Degeneracy Loci in CR Geometry

.1 Introduction

The geometry of complex subspaces contained within real subspaces of a complex vector space is described in detail in Section 2. This leads to an analysis of real subbundles of a complex vector bundle in Section 3, using a generalized Gauss map. These geometric constructions lend themselves to a cohomological formalism of Thom and [Porteous], and consequences of the theory are examined in Section 4. A general formula unifies relations described by [Wells], [Lai], [W_{86}], [HL_{93}], and [Domrin] among the pontrjagin classes of a real oriented msubbundle T, the chern classes of the ambient complex n-bundle F, and chern numbers of complex bundles over the CR-singular sets in a smooth base space M. The initial study of the grassmannian as a universal case shows that the codimensions appearing in these cohomological formulas are indeed the numbers “expected” in geometrically generic configurations.

The theory of degeneracy loci is classical in algebraic geometry ([Fulton_{84}]); a differential-topological approach is considered in a recent article of of Harvey and Lawson [HL_{95}]. In particular, [HL_{95}] also arrives at a more general version of Theorem 4.5, and phrases it in terms of singular differential forms and approximating families. Here, Theorem 4.5 is generalized in different directions, requiring that degeneracy loci be, in a certain sense, submanifolds arising from transverse intersections. The examples make explicit how some formulas regarding complex tangencies follow from [HL_{95}]. Some new formulas (Theorems 4.12, 4.18) follow from theorems of [HT] and [Pragacz] on chern numbers of kernel bundles, and of [Fulton_{91}] on degeneracy loci for flags of bundles.

The theory is applied to smooth maps between complex bundles, and the loci where such maps are complex-linear on subspaces. Corollary 5.2 generalizes a lemma of [EW] and the invariants for pairs of complex structures of [HL_{93}] and [HL_{95}].

.2 Planes in Planes in Planes

Consider a complex vector space F^{n} as an oriented real vector space F_{ℝ} of real dimension 2n, with a complex structure J, J^{2} = −1 and a compatible positive definite metric g. The compatibility condition is the equation g(X, Y) = g(JX, JY), and if (J, g) does not already have this property, then g can be replaced by a positive multiple of g(X, Y) + g(JX, JY). For 0 ≤ m ≤ 2n, denote by SG(m, F) the manifold of all oriented real-linear m-subspaces in F_{ℝ}. SG(m, F) has real dimension m(2n − m).

Definition 2.1 V ∈ SG(m, F) has (exactly) j complex directions if dim_{ℝ} V ∩ JV ≥ 2j (= 2j). Define j_{0} = max{0, m − n} and k = ⌊⌊m/2⌋⌋.

Lemma 2.2

j ∈ {j_{0}, ..., k} ⟺ ∃V ∈ SG(m, F) with exactly j complex directions.

Proof: Note that H := V ∩ JV is the largest J-invariant subspace of F_{ℝ} contained in V and J|_{H} is a complex structure. The inequality j_{0} ≤ j ≤ k follows from simple linear algebra.

Definition 2.3 If V has j = k = m/2 complex directions, then V is J-invariant, and is called a “complex j-subspace” of F if its orientation is compatible with its induced complex structure, or an “anticomplex j-subspace” otherwise.

Definition 2.4 In the case where V has exactly zero complex directions, V is said to be totally real. V with exactly j_{0} complex directions will be called CR-regular.

For j_{0} ≤ j ≤ k, define D_{j} = {V ∈ SG(m, F) with j complex directions}, and C_{j} = {V ∈ SG(m, F) with exactly j complex directions}; then

• C_{k} = D_{k}, and for j_{0} ≤ j ≤ k − 1, C_{j} = D_{j} \ D_{j+1}.

• SG(m, F) = D_{j_{0}} ⊇ … ⊇ D_{j} ⊇ … ⊇ D_{k}

• SG(m, F) is the disjoint union C_{j_{0}} ∪ … ∪ C_{j} ∪ … ∪ C_{k}.

It is convenient to define C_{k+1} = ∅ and D_{j} = SG(m, F) for j < j_{0}.

D_{j} is a subvariety of SG(m, F), with singular locus C_{j+1}, and can be described as follows. Given a complex j-plane (an element of ℂG(j, n)), one could consider the set of real m-planes containing it; this set forms a SG(m−2j, n−2j). This process gives a bundle of grassmannians over ℂG(j, n), but if an m-plane has j +1 complex directions then it appears in this grassmann bundle more than once. D_{j} is the set obtained by making the appropriate identifications on this manifold. The smooth locus of D_{j} is C_{j}, which is a fibered manifold with base ℂG(j, n) and fiber a dense open subset of SG(m − 2j, n − 2j). In particular, C_{j_{0}} is a dense open subset of SG(m, F). If m is even, C_{k} = ℂG(k, n) × S^{0}, and if m is odd, then C_{k} has base ℂG(k, n) and fiber SG(1, 2n − m + 1) = S^{2n−m}.

Lemma 2.5 A brief calculation gives the codimension of C_{j} in SG(m, F),

codim_{ℝ}C_{j} = 2j(n − m + j),  (1)

and that this number is ≥ 0 and strictly increasing with j.

Example 2.6 m = n = 2 ⇒ 0 ≤ j ≤ 1. [Chern-Spanier] show that SG(2, ℝ^{4}) = S^{2} × S^{2} ⊆ Λ^{2}ℝ^{4} = ℝ^{6}, given by x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = 1. If the ℝ^{4} has complex structure operator

J = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \\ −1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) ,

then C_{1} = ℂP^{1} × S^{0} is the parallel pair of 2-spheres given by x_{3} = ±1. (cf also [Banchoff-Farris], [IO], and [Chen-Morvan])

Example 2.7 The metric on F_{ℝ} gives the familiar diffeomorphism from SG(m, F) to SG(2n − m, F), where V ↦ V^{⊥}. Consider V ∈ C_{j}:

dim V ∩ JV = 2j ⇒ dim V^{⊥} ∩ JV = dim JV^{⊥} ∩ V = m − 2j
⇒ dim V^{⊥} ∩ JV^{⊥} = (2n − m) − (m − 2j) = 2(n − m + j)
⇒ V^{⊥} ∈ C′_{j′} ⊆ SG(m′, F),

where, denoting j′ = n − m + j, m′ = 2n − m, etc., C′_{j′} = {V ∈ SG(m′, F) with exactly j′ complex directions}. The inequality j′_{0} ≤ j′ ≤ k′ follows from j_{0} ≤ j ≤ k, and the codimension of C′_{j′} in SG(m′, F) is 2j′(n − m′ + j′) = 2j(n − m + j). So, the diffeomorphism ⊥ preserves the decomposition into C_{j}’s, possibly re-indexing them.

