliation without complex tangencies of dimension r +max{0, m− n}.

§13. Higher self-intersections of plane fields and invariants for pairs of foliations.

Let F → X be a smooth bundle of rank n and consider two subbundles

A ↪^{i} F and B ↪^{j} F

of ranks a and b respectively. We assume all bundles to be simultaneous real or simultaneously complex. From this data we get a bundle map

A ⟶^{α} F/B

given by restricting the projection π : F → F/B to A. Note that

ker α_{x} = A_{x} ∩ B_{x}

whose “expected” or “generic” dimension is

e = max{a + b − n, 0}.

Fix integers r > 0 and k ≥ 0 with e+r = a−k, or equivalently r+k = min{a, n−b}. Note that rank(α_{x}) = a − dim(A_{x} ∩ B_{x}) ≤ k if and only if dim(A_{x} ∩ B_{x}) ≥ a − k = e + r. We say that A and B make good r-contact if α is k-atomic. Under this assumption we define the r-contact current

ℂt_{r}(A, B) = 𝔻_{k}(α)

and note that

supp ℂt_{r}(A, B) ⊆ {x ∈ X : dim(A_{x} ∩ B_{x}) ≧ e + r}.

This current measures the contact degeneracies of rank r, i.e., the set where A meets B in at least r dimensions more than expected. Setting e^{∗} = max{n − (a + b), 0} and applying §§4 and 5 give the following.

Theorem 13.1. Suppose A, B and F are complex and that F is provided with a complex connection and metric. If A and B make good r-contact, then there is a canonical L_{loc}^{1}-form T such that

ℂt_{r}(A, B) = Δ_{e^{∗}+r}^{(e+r)}{c(R^{F})/c(R^{A})c(R^{B})} + dS.

Theorem 13.2. Suppose A, B and F are real bundles of rank 2a_{0}, 2b_{0} and 2n_{0} respectively and let e = 2e_{0}, e^{∗} = 2e_{0}^{∗}. Suppose F is provided with an orthogonal connection. Then if A and B make good r-contact where r = 2r_{0} is even, there is a canonical L_{loc}^{1}-form S such that

ℂt_{r}(A, B) = Δ̃_{e_{0}^{∗}+r_{0}}^{(e_{0}+r_{0})}{p(R^{F})/p(R^{A})p(R^{B})} + dS.

Remark 13.3. One could equivalently formulate the problem by considering the adjoint map α^{∗} : (F/B)^{∗} → A^{∗}. Note that at x ∈ X, we have

ker(α_{x}^{∗}) = A_{x}^{⊥} ∩ B_{x}^{⊥}

whose expected dimension is

e^{∗} = max{n − (a + b), 0}.

Applying 4.4 gives us “dual” formulas which are equivalent to those of 13.1 and 13.2. To see this, observe that by the Shur relations (4.3) and (4.2) we have

Δ_{e^{∗}+r}^{(e+r)}{c(R^{F})/c(R^{A})c(R^{B})} = Δ_{e+r}^{(e^{∗}+r)}{c(R^{A^{∗}})c(R^{B^{∗}})/c(R^{F^{∗}})}

and

Δ̃_{e_{0}^{∗}+r_{0}}^{(e_{0}+r_{0})}{p(R^{F})/p(R^{A})p(R^{B})} = Δ̃_{e_{0}+r_{0}}^{(e_{0}^{∗}+r_{0})}{p(R^{A})p(R^{B})/p(R^{F})}.

Example 13.4. (Higher self intersection classes). Given A ⊂ F as above one can take B to be a generic displacement of A (which will be r-atomic for all relevant r) and compute the “higher self intersections” of the plane field. Consider for example the complex line field λ ⊂ T P^{3} on complex projective 3-space which is tangent to the fibres of the twistor map P^{3} → S^{4}. Now we have c(λ) = 1 + 2ω where ω ∈ H^{2}(P^{3}, Z) ≅ Z is the canonical generator. The self-intersection class coming from taking A = B = λ, F = T P^{3} and r = 1 in 13.1 is

Δ_{2}^{(1)}{c(T P^{3})/c(λ)^{2}} = Δ_{2}^{(1)}{(1 + ω)^{4}/(1 + 2ω)^{2}} = 2ω^{2}.

This implies that a generic deformation of λ (as a smooth complex subbundle) in T P^{3} will coincide with λ along a cycle in P^{3} which represents the homology class 2[P^{1}] ∈ H_{2}(P^{3} ; Z) ≅ Z.

There are analogous complex line fields λ on every P_{C}^{2n+1} tangent to the fibres of the bundle π : P_{C}^{2n+1} → P_{H}^{n}. Here the generic self-intersection locus satisfies

ℂt_{1}(λ) = Δ_{2n}^{(1)}{(1 + ω)^{2n+2}/(1 + 2ω)^{2}} + dS = (n + 1)ω^{2n} + dS.

Example 13.5. (Relative foliation cycles). Suppose we are given two foliations F and F′ of dimensions 2a_{0} and 2b_{0} respectively in a Riemannian manifold of dimension 2n_{0}. Then we can apply 13.2 to A = T F and B = T F′ to produce canonical cohomologies between the current of (e + r)-dimensional contact points of F and F′, i.e., between

ℂt_{r}(F, F′) =^{def} ℂt_{r}(T F, T F′),

and the characteristic forms defined in 13.2.

APPENDIX A Orientations and tensor products

In this appendix we discuss questions of orientations and characteristic classes of tensor product bundles. We begin with some elementary definitions. Let V → Y be a real vector bundle of rank n over a manifold Y. By definition there is an open covering U = {U_{α}}_{α∈A} of Y and local trivializations

f_{α} : V |_{U_{α}} ⟶^{≈} U_{α} × R^{n}.

The compositions give transition functions

g_{αβ} : U_{α} ∩ U_{β} ⟶ GL_{n}(R)

by setting g_{αβ}(x) = f_{α} ∘ f_{β}^{−1}(x, ·). They have the property that g_{αβ}(x) = g_{βα}(x)^{−1} and satisfy the cocycle condition

(A.1)  g_{αβ}g_{βγ}g_{γα} ≡ 1  on U_{α} ∩ U_{β} ∩ U_{γ}.

