Geometric residue theorems.

O. 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 modem 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 [omega], [omega], on a bundle, the difference of the characteristic forms can be written as a coboundary p([omega]) - p([omega]) = dT where T = T([omega],[omega]') 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 [HL2] 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 a between bundles with connection there are formulas

p([omega] - [sigma]([alpha]) = dT

where: p([omega]) is a canonical characteristic form as above, [sigma]([alpha]) is a current defined purely in terms of the singularities of [alpha], and T is a canonical transgression form (with [L.sup.1.sub.loc]-coefficients). This enables us to define secondary invariants for certain connections and singularities.

Furthermore, our theory generates canonical smooth families

p([omega] - p([[omega].sub.s]) = [dT.sub.s] for 0 < s [less than or equal to] [infinity]

where [T.sub.[infinity]] = 0 and where one has convergence [T.sub.s] [right arrow] T everywhere in [L.sup.1.sub.loc] as s [right arrow] 0. In particular, the families of smooth characteristic forms p([[omega].sub.s]) converge to the singular current, i.e., p([[omega].sub.s]) [right arrow] [sigma]([alpha]) as s [right arrow] 0. In certain "approximation modes" p([[omega].sub.s]) will be supported in the s-tubular neighborhood of [sigma]([alpha]. The virtue of the explicit nature of these formulas was seen in [HL2], 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 the 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. How@ ever, 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 Poincare-Lelong type between Chern/Pontrjagin 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 Poincare-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 "nondegenerate vanishing."

Definition 1.1. Let f : U [right arrow] [R.sup.p] be a map where U [subset or equal to] [R.sup.n] is an open set. Then f is said to be atomic if f* [Mathematical Expression Omitted] such that [Mathematical Expression Omitted ]

Let [rho] : [R.sup.p] - {0} [right arrow] [S.sup.p-1] denote radial projection onto the unit sphere, and define

[Mathematical Expression Omitted]

where [c.sub.p] = vol([S.sup.p-1]). The coefficients of this form are integrable in bounded neighborhoods of 0, and satisfies the current equation d = [0] in [R.sup.p]. Note that if f : U [right arrow] [R.sup.p] is atomic, then [Mathematical Expression Omitted].

Definition 1.3. Let f : U [right arrow] [R.sup.p] be atomic. Then the divisor of f is the current of degree p (and dimension n - p) on U defined by Div (f) = d(f* ).

This current has the following properties.

(1.4) d Div (f) = Z(f) (1.5) suppDiv(f) [subset or equal to] {x [element of] U : f(x) = 0} [equivalent] 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 oriented manifold Z (f).

Theorem 1.7. ([HS]) Let f(x) = g(x)f([phi](x)) where [phi] : U [right arrow] U is a diffeomorphism and g : U [right arrow] [GL.sub.p] (R) is a smooth map. Then f is atomic if and only if f is atomic. Furthermore, if det (g) > 0 on U and [phi] is orientation preserving, then [phi]* Div (f) = Div (f).

Definition 1.8. Let E [right arrow] X be a smooth vector bundle over an n-manifold X. A smooth section [mu] [element of] [gamma](E) is said to be atomic if each point x [element of] X has a neighborhood with local coordinates and a local trivialization of E with respect to which [mu] is an atomic [R.sup.p]-valued function. If E and X are oriented, and if [mu] [element of] [gamma](E) is atomic, then Div [mu, is a well-defined current of degree p (and dimension n-p) on X, called the divisor of [mu].

Remark 1.9. Note that Div ([mu]) is well-defined in the nonorientable case provided that the first Stiefel-Whitney classes satisfy [w.sub.1] (E) = [w.sub.1] (X) in [H.sup.1] (X; [Z.sub.2]). This condition guarantees that we can choose local trivializations of E over a coordinate covering so that the changes of trivialization [g.sub.[alpha][beta]]) and the Jacobian matrices of the changes of local coordinates ([[phi].sub.[alpha][beta]) satisfy det (g.sub.[alpha][beta]).det ([[phi].sub.[alpha][beta]]) > 0. (See [Z].)

Theorem 1.10. ([HS]) If f : U [right arrow] [R.sup.p] is real analytic and dim Z(f) n - p, then f is atomic.

Theorem 1.11 ([HS]) Suppose f : U [right arrow] [R.sup.n] satisfies:

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

[Mathematical Expression Omitted]

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

Then f is atomic.

Theorem 1.12. ([HS]) Let f : U [right arrow] [R.sup.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.sup.th] degeneracy current of a bundle map. This is a current associated to the condition rank k.