The following table describes some configurations allowed by Lemma 2.2 and (1), arranged by the codimension of C_{j} in SG(m, F).

j  m  n codim j m  n codim j m  n  codim
j_{0} 0 ≤ m ≤ 2n ≥ 0  0  1 n − 4 ≥ 6 10 1 n − 7 ≥ 9  16
1  n  ≥ 2  2  5 n + 4 ≥ 6 10 2 n − 2 ≥ 6  16
1  n − 1  ≥ 3  4  1 n − 5 ≥ 7 12 4 n + 2 ≥ 6  16
2  n + 1  ≥ 3  4  2 n − 1 ≥ 5 12 8 n + 7 ≥ 9  16
1  n − 2  ≥ 4  6  3 n + 1 ≥ 5 12 1 n − 8 ≥ 10 18
3  n + 2  ≥ 4  6  6 n + 5 ≥ 7 12 3  n  ≥ 6  18
1  n − 3  ≥ 5  8  1 n − 6 ≥ 8 14 9 n + 8 ≥ 10 18
2  n  ≥ 4  8  7 n + 6 ≥ 8 14
4  n + 3  ≥ 5  8  ⋯

Another description of D_{j} detects J-invariant subspaces as subspaces of a certain kernel. Fix V ∈ SG(m, F), containing H = V ∩ JV, and denote the inclusion ι : V ↪ F_{ℝ}. Complexifying (applying ⊗_{ℝ}ℂ to) H, V, and F_{ℝ} extends ι to ι_{ℂ}, and J to J_{ℂ} : F_{ℝ} ⊗ ℂ → F_{ℝ} ⊗ ℂ. F_{ℝ} ⊗ ℂ splits into the +i and −i eigenspaces of J_{ℂ}, as F ⊕ \overline{F}. The “projection onto the holomorphic part” φ : F ⊕ \overline{F} → F is given by the operator (\frac{1}{2} − \frac{i}{2}J_{ℂ}). Since H ⊗ ℂ ↪ V ⊗ ℂ is J_{ℂ}-invariant, it similarly is isomorphic to H_{ℂ} ⊕\overline{H}_{ℂ}, where H_{ℂ} := ι_{ℂ}(V ⊗ℂ)∩F, and \overline{H}_{ℂ} := ι_{ℂ}(V ⊗ ℂ) ∩ \overline{F}. The composite φι_{ℂ} : V ⊗ ℂ → F has kernel \overline{H}_{ℂ}.

Lemma 2.8 V ∈ D_{j} ⟺ some complex j-subspace of V ⊗ ℂ is mapped to {0 ∈ F } by φι_{ℂ}.

Proof: A subspace, regardless of its orientation, is mapped to zero if and only if it is contained in the kernel of the map, \overline{H}_{ℂ}. \overline{H}_{ℂ} has the property dim_{ℝ} V ∩ JV = 2 dim_{ℂ} \overline{H}_{ℂ} ≥ 2j.

.3 Degeneracy Loci and Grassmann Bundles

Now let V be the real oriented tautological m-bundle over SG(m, F), and abbreviate as F the trivial complex n-bundle SG(m, F) × F. Taking all complex j-subspaces of fibers of the complexified bundle V ⊗ ℂ, form the complex grassmann bundle π_{j} : ℂG(j, V ⊗ ℂ) → SG(m, F). Let U^{j} be the tautological j-bundle over ℂG(j, V ⊗ ℂ), and form the pullbacks 𝕍 := π_{j}^{∗}(V ⊗ ℂ) and 𝔽 := π_{j}^{∗}F.

The inclusion of U^{j} into 𝕍, followed by the map φι_{ℂ} on fibers is a bundle map U^{j} → 𝔽, and defines a section s_{j} of Hom(U^{j}, 𝔽). By Lemma 2.7, if this section vanishes at x then π_{j}(x) ∈ D_{j}, and if x ∈ D_{j}, then s_{j} = 0 for some element of π_{j}^{−1}(x). Denote the zero set Div(s_{j}) := {x|s_{j}(x) = 0}; then π_{j}(Div(s_{j})) is called the degeneracy locus of the bundle map φι_{ℂ} : V ⊗ ℂ → F.

Div(s_{j}) ⊆ π_{j}^{−1}D_{j} is a real algebraic subvariety of ℂG(j, V ⊗ ℂ). For x ∈ C_{l}, π_{j}^{−1}(x) ∩ Div(s_{j}) = ℂG(j, l). If Z_{j} := Div(s_{j}) \ π_{j}^{−1}D_{j+1}, then π_{j}|_{Z_{j}} is a diffeomorphism onto C_{j}, and so

codim_{ℝ}Div(s_{j}) = 2j(n − m + j) + 2j(m − j) = 2jn,

as expected for a section of this bundle. The projection of Div(s_{j}) onto D_{j} is a partial desingularization which undoes some of the identifications made in the earlier description of D_{j}.

The above paragraphs and the preceding section generalize to the situation where F is a smooth complex vector bundle over a smooth manifold M. Let F_{ℝ} be the underlying real oriented 2n-bundle, with some Riemannian metric, and μ : SG(m, F) → M the grassmann bundle formed by all the real oriented m-subspaces of fibers (F_{ℝ})_{x}. If the tautological m-bundle over SG(m, F) is ι : V ↪ μ^{∗}F_{ℝ}, then any real oriented m-subbundle T of F_{ℝ} is of the form γ_{T}^{∗} ι : T = γ_{T}^{∗} V ↪ F_{ℝ}, for some smooth section γ_{T} : M → SG(m, F).

Example 3.1 If an m-dimensional manifold is immersed g : M → ℂ^{n}, then the immersion induces a map of real tangent bundles, T M → T ℂ^{n}, and an inclusion of T M in the trivial bundle g^{∗}T ℂ^{n} = M × ℝ^{2n}. Then γ_{T} : M → M × SG(m, 2n) is the graph of the oriented Gauss map of the immersion.

The subsets D_{j} and C_{j} of SG(m, F) are defined in the same way, e. g. D_{j} = {x ∈ SG(m, F)| dim_{ℝ} V_{x} ∩ μ^{∗}J_{x}V_{x} ≥ 2j}, and the codimension formula (1) still applies.

Definition 3.2 The CR-singular set of (j − j_{0})^{th} order, N_{j}, of a subbundle T in F, is the set μ(γ_{T} (M) ∩ D_{j}) ⊆ M. Points of μ(γ_{T} (M) ∩ C_{j_{0}}) are called CR-regular.

If T is the tangent bundle of M and F is the pullback bundle by an immersion M ↪ A of T A, the tangent bundle of the almost complex manifold A, then both points in N_{j_{0}+1} and the subspaces H of fibers of T over such points are called “complex tangents” of the immersion.

Definition 3.3 T is generically included in F if γ_{T} (M) and C_{j} intersect transversely in SG(m, F) for each j.

Lemma 3.4 If T is generically included in F, N_{j} \ N_{j+1} is a smooth submanifold of M, of codimension 2j(n − m + j).

Definition 3.5 T is a CR-subbundle of F of CR codimension m − 2j if γ_{T} (M) ⊆ C_{j}. If j = 0 then T is called totally real, and if j = k = m/2, T is a complex (or anticomplex, on each connected component) j-subbundle of F. A generic CR-subbundle has CR-codimension m−2j_{0}; such subbundles are totally real if m ≤ n, or have totally real orthogonal complement if m ≥ n. If T is the tangent bundle as above, similar adjectives apply to the manifold M.