In terms of this data we define a Ĉech 1-cocycle {w_{αβ}} on the cover U by setting

w_{αβ} =^{def} sgn{det(g_{αβ})} : U_{α} ∩ U_{β} ⟶ {1, −1} ≅ Z_{2}.

From (A.1) we see that w_{αβ} satisfies the cocycle condition

(A.2)  w_{αβ}w_{βγ}w_{γα} ≡ 1  in U_{α} ∩ U_{β} ∩ U_{γ}

and therefore determines a class

w_{1}(V) ∈ H^{1}(Y ; Z_{2})

which can be shown to be independent of the choice of local trivializations for V. It is called the first Stiefel-Whitney class of V. Note that the cocycle w_{αβ}, considered as transition functions on U, determines a two-fold covering (i.e., a Z_{2}- bundle) over Y, called the orientation bundle Or(V) of V. One can naturally identity the fibre of Or(V) at x with the two possible orientations of V_{x}. If Y is a connected manifold then Cov_{2}(Y) ≅ Hom(π_{1}Y, Z_{2}) ≅ H^{1}(Y; Z_{2}) (where Cov_{2}(Y) denotes the equivalence classes of 2-fold coverings of Y). This gives an equivalent definition of w_{1}(V) and shows that

(A.3)  V is orientable ⟺ w_{1}(V) = 0.

Proposition A.4. Let U → Y and F → Y be real vector bundles with rank(U) = m and rank(F) = n, and set H = Hom(U, F) = U^{∗} ⊗ F. Then

(A.5)  w_{1}(H) = nw_{1}(U) + mw_{1}(F).

Proof. Let A : R^{m} → R^{m} and B : R^{n} → R^{n} be linear maps, and consider A ⊗ B : R^{nm} → R^{nm}. Then

det(A ⊗ B) = (det A)^{n}(det B)^{m}.

Now let {a_{αβ}} and {b_{αβ}} be transition functions for U and F respectively over an open covering U of Y. Then H has transition functions h_{αβ} = a_{αβ}^{∗} ⊗ b_{αβ}, and so

det(h_{αβ}) = det(a_{αβ}^{∗})^{n} det(b_{αβ})^{m} = det(a_{αβ})^{n} det(b_{αβ})^{m}.

Hence,

sgn(det h_{αβ}) = (sgn det a_{αβ})^{n}(sgn det b_{αβ})^{m}.

Rewriting this additively gives the result. □

Corollary A.6. The bundle H = Hom(U, F) is orientable if any one of the following conditions holds:

(i) rank U and rank F are both even.

(ii) F is orientable and rank F is even.

(iii) U is orientable and rank U is even.

Corollary A.7. Suppose that rank F is odd and that either F is orientable or rank U is even. Then

w_{1}(H) = w_{1}(U).

Let G_{r}(R^{m}) denote the Grassmannian of (unoriented) r- planes in R^{m}, and let U ⟶ G_{r}(R^{m}) denote the tautological r-plane bundle. There is a natural embedding U ↪ \underline{R}^{m} into the trivialized m-plane bundle, and this gives a splitting

\underline{R}^{m} = U ⊕ U^{⊥}.

Corollary A.8. If r and m are both even, then G_{r}(R^{m}) is orientable.

Proof. Apply A.6, part (i), to T G_{r}(R^{m}) = Hom(U, U^{⊥}). □

Consider a smooth real vector bundle E → X of rank m, and let π : G_{r}(E) → X be the Grassmann bundle whose fibre at x ∈ X consists of all (unoriented) rplanes in E_{x}. Let U → G_{r}(E) be the tautological r-plane bundle with canonical embedding U ↪ π^{∗}E. After a choice of metric in E we have a splitting

π^{∗}E = U ⊕ U^{⊥}

and there is a bundle equivalence

(A.9)  T G_{r}(E) ≅ π^{∗}T X ⊕ Hom(U, U^{⊥}).

Corollary A.10. Suppose X is orientable. If r and m are even, then G_{r}(E) is orientable. If r is even and m is odd, then

w_{1}(U) = w_{1}(G_{r}(E)).

Proof. By A.9 and Proposition A.4 we have

w_{1}(G_{r}(E)) = π^{∗}w_{1}(X) + w_{1}(U^{∗} ⊗ U^{⊥}) = (m − r)w_{1}(U) + rw_{1}(U^{⊥}). □

We now consider integration over the fibres of π. To begin recall that an rform twisted by the orientation bundle on a manifold Y is a section of the bundle Λ^{r}T^{∗}Y ⊗_{Z_{2}} Or(Y) where Or(Y) = Or(T Y), and where Z_{2} = {1, −1} acts multiplicatively. A density on Y is a d-form twisted by Or(Y) where d = dim(Y). Densities with compact support can be integrated over Y (cf [St]).

Proposition A.11. Let X, G_{r}(E) and U be as in Corollary A.10, and assume that r is even and m is odd. Then for any orthogonal connection on U with curvature R^{U}, the Euler form of U on G_{r}(E)

χ(R^{U}) = Pfaff (−\frac{1}{2π} R^{U})

is an r-form twisted by the orientation bundle of G_{r}(E). For any q-form ω on G_{r}(E) the product ω ∧ χ(R^{U}) can be integrated over the fibres of π : G_{r}(E) → X, yielding a smooth form on X.