For the definitions we must fix some notation. Let E [right arrow] X and F [right arrow] 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) [pi] : [G.sub.r](E) [right arrow] X

be the smooth bundle whose fibre at [chi] [element of] X is the set of all r-dimensional linear subspaces of [E.sub.x] (the fibre of E at [chi]). Over [G.sub.r](E) there is a tautological vector bundle U of rank r whose fibre at P [element of] [G.sub.r](E) consists of all vectors [nu] [element of] P. There is a natural bundle embedding

(2.2) U [element of] [[pi].sup.*]E

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

[Mathematical Expression Omitted]

Suppose now that we are given a smooth bundle map [alpha] : E [right arrow] F. Then this lifts to a mapping [Mathematical Expression Omitted] over [G.sub.r](E), and restricting to U gives a map

[Mathematical Expression Omitted]

Definition 2.5. The bundle map [alpha] is said to be k-atomic if [alpha] is an atomic section of the bundle Hom [Mathematical Expression Omitted] over [G.sub.r](E)

Definition 2.6. For a bundle map [alpha]a which is k-atomic, we define its [k.sup.th] degeneracy current on X to be

[D.sub.k]([alpha] = [pi]*Div([alpha]

where [pi]* denotes the push-forward of currents by [pi] : [G.sub.r](E) [right arrow] X.

Note that when E and F are real bundles, Div ([alpha], and therefore [D.sub.k] ([alpha], only make sense when [Mathematical Expression Omitted]. This condition holds when m [equivalent] n [equivalent] k (mo 2) (See Appendix A, A.6-A.10).

The codimension of Div ([alpha]) is rn (or 2rn in the complex case), and the fibre dimension of [G.sub.r](E) is rk (or 2rk respectively). Hence, codim [D.sub.k] = r(n - k) = (m - k)(n - k) in the real case, and codim [D.sub.k] = 2(m - k)(n - k) in the complex case.

Lemma 2.7. For any k-tomic section [alpha], one has

supp[D.sub.k]([alpha] [subset of] {x [element of] X : rank [[alpha].sub.x k}

Proof. If x [element of] supp [D.sub.k]([alpha]), then there exists a subspace U [subset of] [E.sub.x], of dimension r such that [[alpha].sub.x] [vertical bar]U = 0. Hence, rank [[alpha].sub.x] k.

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

Proposition 2.8. Suppose [alpha] vanishes nondegenerately. Then [RK.sub.k] ([alpha]) {[chi] [element of] X : rank [[alpha].sub.x] = k} is a locally rectifiable set, and

[D.sub.k]([alpha]) = [[RK.sub.k]([alpha])]

i. e., [D.sub.k] ([alpha]) is the current given by integration over this set.

Proof. By hypothesis we know that Z([alpha]) is a smooth proper submanifold of [G.sub.r](E), and that Div ([alpha]) = [Z([alpha])]. Therefore, [D.sub.k]([alpha]) = [pi]*[Z([alpha])], i.e., [D.sub.k]([alpha]) is the d-closed locally rectifiable current given by the push-forward of the manifold Z([alpha]). This current has dimension N = dim X - (m - 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 [pi][[vertical bar].sub.Z([alpha]), from Z([alpha]) to X, has Hausdorff

N-dimensional measure zero. Hence [pi]*[Z([alpha])] = q[R] where R is the set of regular values and q is an integer. It remains to show that

(2.9) R [element of] [RK.sub.k]([alpha]),

since, as noted above,

[Mathematical Expression Omitted]

is one to one. To see (2.9), we observe that if rank [[alpha].sub.x] = k - p, then

[Mathematical Expression Omitted]

is a submanifold of [G.sub.r]([E.sub.x]) diffeomorphic to the Grassmannian of r-planes in (r + p)-space. Thus, the preimage under m of each point x with rank [[alpha].sub.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 nondegenerately if its graph is transversal to the zero-section. The following proposition 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 a for which [alpha] vanishes nondegenerately is open and dense in the [C.sup.1]-topology. Consequently, for an y manifold X the set of such [alpha] 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 [alpha] and a point [x.sub.0] [element of] X. Choose trivializations of E and F in a neighborhood U of [x.sub.0]. Then we have a family of sections of Hom(E,F) over U given by [[alpha].sub.L] = [alpha] [vertical bar]U +L for L [element of ] Hom [Mathematical Expression Omitted]. This gives a family of sections [[alpha].sub.L] of the bundle [pi]: Hom (U,[pi]*F) [right arrow] U where over [Mathematical Expression Omitted]. We think of this as a map of manifolds

[Mathematical Expression Omitted]

where V = Hom ([E.sub.[x.sub.0]],[F.sub.[x.sub.0]]). This map (2.11) is actually a submersion. To see this, note first that [pi] [[alpha].sub.L](u) = u, and so the image of [T.sub.u]U x {L} [subsection of] [T.sub.(u.L)](U x V) is a transversal to the fibre of [pi] at all points. However, the map {u} x V [right arrow] Hom(U,[pi]*F) is a surjective linear map at each u (which sends L to "L [vertical bar]U"). This shows that (2.11) is a submersion.

Using a partition of unity …