Example 3.6 If 0 < m < 2n then a generically included subbundle is not complex or anticomplex. Otherwise, if m = 0 or 2n, every subbundle is trivially both generically included and complex (or anticomplex, on connected components). If m = 2n − 1, then j_{0} = k = n − 1, and every subbundle T is a generically included CR-subbundle of CR codimension 1. Every real line subbundle (m = 1) is totally real. If dim_{ℝ} M < 2(n − m + 1), then a generically included T is totally real. In the case T = T M^{m}, the tangent bundle of a real m-manifold, this inequality is similar to bounds in embedding theorems of [Whitney_{44}] and [Haefliger]: if m < \frac{2}{3}(n + 1), then any m-manifold immersed in ℂ^{n} can be perturbed so that there are no complex tangents.

To describe D_{j} as a degeneracy locus inside the grassmann bundle SG(m, F), again form the grassmann bundle π_{j} of complex j-subspaces of the bundle V ⊗ℂ. The tautological complex j-bundle U^{j} over the total space ℂG(j, V ⊗ ℂ) is included in the pullback bundle 𝕍 := π_{j}^{∗}(V ⊗ℂ). This inclusion, composed with the map given on fibers by φι_{ℂ}, is a bundle map U^{j} → 𝔽 := π_{j}^{∗}μ^{∗}F, defining a section s_{j} of the bundle Hom(U^{j}, 𝔽). Div(s_{j}) projects by π_{j} to D_{j}.

\xymatrix{Hom(U^{j}, 𝔽) \ar[d] & V \ar[d] & F \ar[d] \\ ℂG(j, V ⊗ ℂ) \ar[u]^{s_{j}} \ar[r]^{π_{j}} & SG(m, F) \ar[r]^{μ} & M \ar[l]^{γ_{T}}}

Over the set N_{j} \N_{j+1} lies an obvious complex j-bundle, defined at a point x by T_{x} ∩ J_{x}T_{x}, with the complex structure induced by J_{x}. In terms of the above construction, it is a pullback bundle, as follows. γ_{T} |_{N_{j}\N_{j+1}} is a diffeomorphism onto γ_{T} (M) ∩ C_{j}, and π_{j}|_{Z_{j}}^{−1} is a diffeomorphism C_{j} → Z_{j}. Finally, include Z_{j} ↪ ℂG(j, V ⊗ ℂ), and denote the (injective) composition κ_{j} : N_{j} \ N_{j+1} → ℂG(j, V ⊗ℂ). The fiber of U^{j} above an element y of Z_{j} is the j-plane mapped to 0 in 𝔽_{y}, which is, by Lemma 2.7, of the form (\overline{H}_{ℂ})_{y}, where (H_{ℂ})_{y} is isomorphic to the π_{j}^{∗}μ^{∗}J_{y}-invariant real 2j-plane contained in π_{j}^{∗}V_{y}. So, define the complex j-bundle H^{j} over N_{j} \ N_{j+1} by κ_{j}^{∗}\overline{U^{j}}. If m is even, the connected components of H^{k} are complex or anticomplex k-subbundles of components of F |_{N_{k}}.

Remark: At this point, assume M is a smooth compact oriented manifold without boundary, of real dimension d. The orientations on T, F, and T M, the real tangent bundle of M, induce orientations on almost everthing else. At a point x of the grassmann bundle SG(m, F), the tangent space is T_{μ(x)}M ⊕V_{x}^{∗} ⊗ V_{x}^{⊥}, where V is the oriented tautological bundle. At a point y of ℂG(j, V ⊗ ℂ), the tangent space is T_{π_{j}(y)}SG(m, F) ⊕ U_{y}^{j∗} ⊗ (𝕍_{y}/U_{y}^{j}). The tangent space to Div(s_{j}) is oriented so that Hom(U^{j}, 𝔽)|_{Div(s_{j})} ⊕ T Div(s_{j}) = T ℂG(j, V ⊗ ℂ)|_{Div(s_{j})}. This orientation induces orientations for the submanifolds C_{j} and N_{j} \ N_{j+1}.

Remark: In the generic case, the normal bundle of the submanifold N_{j} \ N_{j+1} is isomorphic to Hom(ker φι_{ℂ}, coker φι_{ℂ}) ≅ H̄^{j} ⊗_{ℂ} H_{⊥}^{n−m+j}, where H_{⊥} is the complex tangent space T^{⊥} ∩ JT^{⊥} of the bundle normal to T in F. ([GG], p. 145) This description of the normal bundle is another geometric feature of the codimension formula (1). A complex structure on ν(N_{1} \ N_{2}) was observed in the m = n case in [W_{85}].

Remark: If X is connected without boundary, but not orientable, then it admits a connected, orientable two-fold cover X^{o} → X. CR singularities of a bundle T over X induce CR singularities when T and F are pulled back to X^{o}.

It will frequently be the case that CR-singularities of high order are to be avoided; this is denoted by “N_{j+1} = ∅.” This condition may follow from the hypothesis “T generic,” (for large j), and otherwise is not inconsistent with genericity. If T is generically included in F and N_{j+1} = ∅, then γ_{T} (M) and C_{l} intersect transversely in SG(m, F) for each l ≤ j, and γ_{T} (M) ∩ D_{j+1} = ∅.

Example 3.7 If d < 2(j +1)(n−m+j +1), then N_{j+1} = ∅ for T generically included in F. If N_{j+1} = ∅ and T is generic, then N_{j} is a closed submanifold of M. Example 3.6 is a special case of this phenomenon.

.4 Determinantal Formulas

Definition 4.1 For two complex vector bundles R, S over a manifold M, let c(R − S) denote the quotient of total chern classes cR(cS)^{−1} in the cohomology ring H^{∗}(M; ℂ), with c_{i}(R − S) the degree 2i part.

Example 4.2 Abbreviate F = μ^{∗}F over SG(m, F). Then

c(F − V ⊗ ℂ)

:= cF (c(V ⊗ ℂ))^{−1}  (2)
≡ (∑_{a=0}^{n} c_{a}F)(∑_{b=0}^{k}(−1)^{b}p_{b}V)^{−1}  (3)

= (∑_{a=0}^{n} c_{a}F)(∑_{i=0}^{∞}(∑_{b=1}^{k}(−1)^{b+1}p_{b}V)^{i})  (4)
= 1 + c_{1}F + (c_{2}F + p_{1}V) + (c_{3}F + c_{1}F p_{1}V) +  (5)

(c_{4}F + c_{2}F p_{1}V − p_{2}V + p_{1}^{2}V) +

(c_{5}F + c_{3}F p_{1}V − c_{1}F p_{2}V + c_{1}F p_{1}^{2}V) +

(c_{6}F + c_{4}F p_{1}V − c_{2}F p_{2}V + c_{2}F p_{1}^{2}V + p_{3}V − 2p_{1}V p_{2}V + p_{1}^{3}V) + …

(3) is the definition of pontrjagin classes. (4) is a formal power series expansion. The terms of (5) are grouped by total (even) degree.