Proof. Let r = 2r_{0}. For a skew-symmetric transformation A : R^{r} → R^{r} we have the equation

(A.12)  (\frac{1}{2}e^{t}Ae)^{r_{0}} = r_{0}! Pfaff(A)e_{1} ∧ ⋯ ∧ e_{r}

in ∧^{r}R^{r}, where e_{1}, …, e_{r} denotes any orthonormal basis of R^{r}, and where e^{t}Ae = ΣA_{ij}e_{i} ∧ e_{j}. The left hand side of (A.12) is independent of orientation. Hence from (A.12) we see that the Pfaffian of R^{U} is twisted by Or(U). However, from (A.9) we have

Or(G_{r}(E)) = π^{∗}Or(X) ⊗ Or(U^{∗} ⊗ U^{⊥}) = Or(U^{∗}) = Or(U)

under our assumption that X is orientable and r is even. Now any Or(G_{r}(E))- twisted form α on G_{r}(E) (for example, χ(R^{U}) ∧ ω) defines a current on G_{r}(E) and hence can be pushed forward by π as a current to X. To see this push forward is well defined let β be a form with compact support on X of degree d′ = dim(G_{r}(E)) − deg α. Then

[π_{∗}α](β) = ∫_{G_{r}(E)} α ∧ π^{∗}β

which is well-defined since α ∧ π^{∗}β is a density on G_{r}(E). Finally since α is smooth this push forward can be computed using fibre integration yielding a smooth untwisted form on X, since X is oriented. □

We have observed above that if U and F are real vector bundles of even rank, then U^{∗} ⊗ F is orientable. However, it remains to choose a canonical orientation.

Convention A.13. Let V and W be finite dimensional vector spaces with ordered bases (e_{1}, …, e_{m}) and (f_{1}, …, f_{n}) respectively. Then the canonical orientation on V ⊗ W with respect to these bases is given by

(e_{1} ⊗ f_{1}, e_{2} ⊗ f_{1}, …, e_{m} ⊗ f_{1}, e_{1} ⊗ f_{2}, …, e_{m} ⊗ f_{2}, …, e_{m} ⊗ f_{n}).

This depends only on the orientations of V and W determined by (e_{1}, …, e_{m}) and (f_{1}, …, f_{m}). If n is even, it is independent of the orientation of V (and if m is even it is independent of the orientation of W). If n ≡ m ≡ 0 (mod 2), it is independent of the orientations of both V and W.

Remark A.14. Given an inner product on V there are natural isomorphisms V ⊗ W ≅ V^{∗} ⊗W = Hom(V, W) which transfer the canonical orientation to Hom(V, W) (independently of the choice of inner product).

Suppose dim_{R}(V) = dim_{R}(W) = 2 and both are oriented and equipped with inner products so that V ≅ W ≅ ℂ. Then we have a natural splitting

(A.15)  Hom_{R}(V, W) = Hom_{ℂ}(V, W) ⊕ Hom_{ℂ}(V, \overline{W})

given by writing

A = \frac{1}{2}(A − J_{W} AJ_{V}) + \frac{1}{2}(A + J_{W} AJ_{V})

where J_{V}, J_{W} are the complex structures corresponding to the orientations on V and W.

Lemma A.16. The canonical orientation on Hom_{R}(V, W) is opposite to the one corresponding to the complex structure induced via (A.15).

Proof. Let (v_{1}, v_{2}), (w_{1}, w_{2}) be oriented bases for V and W, and let h_{ij} = v_{j}^{∗} ⊕ w_{i} 1 ≤ i, j ≤ 2 be the corresponding basis of H = Hom_{R}(V, W). Then Hom_{ℂ}(V, W) has an oriented basis (ϵ_{1}, ϵ_{2}) where ϵ_{1} = h_{11} + h_{22} and ϵ_{2} = J(ϵ_{1}) = ϵ_{1} ∘ J_{V} = h_{21} −h_{12}. Similarly Hom(V, \overline{W}) has basis ϵ′_{1} = h_{11} −h_{22} and ϵ′_{2} = J(ϵ′_{1}) = ϵ′_{1} ∘J_{V} = −h_{21} − h_{12}. Taking the wedge product over R gives

\begin{aligned}ϵ_{1} ∧ ϵ_{2} ∧ ϵ_{3} ∧ ϵ_{4} & = (h_{11} + h_{22}) ∧ (h_{21} − h_{12}) ∧ (h_{11} − h_{22}) ∧ (−h_{21} − h_{12}) \\ & = −4h_{11} ∧ h_{21} ∧ h_{12} ∧ h_{22}. □ \end{aligned}

We shall now compute the Euler form of the bundle Hom(U, F) over a manifold Y. In practice we will set Y = G_{r}(E) and take U to be the tautological bundle.

Theorem A.17. Let U → Y and F → Y be smooth vector bundles with orthogonal connections. Give H =^{def} Hom(U, F) the induced tensor product connection. Suppose

rank(U) = 2r_{0} and rank(F) = 2n_{0}

for integers r_{0}, n_{0} > 0. Then the Euler form of H for its canonical orientation is given by

(A.18)  χ(R^{H}) = Δ̃_{n_{0}}^{(r_{0})}{p(R^{F})p(R^{U})^{−1}}

where Δ̃_{a}^{(b)} is the Shur polynomial introduced in §5.

Proof. We begin by proving equation A.18 at the cohomology level. Consider first the case where dim U = dim F = 2 and both U and F are oriented. Set

a = χ(U) = c_{1}(U) and b = χ(F) = c_{1}(F)

in H^{2}(Y; R). Then from the equation

Hom_{R}(U, F) = Hom_{ℂ}(U, F) ⊕ Hom_{ℂ}(U, \overline{F})

and Lemma A.16 we get

χ(H) = −χ(U^{∗} ⊗_{ℂ} F)χ(U^{∗} ⊗_{ℂ} \overline{F})
= −(b − a)(−b − a)
= b^{2} − a^{2}
= p_{1}(F) − p_{1}(U).