.4.1 Formulas for isolated complex tangents

A determinantal formula developed over a long history by Giambelli, Thom, and [Porteous] about degeneracy loci of bundle maps applies to the geometry of complex tangents as developed previously. Both of the following propositions are from [Porteous]:

Proposition 4.3 The equation of degree 2jn cohomology classes

[Div(s_{j})] = c_{jn}Hom(U^{j}, 𝔽)  (6)

pushes forward by π_{j} to the equation

π_{j∗}[Div(s_{j})] = Δ_{n−m+j}^{(j)}(c(μ^{∗}F − V ⊗ ℂ))  (7)
:= det(c_{n−m+j+a−b}(μ^{∗}F − V ⊗ ℂ))_{1≤a,b≤j},  (8)

of degree 2j(n − m + j) cohomology classes.

Lemma 4.4 π_{j∗}[Div(s_{j})]|_{SG(m,F)\D_{j+1}} = [D_{j}]|_{SG(m,F)\D_{j+1}}.

Theorem 4.5 If d = dim_{ℝ} M = 2j(n−m+j), and T is generically included in F, then

∑_{x∈N_{j}} ind(x) = ∫_{M} Δ_{n−m+j}^{(j)}(c(F − T ⊗ ℂ)),  (9)

where ind(x) is the oriented intersection number of γ_{T} (M) and C_{j} at γ_{T} (x).

Proof: This equation is the integral of the product of both sides of (7) with [γ_{T} (M)]. The chern and pontrjagin classes on the RHS pull back to M by functoriality. The hypothesis T generic assures that γ_{T} (M) is transverse to C_{j}, the smooth part of the support of π_{j∗}[Div(s_{j})], and the codimension formula (1) implies the intersection is only in isolated points of C_{j}.

Example 4.6 ([W_{86}]) Consider the situation where T is a real 2-subbundle of a complex 2-bundle F over a smooth compact oriented surface M. (This will be described by d = m = n = 2.) If T is generically included in F, the set N_{1} of points x where the fiber T_{x} is complex or anticomplex is a codimension 2 submanifold, i.e., finite. Formula (9) then reads:

∑_{x∈N_{1}} ind(x) = ∫_{M} c_{1}F.  (10)

In fact, this holds for d = 2, m = n ≥ 2. The local geometry of this index when T = T M is also considered by [IO].

Example 4.7 In the case d = m = 4, n = 5, [Domrin] calculated the index sum

∑_{x∈N_{1}} ind(x) = ∫_{M} c_{2}F + p_{1}T.

For example, the complex projective plane can be embedded in ℂ^{3} ⊕ ℝ^{3} by the map P : [z_{1} : z_{2} : z_{3}] ↦ \frac{1}{N} (z_{2}z̄_{3}, z̄_{1}z_{3}, z_{1}z̄_{2}, z_{1}z̄_{1}, z_{2}z̄_{2}, z_{3}z̄_{3}), with N(z_{1}, z_{2}, z_{3}) = |z_{1}|^{2} + |z_{2}|^{2} + |z_{3}|^{2}. If the target space ℝ^{9} is included in ℂ^{5}, the image ℂP^{2}, or generic smooth perturbations of it, has isolated points at which the tangent space contains a complex line. The index sum is p_{1}ℂP^{2} = 3.

Example 4.8 m = n ≥ 4, j = 2, d = 8. Abbreviate by c_{i} the 2i-degree part of c(F − T ⊗ ℂ), corresponding to the i^{th} term of (5).

∑_{x∈N_{2}} ind(x) = ∫_{M} | \begin{matrix}c_{2} & c_{1} \\ c_{3} & c_{2} \end{matrix} |  (11)

= ∫_{M}(c_{2}F + p_{1}T)^{2} − c_{1}F (c_{3}F + c_{1}F p_{1}T).  (12)

In particular, if F is trivial, as in the case where M is immersed in ℂ^{8}, T = T M, and F is the pullback by the immersion of T ℂ^{8}, then ([Lai])

∑_{x∈N_{2}} ind(x) = ∫_{M} p_{1}^{2}T.  (13)

Example 4.9 m − n + 1 = j = 2l > 0, d = 2j.

∑_{x∈N_{j}} ind(x) = ∫_{M} Δ_{1}^{(j)}(c(F − T ⊗ ℂ)),  (14)

This can be a complicated polynomial. One of several useful determinantal identities in [Fulton_{84}] applies,

Δ_{a}^{(b)}(c(R − S)) = (−1)^{ab}Δ_{b}^{(a)}(c(S − R)),

so the characteristic class can be rewritten Δ_{j}^{(1)}(T ⊗ ℂ − F), which simplifies to (−1)^{l}p_{l}T if F is trivial. Equation (14) becomes this corollary of Theorem 4.5: Let M be a 4l-dimensional manifold, and let T be a generic (n + 2l − 1)- subbundle of the trivial complex n-bundle. Then (note j_{0} = 2l − 1, so N_{2l} is the first CR-singular set)

∑_{x∈N_{2l}} ind(x) = (−1)^{l} ∫_{M} p_{l}T,  (15)

and, letting n = 2l+1, and T = T M yields a formula ([HL_{93}]) about 2l-complex and anticomplex tangencies of generic immersions M^{4l} → ℂ^{2l+1}:

∑_{x∈N_{2l}} ind(x) = (−1)^{l} ∫_{M} p_{l}M.  (16)

Example 4.10 m = 2, j = 1, d = 2(n − 1) ≥ 0.

∑_{x∈N_{1}} ind(x) = ∫_{M} Δ_{n−1}^{(1)}(c(F − T ⊗ ℂ)) = ∫_{M} c_{n−1}(F − T ⊗ ℂ)  (17)

= ∫_{M}((cF)(1 − p_{1}T)^{−1})_{2n−2} = ∫_{M} ∑_{b=0}^{∞} c_{n−1−2b}F (p_{1}T)^{b}.(18)

Note that Example 4.6 is the special case n = 2. If T = νM, the normal 2bundle of a generic immersion of a 2(n − 1)-manifold M in an almost complex n-manifold, and F is the restriction to M of the ambient tangent bundle, then p_{1}νM = Ω̃^{2}, where Ω̃ is the euler class of the normal bundle. In this scenario,

(18) is a theorem of [Lai]:

∑_{x∈N_{1}} ind(x) = ∫_{M} ∑_{b=0}^{∞} Ω̃^{2b}c_{n−1−2b}F.  (19)

If n is even and F is trivial, then (RHS 18) = 0. If n = 2l + 1, and F is trivial, such as when M is generically immersed in ℂ^{n}, then (19) becomes

∑_{x∈N_{1}} ind(x) = ∫_{M} Ω̃^{2l}.  (20)

For a given immersion of M, equations (16) and (20) give the same information. The gauss maps of the tangent and normal bundle are related by γ_{T M} =⊥ ∘γ_{νM}; by Example 2.6, the CR-singular sets are the same, and so are the intersection indices. Since pT MpνM = pℂ^{2l+1} = 1, pT M = (1 + p_{1}νM)^{−1} ⇒ p_{l}T M = (−1)^{l}(p_{1}νM)^{l}.