Suppose now that U and F are oriented and apply the Splitting Principle to write them formally as

U = U_{1} ⊕ ⋯ ⊕ U_{r_{0}} and F = F_{1} ⊕ ⋯ ⊕ U_{n_{0}}

where U_{i} and F_{j} are oriented 2-plane bundles. We formally set

a_{i} = χ(U_{i}) and b_{j} = χ(F_{j})

for all i, j. Then

Hom(U, F) = ⨁_{j=1}^{n_{0}} ⨁_{i=1}^{r_{0}} Hom(U_{i}, F_{j})

and so

(A.19)  χ(Hom(U, F)) = ∏_{j=1}^{n_{0}} ∏_{i=1}^{r_{0}}(b_{j}^{2} − a_{i}^{2}) = Δ̃_{n_{0}}^{(r_{0})}{1 + η_{1} + η_{2} + ⋯ }

where η_{k} ∈ H^{4k}(Y; R) is defined by

\frac{p(F)}{p(U)} = \frac{∏_{j=1}^{n_{0}}(1 + b_{j}^{2})}{∏_{i=1}^{r_{0}}(1 + a_{i}^{2})} = 1 + η_{1} + η_{2} + · · · .

See [Fu, page 419] for the second equality in (A.19). This establishes formula (A.18) at the cohomology level when U and F are orientable.

To establish the formula at the level of forms we recall that by [NR] any k-plane bundle with connection over a manifold Y is induced from the universal bundle U_{k} with its connection over G_{k}(R^{N}), for N sufficiently large, by a smooth map f : Y → G_{k}(R^{N}). Suppose now that f : Y → G_{2r_{0}}(R^{N}) and g : Y → G_{2n_{0}}(R^{N}) classify U and F with their connections, i.e., f^{∗}(U_{2r_{0}}^{∗}) ≅ U^{∗} and g^{∗}(U_{2n_{0}}) ≅ F. Then f × g : Y → G_{2r_{0}}(R^{N}) × G_{2n_{0}}(R^{N}) has the property that

(f × g)^{∗}(U_{2r_{0}}^{∗} ⊗ U_{2n_{0}}) ≅ U^{∗} ⊗ F

as bundles with connection. Consequently if formula (A.18) holds for U_{2r_{0}}^{∗} ⊗ U_{2n_{0}}, it holds in general, since

χ (R^{U^{∗}⊗F} ) = (f × g)^{∗}χ (R^{U_{2r_{0}}^{∗} ⊗U_{2n_{0}}} )
= (f × g)^{∗}Δ_{n_{0}}^{(r_{0})} {p(R^{U_{2n_{0}}} )p(R^{U_{2r_{0}}} )^{−1}}
= Δ̃_{n_{0}}^{(r_{0})} {p(R^{F})p(R^{U})^{−1}} .

Hence it suffices to prove (A.18) in the universal case. However, here (A.18) is a consequence of the cohomology calculation. To see this note that both χ(R^{U_{2r_{0}}^{∗} ⊗U_{2n_{0}}} ) and Δ̃_{n_{0}}^{(r_{0})}{p(R^{U_{2n_{0}}} )p(R^{U_{2r_{0}}} )^{−1}} are invariant under the full isometry group of G_{2r_{0}}(R^{N}) × G_{2n_{0}}(R^{N}) and are therefore harmonic by a standard result in the theory of symmetric spaces.

When U and F are possibly non-orientable we pass to the 2-fold or 4-fold covering where they are orientable. The equation of forms (A.18) is invariant under the covering group and therefore descends to Y. □

Theorem A.20. Let U, F and H = Hom(U, F) be as in Theorem A.17 except that

rank(U) = 2r_{0} and rank(F) = 2n_{0} + 1

for integers r_{0}, n_{0} > 0. Then the canonical orientation gives an isomorphism Or(H) ≅ Or(U), and the Or(H)-twisted Euler form of H is equal to the following Or(U)-twisted form

(A.21)  χ(R^{H}) = χ(R^{U})Δ̃_{n_{0}}^{(r_{0})} {p(R^{F})p(R^{U})^{−1}} .

Proof. Note that by Proposition A.4 we have w_{1}(H) = w_{1}(U), and so Or(H) ≅ Or(U) under the identification Cov_{2}(Y) ≅ H^{1}(Y; Z_{2}). We lift to the 2-fold covering of Y where H and U are orientable and establish (A.21) there. Since both sides of (A.21) transform by −1 under the (non-trivial) deck-transformation, the result will follow.

We assume therefore that U is orientable. We may assume also that F is orientable since if not we pass to a 2-fold covering where it is, and then observe that the equation is invariant, as in the proof of A.17. The proof now proceeds as before. We apply the Splitting Principle and formally write

U = U_{1} ⊕ ⋯ ⊕ U_{r_{0}} and F = F_{1} ⊕ ⋯ ⊕ F_{n_{0}} ⊕ R

where U_{i}, F_{j} are oriented 2-plane bundles with a_{i} = χ(U_{i}) and b_{j} = χ(F_{j}). Then

H = U^{∗} ⊗ F = U^{∗} ⊕ ⨁_{j=0}^{n_{0}} ⨁_{i=0}^{r_{0}} U_{i}^{∗} ⊗ F_{j}

and so

χ(H) = χ(U) ∏_{j=1}^{n_{0}} ∏_{i=1}^{r_{0}}(b_{j}^{2} − a_{i}^{2}) = χ(U)Δ_{n_{0}}^{(r_{0})}{p(F)p(U)^{−1}}.

As above, this cohomology formula implies the formula at the level of forms via [NR] and the uniqueness of invariant forms in the cohomology of G_{2r_{0}}(R^{N}) × G_{2n_{0}}(R^{N}). □

We conclude this section by presenting the analogue of the last two theorems for the complex case.

Theorem A.22. Let U → Y and F → Y be smooth complex vector bundles with complex connections and with

rank(U) = r and rank(F) = n.

Give H = Hom(U, F) = U^{∗} ⊕ F the induced tensor product connection. Then the top Chern form of H is given by

(A.23)  c_{nr}(R^{H}) = Δ_{n}^{(r)}{c(R^{F})c(R^{U})^{−1}}

where c(R) = det (1 + \frac{i}{2π} R) denotes the top Chern form of the connection.

Proof. At the cohomology level this result is well known (cf. [Fu, 14.4.12]), and one can deduce the formula at the level of forms by passing to the classifying spaces as in the proof of A.17. □

Alternatively one can prove (A.23) directly by establishing the following matrix identity. Let M_{k×k} denote the vector space of complex k × k matrices. For any commutative ring R with unit, and for integers a, b > 0, let

(A.24)  Δ_{b}^{(a)} : R[[t]] ⟶ R

be the function given by

Δ_{b}^{(a)} (∑_{k≥0} r_{k}t^{k}) = det_{a×a}((r_{b−i+j}))

where ((r_{b−i+j})) is the a × a matrix whose (i, j)^{th} entry is r_{b−i+j} when r − i + j ≧ 0 and 0 otherwise.

Lemma A.25. For A ∈ M_{a×a} and B ∈ M_{b×b}, consider the matrix A⊕1_{b}−1_{a}⊕B ∈ M_{ab×ab}. Then

det {A ⊕ 1_{b} − 1_{a} ⊕ B} = Δ_{b}^{(a)} {det(1_{b} + tB) det(1_{a} + tA)^{−1}} .

Proof. It suffices to restrict attention to the Zariski open dense subset of diagonalizable matrices (A, B) in M_{a×a} × M_{b×b}. Thus we may assume that A = diag(x_{1}, …, x_{a}) and B = diag(y_{1}, …, y_{b}). The lemma is now an immediate consequence of the following.

Fact A.26. ([Fu, page 419]) In the polynomial ring R = ℂ[x_{1}, …, x_{a}, y_{1}, …, y_{b}] in (a + b) indeterminants, one has the identity

∏_{j=1}^{b} ∏_{i=1}^{a}(y_{j} − x_{i}) = Δ_{b}^{(a)} \{\frac{∏_{j=1}^{b}(1 + ty_{j})}{∏_{i=1}^{a}(1 + tx_{i})}\} .

Note. Using A.26 one can give alternative proofs of Propositions A.17 and A.20 above.

APPENDIX B Integration over the fibres in G_{r}(E)

In this appendix we establish the topological formulas required to complete the computations in §§4 and 5.

Proposition B.1. Let E → X be a smooth hermitian vector bundle of rank m. Consider the Grassmann bundle π : G_{r}(E) → X of r-planes in E, where 0 < r < m, and let U → G_{r}(E) be the tautological complex r-plane bundle. Write

π^{∗}E = U ⊕ U^{⊥},

and let c_{m−r}(U^{⊥}) be the top Chern class of U^{⊥}. Then under the Gysin map π_{∗} : H^{2r(m−r)}(G_{r}(E)) → H^{0}(X) we have

(B.2)  π_{∗} {c_{m−r}(U^{⊥})^{r}} = 1.

Proof. It suffices to prove (B.2) in the case that X is a point, i.e., to prove that

(B.3)  ⟨c_{m−r}(U^{⊥})^{r}, G_{r}(C^{m})⟩ = 1.

To see this we consider the Poincaré dual of c_{m−r}(U^{⊥}) i.e., the divisor Div(α_{v}) of an atomic section α_{v} ∈ Γ(U^{⊥}) defined by fixing a vector v ∈ C^{m} and setting

α_{v}(U) = π_{U^{⊥}}(v)

at U ∈ G_{r}(C^{m}) where π_{U^{⊥}} : C^{m} = U ⊕ U^{⊥} → U^{⊥} is orthogonal projection. Now α_{v} vanishes non-degenerately and

Div(α_{v}) = {U ∈ G_{r}(C^{m}) : v ∈ U}.

We now choose r vectors v_{1}, …, v_{r} ∈ C^{m} which are linearly independent. Then these divisors meet transversely and

(B.4)  Div(α_{v_{1}}) ∩ ⋯ ∩ Div(α_{v_{r}}) = {U ∈ G_{r}(C^{m}) : v_{1}, …, v_{r} ∈ U}
= {span(v_{1}, …, v_{r})} .

Under Poincaré duality cup product followed by evaluation on the fundamental class becomes intersection product. Hence, (B.4) ⇒ (B.3). □

Proposition B.5. Let E → X be a smooth riemannian vector bundle of rank m. Consider the Grassmann bundle π : G_{r}(E) → X of r-planes in E, where 0 < r < m, and let U → G_{r}(E) be the tautological real r-plane bundle. Suppose

r = 2r_{0} and m = 2m_{0}

for positive integers r_{0} and m_{0}. Write

π^{∗}E = U ⊕ U^{⊥}

and let p_{m_{0}−r_{0}}(U^{⊥}) be the top Pontrjagin class of U^{⊥}. Then under the Gysin map π_{∗} : H^{4r_{0}(m_{0}−r_{0})}(G_{2r_{0}}(E)) → H^{0}(X) we have

(B.6)  π_{∗} {p_{m_{0}−r_{0}}(U^{⊥})^{r_{0}}} = 1.

Proof. It suffices to consider the case where X is a point, i.e., it suffices to prove that

(B.7)  ⟨p_{m_{0}−r_{0}}(U^{⊥})^{r_{0}}, G_{2r_{0}}(R^{2m_{0}})⟩ = 1.

To do this we lift to the Grassmannian G̃_{2r_{0}}(R^{2m_{0}}) of oriented 2r_{0}-planes with tautological bundle Ũ, and prove that

(B.8)  ⟨p_{m_{0}−r_{0}}(Ũ^{⊥})^{r_{0}}, G̃_{2r_{0}}(R^{2m_{0}})⟩ = 2.

Since G̃_{2r_{0}}(R^{2m_{0}}) → G_{2r_{0}}(R^{2m_{0}}) is a 2-sheeted covering whose deck transformation preserves the orientation of G̃_{2r_{0}}(R^{2m_{0}}) and the class p_{m_{0}−r_{0}}(Ũ^{⊥}), we see that (B.8) ⇒ (B.7).

To prove (B.8) we first establish the following.

Lemma B.9.

p_{m_{0}−r_{0}}(Ũ^{⊥}) = χ(Ũ^{⊥})^{2}.