More generally, If T^{m} is generic in F^{n}, then so is T^{⊥}; the set N_{j} of CRsingularities of T is the same set as the N_{n−m+j} corresponding to T^{⊥} in F, and either sum of indices is given by Theorem 4.5. The equality of the chern numbers is also a consequence of determinantal identities:

∫_{M} Δ_{j}^{(n−m+j)}(c(F − T^{⊥} ⊗ ℂ)) = ∫_{M} Δ_{j}^{(n−m+j)} (\frac{cF}{c((F/T) ⊗ ℂ)})

= ∫_{M} Δ_{j}^{(n−m+j)} (\frac{cF}{cF cF̄c(T ⊗ ℂ)^{−1}} ) = ∫_{M} Δ_{j}^{(n−m+j)}(c(T ⊗ ℂ − F̄))

= (−1)^{j(n−m+j)} ∫_{M} Δ_{j}^{(n−m+j)}(c(T ⊗ ℂ − F)) = ∫_{M} Δ_{n−m+j}^{(j)}(c(F − T ⊗ ℂ)).

Example 4.11 In the case of an oriented four-manifold M generically immersed in an almost complex 3-manifold A, the characteristic classes of T = T M, ν = T^{⊥}, and F = T A|_{M} are related by p_{1}T + p_{1}ν = c_{1}^{2}F − 2c_{2}F. The sum of the indices of complex tangents for T is ∫_{M} c_{1}^{2}F − c_{2}F − p_{1}T, and for ν is ∫_{M} c_{2}F + p_{1}ν. These numbers are equal, and zero if M is a CR submanifold of A. For any such M^{4}, the pontrjagin number ∫_{M} p_{1}T is three times the signature ([Hirzebruch]), and if A = ℂ^{3}, then ∫_{M} p_{1}ν is three times the algebraic number of triple points of the immersion ([Herbert]). If M is isotopic to a complex submanifold Y, the chern classes of Y pull back to M; adjunction formulas for almost complex hypersurfaces can be used ([GH], p. 601). For example, if Y is a smooth, degree d complex hypersurface in A = ℂP^{3}, and H is the hyperplane class in the cohomology of ℂP^{3}, c_{2}(T ℂP^{3}|_{Y}) = 6H^{2}, and c_{1}T^{⊥}Y = dH. Since ∫_{Y} H^{2} = d, the chern number ∫_{Y} c_{2}(T ℂP^{3}|_{Y}) + (c_{1}T^{⊥}Y)^{2} is 6d + d^{3}, which equals ∫_{M} c_{2}F +p_{1}ν if M is isotopic to Y. This number is always positive; there are no CR submanifolds of ℂP^{3} isotopic to smooth complex hypersurfaces. In particular, a generically immersed submanifold isotopic to a complex hyperplane ℂP^{2} has seven complex tangents, counted with multiplicity.

.4.2 Formulas for non-isolated complex tangents

A procedure for finding the chern numbers of a kernel bundle was given by [HT]. Later, [Pragacz] gave a closed formula in terms of the segre class s(F −T ⊗ℂ) = 1 + s_{1} + s_{2} + …, where the first few terms of the segre class are as follows:

s = (1 − c_{1} + c_{2} − c_{3} + c_{4} − c_{5} + …)^{−1}
= 1 + c_{1} + (c_{1}^{2} − c_{2})
+(c_{1}^{3} − 2c_{1}c_{2} + c_{3})
+(c_{1}^{4} − 3c_{1}^{2}c_{2} + 2c_{1}c_{3} + c_{2}^{2} − c_{4}) + …
= 1 + c_{1}F + (c_{1}^{2}F − c_{2}F − p_{1}T)
+(c_{1}^{3}F − 2c_{1}F c_{2}F + c_{3}F − c_{1}F p_{1}T)

+(c_{1}^{4}F − 3c_{1}^{2}F c_{2}F + 2c_{1}F c_{3}F + c_{2}^{2}F − c_{4}F − c_{1}^{2}F p_{1}T + c_{2}F p_{1}T + p_{2}T) + ….

The segre classes will appear in determinants labeled with a sequence of integers J = (j_{1}, j_{2}, …, j_{r}), with

Δ_{J}(s(R − S)) = det(s_{j_{p}−p+q})_{1≤p,q≤r}.

Theorem 4.12 If T is generic in F → M, N_{j+1} = ∅, and the product of chern classes ∏ c_{i}(K)^{β_{i}} is expressed as a sum with nonnegative integer coefficients ∏ c_{i}(K)^{β_{i}} = ∑ m_{J}Δ_{J}(s(K)), where ∑ iβ_{i} = \frac{1}{2} dim_{ℝ}(N_{j}) = d, then

∫_{N_{j}} ∏_{i} c_{i}(H)^{β_{i}} = ∫_{M} ∑_{J} m_{J}Δ_{j^{n−m+j},J̃}(s(F − T ⊗ ℂ)),

where J̃ denotes the conjugate partition (i_{1}, i_{2}, …), i_{a} = card{h : j_{h} ≥ a}.

Proof: Recall that M is assumed smooth, compact, and oriented, and by Example 3.7, N_{j} is a closed, oriented submanifold. The formula follows from the equality of cohomology classes ([Pragacz] Lemma 5.1 and Prop. 5.3)

Δ_{J}(s(−K))[N_{j}] = Δ_{j^{n−m+j},J}(s(F − T ⊗ ℂ)),

which by a determinantal identity implies

Δ_{J}(s(K))[N_{j}] = (−1)^{d}Δ_{j^{n−m+j},J̃}(s(F − T ⊗ ℂ)),

where K is the kernel bundle over N_{j} of the map T ⊗ ℂ → F. The sign (−1)^{d} is cancelled when writing formulas for the chern number of H = K̄.

The segre classes are sometimes more natural than the chern classes in enumerative constructions such as these. The determinants in this formula could immediately be rewritten in terms of chern classes using the determinantal identities Δ_{I}(s(E − F)) = (−1)^{|I|}ΔĨ(s(F − E)) = ΔĨ(c(E − F)).

Example 4.13 For T generic, d = 12, j = 2, m = n ≥ 4, the characteristic numbers are expressed as:

∫_{N_{2}} c_{2}H^{2} = ∫_{M} | \begin{matrix}s_{2} & s_{3} & s_{4} \\ s_{1} & s_{2} & s_{3} \\ 1 & s_{1} & s_{2} \end{matrix} | = ∫_{M} c_{3}^{2} − c_{2}c_{4},

∫_{N_{2}} c_{1}^{2}H^{2} = ∫_{M} | \begin{matrix}s_{2} & s_{3} & s_{4} \\ s_{1} & s_{2} & s_{3} \\ 1 & s_{1} & s_{2} \end{matrix} | + | \begin{matrix}s_{2} & s_{3} & s_{4} & s_{5} \\ s_{1} & s_{2} & s_{3} & s_{4} \\ 0 & 1 & s_{1} & s_{2} \\ 0 & 0 & 1 & s_{1} \end{matrix} | = ∫_{M} c_{3}^{2} − c_{1}c_{5}.

These are two possibly different obstructions to N_{2} = ∅.