Proof. We apply the Splitting Principle and write Ũ^{⊥} formally as a direct sum of oriented 2-plane bundles

(B.10)  Ũ^{⊥} = U_{1} ⊕ ⋯ ⊕ U_{k_{0}}

where k_{0} = m_{0} − r_{0}, and set a_{j} = χ(U_{j}). Then by definition

p_{k_{0}}(Ũ^{⊥}) = σ_{k_{0}}(a_{1}^{2}, …, a_{k_{0}}^{2})
= a_{1}^{2} ⋯ a_{k_{0}}^{2} = (a_{1} ⋯ a_{k_{0}})^{2}
= χ(Ũ^{⊥})^{2}. □

We now proceed as in the proof of (B.1). The Poincaré dual of χ(Ũ^{⊥}) is represented by the divisor of the atomic section α_{v} ∈ Γ(Ũ^{⊥}) defined by fixing a vector v ∈ R^{2m_{0}} and setting

α_{v}(Ũ) = πŨ^{⊥}(v)

at Ũ ∈ G̃_{2r_{0}}(R^{2m_{0}}) where πŨ^{⊥} : R^{2m_{0}} = Ũ ⊕ Ũ^{⊥} → Ũ^{⊥} is orthogonal projection. In fact α_{v} vanishes non-degenerately and

Div(α_{v}) = {Ũ ∈ G̃_{2r_{0}}(R^{2m_{0}}) : v ∈ Ũ}.

We choose 2r_{0} linearly independent vectors v_{1}, …, v_{2r_{0}} ∈ R^{2m_{0}}. Then the oriented submanifolds Div(α_{v_{j}}) meet transversely and

Div(α_{v_{1}}) ∩ ⋯ ∩ Div(α_{v_{2r_{0}}} ) = {Ũ ∈ G̃_{2r_{0}}(R^{2m_{0}}) : v_{1}, …, v_{2r_{0}} ∈ Ũ}
≅ {v_{1} ∧ ⋯ ∧ v_{2r_{0}}, −v_{1} ∧ ⋯ ∧ v_{2r_{0}}} ,

i.e., this intersection consists exactly of the two points corresponding to the plane span{v_{1}, …, v_{2r_{0}}} with its two possible orientations. Using invariance under the deck transformation group and a local calculation one concludes that

⟨p_{m_{0}−r_{0}}(Ũ^{⊥})^{r_{0}}, G̃_{2r_{0}}(R^{2m_{0}})⟩ = ⟨χ(Ũ^{⊥})^{r_{0}}, G̃_{2r_{0}}(R^{2m_{0}})⟩
= #{Div(α_{v_{1}}) ∩ ⋯ ∩ Div(α_{v_{2r_{0}}} )}
= 2.

This establishes (B.8) and proves (B.5). □

We now recall that if r is even and m is odd, then G_{r}(R^{m}) is non-orientable and Or(G_{r}(R^{m})) = Or(U). More generally if E → X is a oriented bundle of odd rank over an oriented manifold, and if r is even, then G_{r}(E) is non-orientable and Or(G_{r}(E)) = Or(U), where U is the tautological bundle as above. As seen in (A.11), integration over the fibre gives a Gysin map π_{∗} : H^{k}(G_{r}(E), Or(U)) → H^{k−r(m−r)}(X; R) where H^{k}(G_{r}(E); Or(U)) denotes the cohomology of Or(U)- twisted forms, i.e., cohomology with coefficients in the local system Or(U). As seen also in (A.11), χ(U) ∧ Ω is an Or(U)-twisted class for any Ω ∈ H^{∗}(G_{r}(E); R).

Proposition B.11. Let E → X be an oriented riemannian bundle of rank m over an oriented manifold. Let G_{r}(E), U and U^{⊥} be as in Proposition B.5, but assume that

r = 2r_{0} and m = 2m_{0} + 1

for positive integers r_{0} and m_{0}. Then under the Gysin map π_{∗} : H^{r(m−r)}(G_{r}(E); Or(U)) → H^{0}(X; R) we have

(B.12)  π_{∗} {χ(U)p_{m_{0}−r_{0}}(U^{⊥})^{r_{0}}} = 1.

Proof. It suffices to consider the case where X is a point. Fix v_{0} ∈ R^{m} and let β be the cross-section of U given by

β(U) = π_{U}(v_{0})

where π_{U} : R^{m} = U ⊕ U^{⊥} → U is orthogonal projection. Now the Poincaré dual of χ(U) is represented by the oriented cycle

Div(β) = The Grassmannian of r-planes in v_{0}^{⊥}
≅ G_{2r_{0}}(R^{2m_{0}}).

By the standard formula for the Poincaré dual we have

⟨χ(U)p_{m_{0}}(U^{⊥})^{r_{0}}, G_{r}(R^{m})⟩ = ⟨p_{m_{0}}(U^{⊥})^{r_{0}}, χ(U) ∩ G_{r}(R^{m})⟩

= ⟨p_{m_{0}}(U^{⊥})^{r_{0}}, Poincaré dual of χ(U)⟩

= ⟨p_{m_{0}}(U^{⊥})^{r_{0}}, G_{2r_{0}}(R^{2m_{0}})⟩

= 1

where the last equality comes from (B.5). □

APPENDIX C Explicit formulae

Here we collect together explicit algebraic expressions for the various forms occuring in the equations of Section 3. These explicit expressions are taken from [HL_{2}]. Suppose that V is either a complex vector bundle of complex rank n or a real vector bundle of even rank 2n which is oriented. Equip V with an inner product ⟨ , ⟩_{V} and a connection D_{V}. The equations are:

Complex Case:  c_{n}(Ω_{V}) − τ_{s} = dσ_{s}
τ_{s} − [X] = dr_{s}
c_{n}(Ω_{V}) − [X] = dσ.

Real Case:  χ(Ω_{V}) − τ_{s} = dσ_{s}
τ_{s} − [X] = dr_{s}
χ(Ω_{V}) − [X] = dσ.