Example 4.14 d = 2j(n + q − m + j), q ≥ 0: suppose T is a generic subbundle of F such that N_{j+1} = ∅. To find a power of the top chern class of H^{j}, the determinantal formula can be written in terms of chern classes,

∫_{N_{j}} c_{j}^{q}H^{j} = ∫_{M} Δ_{n+q−m+j}^{(j)}(c(F − T ⊗ ℂ)),  (21)

and the special case q = 0 is Theorem 4.5.

Example 4.15 j = 1, m = n = 4, q = 1, d = 4, T generic ([W_{86}])

∫_{N_{1}} c_{1}H^{1} = ∫_{M} Δ_{2}^{(1)}(c(F − T ⊗ ℂ)) = ∫_{M} c_{2}F + p_{1}T.

More generally, for T generic, j = 1, m = n ≥ 2, d = 2(q + 1), and N_{2} = ∅ (which is implied by generic for d = 4 or 6),

∫_{N_{1}} c_{1}^{q}H^{1} = ∫_{M} Δ_{q+1}^{(1)}(c(F − T ⊗ ℂ)) = ∫_{M} c_{q+1}(F − T ⊗ ℂ).

For example, [Whitney_{44}] showed the embedding of ℂP^{2} from Example 4.7 can be composed with a projection resulting in an embedding into ℝ^{7}. Generic perturbations of this map give a 4-manifold in ℂ^{4} such that N_{1} is a smooth surface and ∫_{N_{1}} c_{1}H^{1} = p_{1}ℂP^{2} = 3.

Example 4.16 Suppose T is a totally real n-subbundle of the trivial bundle F = ℂ^{n}. Then all its pontrjagin classes except p_{0} are zero. This follows directly from the definition of pontrjagin classes since F and T ⊗ ℂ are isomorphic, but is also a consequence of the formula. γ_{T} (M) does not intersect the support of π_{1∗}([Div(s_{1})]c_{1}^{q}U^{1∗}), so

0 = Δ_{q+1}^{(1)}(c(F − T ⊗ ℂ)) = (c(T ⊗ ℂ)^{−1})_{q+1}

for q ≥ 0, so pT = (1 + 0 + 0 + …)^{−1} = 1. This was noticed by Kobayashi and [Wells] in the T = T M case. The same conclusion holds if T is a totally real n − 1-subbundle of ℂ^{n}; in this case 0 = Δ_{q+2}^{(1)}(c(F − T ⊗ ℂ)), and c_{1} = 0 by hypothesis. This also gives a result of [Lai]: if T is a CR-regular n+1-subbundle of ℂ^{n}, then T^{⊥} is a totally real n − 1-subbundle, and pℂ^{n} = pT pT^{⊥} = pT = 1.

Example 4.17 Suppose T is a generic CR-subbundle of F, with m > n, so F_{ℝ} decomposes as a direct sum of smooth subbundles T^{⊥} ⊕ JT^{⊥} ⊕ H^{m−n}. The chern numbers of H can be calculated in terms of cF and pT using the sum formula

cF = c(T^{⊥} ⊗ ℂ)cH,

and the kernel bundle formula gives the same result when applied to N_{m−n} = M:

∏ c_{i}^{β_{i}}H = ∑ m_{J}ΔJ̃(s(F − T ⊗ ℂ))
= ∑ m_{J}Δ_{J}(c(F − T ⊗ ℂ))
= ∑ m_{J}Δ_{J}(s(F − T^{⊥} ⊗ ℂ))
= ∏ c_{i}^{β_{i}}(F − T^{⊥} ⊗ ℂ).

In the case where T is the normal bundle of a totally real submanifold, the topology of H is considered in [Forstnerič].

.4.3 Flags of subbundles

As another generalization of Theorem 4.5, a “flag of real subbundles” inside the complex bundle F is considered. To fix notation, t ≥ 1 subspaces (or subbundles) are indexed:

T_{1}^{m_{1}} ⊆ T_{2}^{m_{2}} ⊆ … ⊆ T_{t}^{m_{t}} ⊆ F^{n},

with 0 ≤ m_{1} ≤ m_{2} ≤ … ≤ m_{r} ≤ 2n. The interesting loci of complex tangents are indexed by sequences of j_{i} = dim_{ℂ} T_{i} ∩ JT_{i}:

0 < j_{1} < j_{2} < … < j_{t}  (22)
m_{1} − j_{1} < m_{2} − j_{2} < … < m_{t} − j_{t} < n.  (23)

Flags with (at least) j_{i} complex directions in T_{i}, 1 ≤ i ≤ t, form a locus D_{(j_{1}...j_{t})} in the (partial) flag manifold SF l(m_{1}, …, m_{n}, F) of oriented real subspaces of F_{ℝ}^{2n}.

It is not too restrictive to consider strictly increasing indices j_{i} and m_{i} −j_{i}— consider T_{1}^{m_{1}} ⊆ T_{2}^{m_{2}} ⊆ F^{n} with m_{2} ≤ n. If T_{1} has exactly j_{1} complex directions, any T_{2} will have at least that many, and generically will have an equal number, j_{2} = j_{1}. So, the locus in SF l where 0 < j_{1} ≤ j_{2} is the inverse image of D_{j_{1}} ⊆ SG(m_{1}, F) under the projection forgetting T_{2}, and enumerative questions about the locus D_{(j_{1}j_{2})} reduce to the single subbundle case if j_{2} = j_{1}. Similarly, if n ≤ m_{1}, the number of complex directions in the normal space, n − m_{i} + j_{i}, generically does not increase for i = 1, 2, so the configurations which do not reduce to the grassmannian case are those with m_{i} − j_{1} < m_{2} − j_{2}.

The formula for the real codimension of the locus D_{(j_{1}...j_{t})} in SF l is

2j_{1}(n − m_{1} + j_{1}) + 2(j_{2} − j_{1})(n − m_{2} + j_{2}) + … + 2(j_{t} − j_{t−1})(n − m_{t} + j_{t}).

Note that this reduces to formula (1) when t = 1, and that each term is non-zero by (22), (23). The proof of the special case t = 2 illustrates the general idea.

Example 4.18 SF l(m_{1}, m_{2}, F^{n}) is a fiber space with base SG(m_{1}, F), the oriented m_{1}-planes in F_{ℝ}^{2n}. The m_{2}-planes containing a fixed T_{1}^{m_{1}} form the fiber over T_{1}, a grassmannian of quotients of m_{2} − m_{1} dimensions inside F/T_{1} The total dimension of SF l is m_{1}(2n − m_{1}) + (m_{2} − m_{1})((2n − m_{1}) − (m_{2} − m_{1})). Those T_{1} that contain any complex j_{1}-dimensional subspace form a codimension 2j_{1}(n−m_{1} +j_{1}) locus in the base space SG(m_{1}, F), and so a dimension 2m_{1}n− m_{1}^{2} − 2j_{1}n + 2j_{1}m_{1} − 2j_{1}^{2} base variety. Those H_{2}^{j_{2}} that contain the complex subspace H_{1}^{j_{1}} form a fiber with real dimension 2(j_{2} − j_{1})(n − j_{1} − (j_{2} − j_{1})). Generically, H_{2} and T_{1} intersect only in H_{1} and the real dimension of their sum (H_{2} + T_{1} as real vector subspaces) is m_{1} + 2j_{2} − 2j_{1}. The space of T_{2}^{m_{2}} that contain both T_{1} and H_{2} forms a fiber of real dimension (m_{2} − (m_{1} + 2j_{2} − 2j_{1}))(2n−(m_{1} +2j_{2} −2j_{1})−(m_{2} −(m_{1} +2j_{2} −2j_{1}))). The total parameter count for the space D_{(j_{1}j_{2})} is 2j_{1}(n − m_{1} + j_{1}) + 2(j_{2} − j_{1})(n − m_{2} + j_{2}) less than the dimension of SF l. This description of D_{(j_{1}j_{2})} omits some identifications that must be made when some of the subspaces intersect non-transversely, causing singularities in the locus but not decreasing the dimension.

A flag of subbundles over M is “generic” if its image under the Gauss map to the flag bundle SF l(m_{1}, …, m_{t}, F) → M meets the smooth parts of the loci D_{j} transversely for each j satisfying (22), (23). The locus in M mapped to D_{j} will be called N_{j}. Define the partition

μ = ((n − m_{1} + j_{1})^{j_{1}}, (n − m_{2} + j_{2})^{j_{2}−j_{1}}, …, (n − m_{t} + j_{t})^{j_{t}−j_{t−1}}),

and, for 1 ≤ a ≤ j_{t}, let ρ(a) = min{s : 1 ≤ s ≤ t, a ≤ j_{s}}.

Theorem 4.19 If T_{1} ⊆ … ⊆ T_{t} ⊆ F is a generic flag of subbundles, then N_{j} has codimension 2|μ| = 2j_{1}(n − m_{1} + j_{1}) + … + 2(j_{t} − j_{t−1})(n − m_{t} + j_{t}), and the current defined by N_{j} has cohomology class

[N_{j}] = det(c_{μ_{a}−a+b}(F − T_{ρ(a)} ⊗ ℂ))_{1≤a,b≤j_{t}}.

The formula follows from [Fulton_{91}]; the real degree of the cohomology class is 2|μ|, and the dimension count in the universal flag bundle is necessary to show that generically N_{j} is a subset with a smooth locus of codimension 2|μ|.

Example 4.20 The simplest example with t = 2 is j_{1} = 1, j_{2} = 2, m_{1} = 2, m_{2} = 4, n = 3. In this case, F^{3} is a complex 3-plane bundle, containing T_{2}^{4} as a real subbundle, which in turn contains T_{1}^{2}. T_{1}^{2} is generically totally real in F^{3}, but is a complex line along a locus of real codimension 4. T_{2}^{4} generically contains a complex line, but is a complex 2-plane along a locus of real codimension 4. T_{1}^{2} and T_{2}^{4} are simultaneously complex along the locus N_{(12)}, which generically has codimension 2 · 1 · (3 − 2 + 1) + 2 · (2 − 1) · (3 − 4 + 2) = 6.

The formula uses the partition μ = (2, 1) and the function ρ(1) = 1, ρ(2) = 2:

[N_{(12)}] = | \begin{matrix}c_{2}(F − T_{1} ⊗ ℂ) & c_{3}(F − T_{1} ⊗ ℂ) \\ 1 & c_{1}(F − T_{2} ⊗ ℂ) \end{matrix} |
= c_{1}F (c_{2}F + p_{1}T_{1}) − (c_{3}F + c_{1}F p_{1}T_{1}) = c_{1}F c_{2}F − c_{3}F.

Consider M = ℂP^{1} ×ℂP^{1} ×ℂP^{1} as a real 6-manifold, with complex tangent bundle F^{3}. If p_{1} : M → ℂP^{1} is projection onto the first factor, and p_{2} : M → ℂP^{1} × ℂP^{1} is projection onto the first two factors, then p_{1}^{∗}T ℂP^{1} ⊆ p_{2}^{∗}T (ℂP^{1} × ℂP^{1}) ⊆ F^{3} is a flag of complex subbundles tangent to the product foliations of M. A real perturbation of the subbundles (or the foliations) can give a generic flag of real subbundles T_{1}^{2} ⊆ T_{2}^{4} ⊆ F^{3}, with N_{(12)} expected to be isolated. Denoting by a, b, c the generators of the cohomology of each of the three factors ℂP^{1} pulled back to M, the total chern class of F is (1+2a)(1+2b)(1+2c). The class c_{1}F c_{2}F − c_{3}F is 16abc, so the index sum of the isolated complex tangents of T_{1} ⊆ T_{2} ⊆ F is 16.

.5 Coincidence as CR Singularities of a Graph

A smooth real-linear map α between complex vector bundles T^{r} = (T_{ℝ}, J^{T}) and F^{n} = (F_{ℝ}, J^{F}) of real dimension 2r = m and 2n over M may be complex-linear on some vectors v⃗_{x} ∈ T_{x}, meaning α_{x}(J_{x}^{T} v⃗_{x}) = J_{x}^{F} α_{x}(v⃗_{x}). In the fiber T_{x}, the set of vectors where α_{x} is complex-linear is a complex subspace K_{x}, and the subset of points x ∈ M where dim_{ℂ} K_{x} is at least j defines the “coincidence locus” Q_{j}.

The map α : T_{ℝ} → F_{ℝ} defines its graph \underline{α} as a real-linear inclusion of the image \underline{α}T in T_{ℝ} ⊕ F_{ℝ} of the map v⃗ ↦ (v⃗, α(v⃗)). T_{ℝ} ⊕ F_{ℝ} has the direct sum complex structure, J_{x}^{⊕}(v⃗, w⃗) = (J_{x}^{T} v⃗, J_{x}^{F} w⃗). Denote T ⊕ F = (T_{ℝ} ⊕ F_{ℝ}, J^{⊕}).

Lemma 5.1 K_{x} ≅ \underline{α}T_{x} ∩ J_{x}^{⊕}\underline{α}T_{x}.

Proof: The claim is that \underline{α}_{x} is a ℂ-linear isomorphism when restricted to K_{x}, and that its image is the maximal J_{x}^{⊕}-complex subspace of \underline{α}T_{x}.

v⃗ ∈ K ⟺ (J^{T} v⃗, J^{F} αv⃗) = (J^{T} v⃗, αJ^{T} v⃗) ⟺ J^{⊕}(v⃗, αv⃗) ∈ \underline{α}T
⟺ (v⃗, αv⃗) ∈ \underline{α}T ∩ J^{⊕}\underline{α}T.

So, a map α such that \underline{α}T is a generically included subbundle of real rank m = 2r in the complex bundle T ⊕ F of complex rank n + r has coincidence loci Q_{j} \ Q_{j+1} of real codimension 2j((n + r) − 2r + j) in M, corresponding to the complex tangent loci of \underline{α}T. The following corollary of Theorem 4.5 describes the the coincidence locus in terms of the chern classes of (T, J^{T}) and (F, J^{F}):

Corollary 5.2

[Q_{j}] = [N_{j}] = Δ_{(n+r)−2r+j}^{(j)}(c(T ⊕ F − T ⊗ ℂ))
= Δ_{n−r+j}^{(j)}(\frac{cT cF}{cT cT̄}) = Δ_{n−r+j}^{(j)}(c(F − T̄)).