In both cases σ_{s} and r_{s} are related by

r_{s} = σ − σ_{s}

and

lim_{s→0} r_{s} = 0  lim_{s→∞} σ_{s} = 0,

in L_{loc}^{1}(V). These are current equations on the total space of V with X ⊂ V the zero section. Alternatively, given a smooth atomic section μ : X → V they can be pulled back to equations on X. The atomic hypothesis ensures that the L_{loc}^{1}(V) form σ pulls back to an L_{loc}^{1}(X) form on X, as well as ensuring that Div(μ) is the appropriate replacement for [X].

In order to describe the global L_{loc}^{1} forms τ_{s}, σ_{s}, and σ explicitely the following notation is useful. Let e_{1}, …, e_{n} denote a local frame for V and let e denote the column with i^{th} entry e_{i}. Then a section μ can be written as μ = ue with u = (u_{1}, …, u_{n}). The equation D_{V} e = ω_{V} e defines the local gauge ω_{V} as an n × n matrix of one forms. The curvature operator R_{V} = D_{V}^{2} has matrix form Ω_{V} = dω_{V} − ω_{V} ∧ ω_{V}. Since D_{V} μ = (du + uω_{V})e, it is convenient to let Du ≡ du + uω_{V}, so that D_{V} μ = (Du)e. Let h_{V} ≡ (⟨e_{i}, e_{j}⟩_{V}) denote the metric matrix, and let u^{∗} ≡ h_{V} ū^{t} so that |μ|^{2} = uh_{V} ū^{t} = uu^{∗}. (We will also find it convenient to let |u|^{2} denote |μ|^{2} = uu^{∗} rather than uū^{t}.) Also, let

Du^{∗} = du^{∗} − ω_{V} u^{∗}.

Just as Du is the matrix form of D_{V} μ, this formula is the matrix form of Dμ^{∗} where μ^{∗} is the adjoint of μ thought of as a bundle map from the trivial bundle \underline{C} to V.

In the real case we only consider oriented orthonormal frames e and hence h_{V} = 1 (the identity matrix) so that u^{∗} = u^{t}, |μ|^{2} = uu^{t} = |u|^{2}, and Du^{t} = du^{t} − ω_{V} u^{t}. As before χ(t) : [0, ∞] → [0, 1] determines the approximation mode. For brevity let

χ_{s} ≡ χ (\frac{|u|^{2}}{s^{2}} ) .

In the complex case, consider the dual frame e^{∗} = (e_{1}^{∗}, …, e_{n}^{∗}), together with the frame e, as elements of the grassmann algebra ∧(V^{∗} ⊕ V). Let λ(e) ≡ e_{1}^{∗} ∧ e_{1} ∧ ⋯ ∧ e_{n}^{∗} ∧ e_{n} denote the volume form. Then for any matrix A, the determinant can be computed from

\frac{1}{n!}(e^{∗}Ae) = (det A)λ(e).

Consequently, the equation

det(A ; B)λ(e) = \frac{1}{(n − 1)!}(e^{∗}Ae)(e^{∗}Ae)^{n−1}

can be used to compute

det(A ; B) ≡ \frac{d}{dt} det(B + tA) |_{t=0} .

Complex Case:

τ_{s} = (\frac{i}{2π} )^{n} (1 − χ_{s}) det (Ω_{V} − χ_{s} \frac{Du^{∗}Du}{|u|^{2}})

+ (\frac{i}{2π} )^{n} (χ_{s}(1 − χ_{s}) − χ′_{s} \frac{|u|^{2}}{s^{2}} ) \frac{d|u|^{2}}{|u|^{2}} det (\frac{u^{∗}Du}{|u|^{2}} ; Ω_{V} − χ_{s} \frac{Du^{∗}Du}{|u|^{2}}) .

σ_{s}λ(e) = −\frac{1}{n!} (\frac{i}{2π} )^{n} \frac{e^{∗}u^{∗}Due}{|u|^{2}} \frac{(e^{∗}Ω_{V}e − χ_{s}e^{∗} \frac{Du^{∗}Du}{|u|^{2}}e)^{n} − (e^{∗}Ω_{F}e)^{n}}{e^{∗} \frac{Du^{∗}Du}{|u|^{2}}e}

σλ = −\frac{1}{n!} (\frac{i}{2π} )^{n} \frac{e^{∗}u^{∗}Due}{|u|^{2}} \frac{(e^{∗}Ω_{V}e − e^{∗} \frac{Du^{∗}Du}{|u|^{2}}e)^{n} − (e^{∗}Ω_{V}e)^{n}}{e^{∗} \frac{Du^{∗}Du}{|u|^{2}}e}

The choice χ(t) ≡ t/1 + t is referred to as the algebraic approximation mode. (See [HL_{2}] and [Z]) for motivation for this choice). Note that σ is independent of the choice of approximation mode χ.

Complex Case with Algebraic Approximation Mode:

τ_{s} = (\frac{i}{2π} )^{n} \frac{s^{2}}{|u|^{2} + s^{2}} det (Ω_{V} − \frac{Du^{∗}Du}{|u|^{2} + s^{2}} ) ,

or equivalently,

τ_{s}λ = \frac{1}{n!} (\frac{i}{2π} )^{n} \frac{s^{2}}{|u|^{2}+s^{2}} (e^{∗}Ω_{V}e − \frac{e^{∗}Du^{∗}Due}{|u|^{2} + s^{2}})^{n} .

σ_{s}λ = −\frac{1}{n!} (\frac{i}{2π} )^{n} e^{∗}u^{∗}Due\frac{(e^{∗}Ω_{V}e − \frac{e^{∗}Du^{∗}Due}{|u|^{2}+s^{2}})^{n} − (e^{∗}Ω_{V}e)^{n}}{e^{∗}Du^{∗}Due} .

Now we consider the real case. The real rank of V is assumed to be even (= 2n).

Real Case: Suppose A is a skew 2n × 2n matrix. Let e_{1}, …, e_{2n} denote an oriented orthonormal (local) frame for V and let λ ≡ e_{1} ∧ ⋯ ∧ e_{2n} ∈ Λ^{2n}V denote the unit volume element. Recall the definition of the pfaffian of A

Pf(A)λ = \frac{1}{n!} (\frac{1}{2}e^{t}Ae)^{n} .

Since V is oriented the unit volume form λ for V is globally defined, and hence Pf(A) is globally defined independently of the choice of oriented orthonormal frame e.

τ_{s}λ = \frac{1}{n!} (\frac{−1}{4π})^{n} (1 − χ_{s}) (e^{t}Ω_{V} e − 2χ_{s} (1 − \frac{χ_{s}}{2} ) \frac{(Due)^{2}}{|u|^{2}} )^{n} +

\frac{2}{(n−1)!} (\frac{−1}{4π})n(χ_{s}(1−χ_{s})(1− \frac{χ_{s}}{2} )−χ′_{s} \frac{|u|^{2}}{s^{2}} ) \frac{d|u|^{2}}{|u|^{2}} \frac{(ue)(Due)}{|u|^{2}}(e^{t}Ω_{V} e−2χ_{s}(1− \frac{χ_{s}}{2} ) \frac{(Due)^{2}}{|u|^{2}} )^{n−1}.

σ_{s}λ = \frac{2}{(n−1)!} (\frac{−1}{4π})^{n} \frac{(ue)(Due)}{|u|^{2}} ∫_{0}^{χ_{s}}(e^{t}Ω_{V} e − 2x (1 − \frac{x}{2}) \frac{(Due)^{2}}{|u|^{2}} )^{n−1} dx.

The choice χ(t) = 1− \frac{1}{\sqrt{1+t}} is referred to as the real algebraic approximation mode (See [HL_{2}]).

Real Case with Algebraic Approximation Mode:

τ_{s}λ = \frac{1}{n!} (\frac{−1}{4π})^{n} \frac{s}{\sqrt{|u|^{2} + s^{2}}} (e^{t}Ω_{V} e − \frac{(Due)^{2}}{|u|^{2} + s^{2}} )^{n} .

σλ = \frac{1}{π^{n}} ∑_{p=0}^{n−1}(−1)^{n−p} \frac{p!}{(n−p−1)!(2p+1)!2^{2n−2p−1}} \frac{(ue)(Due)^{2p+1}}{|u|^{2p+2}} (e^{t}Ω_{V} e)^{n−p−1}.

Thus the part of σ of top degree 2n − 1 in the 1-forms du_{1}, …, du_{2n} is

σ_{2n−1} = vol(S^{2n−1})^{−1}θ(u)

where

θ(u) ≡ ∑_{k=1}^{2n}(−1)^{k−1} \frac{u_{k}du_{1} ∧ ⋯ ∧ d̂u_{k} ∧ ⋯ ∧ du_{2n}}{|u|^{2n}}

denotes the solid angle kernel on R^{2n}.

Remark. These explicit formula have two different interpretations. First they define forms on the total space of the bundle V. In this case u ≡ u_{1}, …, u_{n} is the fiber variable. Second they define forms on X where u ≡ (u_{1}, …, u_{n}) is the n-tuple of C^{∞} functions on X representing the given atomic section μ in the frame e.

References

[BC1] R. Bott and S.S. Chern, Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections, Acta. Math. 114 (1968), 71–112.

[BC2] , Some formulas related to complex transgression, Essays on Topology and Related Topics, Springer-Verlag, 1970, pp. 48–57.

[Ca] H. Cartan, Quelque questions de topologie, Exposé 8, Seminaire E. N. S., 1956-57.

[ChS] J. Cheeger and J. Simons, Ann. Math..

[C] S.-S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. Math. 45 (1944), 747–752.

[CS] S.-S. Chern and J. Simons, Ann. Math..

[Fe] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.

[Fu] W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge, Band 2, Springer Verlag, Berlin-Heidelberg, 1984.

[GG] M. Golubitsky and V. Guillemin, Stable Mappings and their Singularities (1973), Springer Verlag, Berlin-Heidelberg.

[HL_{1}] F.R. Harvey and H. Blaine Lawson, On boundaries of complex analytic varieties I, Ann.

of Math. 102 (1975), 223–290. [HL_{2}] , A Theory of Characteristic Currents Associated with a Singular Connection —

Part II, to appear.

[HS] F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567–600.

[Lai] H. Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. A.M.S. 172 (1972), 1-33.

[M1] R. MacPherson, Singularities of vector bundle maps – Proceedings of Liverpool Singularities Symposium , I, Springer Lecture Notes in Mathematics 192 (1971), 316-318.

[M2] R. MacPherson, Generic vector bundle maps, Dynamical Systems, Proceedings of Symposium – University of Bahia, Salavador 1971), Academic Press, New York, 1973, pp. 165-175.

[NR] Narasimhan and Ragunathan.

[P] I. R. Porteous, Simple singularities of maps – Proceedings of Liverpool Singularities Symposium , I, Springer Lecture Notes in Mathematics 192 (1971), 286-307.

[R] F. Ronga, Le calcul de la classe de cohomologie duale a Σ̄^{k}. – Proceedings of Liverpool Singularities Symposium , I, Springer Lecture Notes in Mathematics 192 (1971), 313-315.

[St] S. Sternberg, Lectures on Differential Geometry, Prentice Hall, 1964.

[S] R. Stingley, Ph.D. Thesis, Stony Brook (1993).

[T] R. Thom, Les ensembles singuliers d’une application différentiables et leurs propriétés homologiques, Seminaire de Topologie de Strasbourg (1957).

[W1] S. Webster, Minimal surfaces in a Kähler surface, J. Diff. Geom. 20 (1984), 463–470.

[W2] , The Euler and Pontrjagin numbers of an n-manifold in C^{n}, Comment. Math. Helv. 60 (1985), 193–216.

[W3] , On the relation between Chern and Pontrjagin numbers, Contemp. Math. 49 (1986), 135–143.

[Wo] J. Wolfson, Minimal surfaces in Kähler surfaces and Ricci curvature, J. Diff. Geom. 29 (1989), 281–294.

[Z] J. Zweck, Compactification problems in the theory of characteristic currents associated with a singular connection, Thesis, Rice University (1993).