The relationship between complex coincidence and the CR structure of the graph seems to be well-known, but not formulated as explicitly as this in the literature. (cf [Freeman] and §4.2, [Chirka])

Example 5.3 If T is the conjugate bundle of F, i.e., T_{ℝ} = F_{ℝ} and J^{T} = −J^{F}, then the identity map is complex-linear on no vector— Q_{j} = ∅ for j > 0. The graph of the identity map is a totally real inclusion and c(F − T̄) = 1 + 0 + 0 + . . ..

Example 5.4 Two complex structures J^{T} and J^{F} on the same real vector bundle are relatively generic if the graph of the identity map is a generic inclusion in the sum. Invariants for such a pair were calculated in [HL_{93}] and [HL_{95}]:

[Q_{j}] = Δ_{j}^{(j)}(c(F − T̄)).

Example 5.5 The graph of a smooth map f : M → A (not necessarily an immersion) defines an embedding \underline{f} : M → M × A. The coincidence locus Q_{j} of df is the same as the locus N_{j} of complex tangents of the image of d\underline{f} = \underline{df} in T (M × A).

Example 5.6 ([EW]) If f is a smooth map between connected, compact, oriented Riemannian surfaces, f : (M, g_{M}) → (A, g_{A}), the degree of f is an

Figure 1: A smooth map ℂP^{1} → ℂ, conformal at two points

integer. Giving M and A complex structures compatible with the metrics, and using Webster’s formula (Example 4.6) for the image of the graph of f in (M × A, J^{⊕}),

∑_{x∈N_{1}} ind(x) = ∫_{M} c_{1}(T M ⊕ f^{∗}T A) = χM + (deg f)χA.

Reversing the orientation and complex structure on A changes the index sum to χM −(deg f)χA, and similarly, reversing the orientation and complex structure on M gives index sum −χM + (deg f)χA. The sign of the index differs from the [EW] formulas, which use a bundle map T^{1,0}M → T^{0,1}A instead of the projection onto the (1, 0) part as in Lemma 2.7.

Example 5.7 As an example of a smooth map between Riemann surfaces, the unit sphere in ℝ^{3} can be projected radially onto the x, y-plane from the point (0, 0, 2), giving a degree zero map with one (index ±2) point each of direct and indirect conformality. Varying the point of projection along the line {(r, 0, 2)} projects the sphere onto a region whose boundary is an ellipse. Not too surprisingly, the map is conformal at exactly those points projected onto the foci of the ellipse. In the figure, the lines connecting the foci to the vertex of the cone meet the sphere at four points, two each of direct and indirect conformality. Two of the four points are the poles (0, 0, ±1). The minor semiaxis of the ellipse, parallel to the y-axis, has constant length 2/\sqrt{3}. The major semiaxis has length \frac{2}{3} \sqrt{r^{2} + 3}, and the foci are at r/3 and −r. It is a well-known theorem that, in this case, the ellipses defined by the intersections of the planes z = 1 and z = −1 with the cone each have a focus at a pole of the sphere, and that they are similar (and so conformal) to any parallel ellipse on the cone.

Example 5.8 If M and A are complex manifolds, with real dimensions 4 and 6, and complex structures (T M, J^{T}), (T A, J^{A}), a smooth map f : M → A generically will satisfy df ∘ J^{T} = J^{A} ∘ df only on complex lines tangent to a discrete set Q_{1}. The graph of f is an embedding of a real 4-manifold in M ×A, a complex 5-manifold, and if f is real-analytic, so is the image f(M). By Example 4.7, the index sum of Q_{1} is

∑_{x∈Q_{1}} ind(x) = ∫_{M} p_{1}M + c_{2}(T M ⊕ f^{∗}T A)

= ∫_{M} c_{1}^{2}M − c_{2}M + c_{1}Mf^{∗}c_{1}A + f^{∗}c_{2}A.

This example shows that the geometry of a generic map f from a complex surface to a complex 3-manifold reduces to the study of isolated complex tangents in the graph of f.

Bibliography

[Aigner] M. Aigner, Combinatorial Theory, GMW 234, Springer-Verlag, New York, 1979.

[Banchoff-Farris] T. Banchoff and F. Farris, Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space, Pacific Journal of Math. (1) 161 (1993), 1–24.

[BC] R. Benedetti and P. Cragnolini, On simultaneous diagonalization of one hermitian and one symmetric form, Linear Algebra and Appl., 57 (1984), 215-226.

[Bishop] E. Bishop, Differentiable manifolds in complex euclidean space, Duke Math. J. (1) 32 (1965), 1–21.

[Boggess] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, 1991.

[Chen-Morvan] B.-Y. Chen and J.-M. Morvan, A cohomology class for totally real surfaces in ℂ^{2}, Japan. J. Math. (1) 21 (1995), 189–205.

[Chern-Spanier] S.-S. Chern and E. H. Spanier, A theorem on orientable surfaces in four-dimensional space, Comment. Math. Helv. 25 (1951), 205–209.

[Chirka] E. M. Chirka, Introduction to the geometry of CRmanifolds, Russian Math. Surveys (1) 46 (1991), 95–197.

[CLO] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.

[Domrin] A. V. Domrin, The number of ℝℂ-singular points on a 4dimensional submanifold in a 5-dimensional complex manifold, Mathematical Notes, (1–2) 57 (1995), 167–170.

[EW] J. Eells and J. C. Wood, Restrictions on harmonic maps of surfaces, Topology (3) 15 (1976), 263–266.

[Forstnerič] F. Forstnerič, Approximation by automorphisms on smooth submanifolds of ℂ^{n}, Math. Ann. (4) 300 (1994), 719– 738.

[Freeman] M. Freeman, Local complex foliation of real submanifolds, Math. Ann. (1) 209 (1974), 1–30.

[Fulton_{84}] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer-Verlag, Berlin, 1984.

[Fulton_{91}] , Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. (3) 65 (1991), 381–420.

[GG] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York, GTM 14, 1973.

[GF] H. Grauert and K. Fritzsche, Several Complex Variables, GTM 38, Springer-Verlag, New York, 1976.

[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.

[Haefliger] A. Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47–82.

[HT] J. Harris and L. Tu, Chern numbers of kernel and cokernel bundles, Invent. Math. (3) 75 (1984), 467–475.

[HL_{93}] F. R. Harvey and H. B. Lawson, Jr., A Theory of Characteristic Currents Associated with a Singular Connection, Astérisque 213, 1993.

[HL_{95}] , Geometric Residue Theorems, American Journal of Mathematics (4) 117 (1995), 829–873.

[Herbert] R. J. Herbert, Multiple Points of Immersed Manifolds, Memoirs AMS (250) 34, 1981.

[Hirzebruch] F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, Berlin, 1978.

[IO] G. Ishikawa and T. Ohmoto, Local invariants of singular surfaces in an almost complex four-manifold, Annals of Global Analysis and Geometry (2) 11 (1993), 125–133.

[Lai] H. F. Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. AMS 172 (1972), 1–33.

[Moser] J. Moser, Analytic surfaces in ℂ^{2} and their local hull of holomorphy, Ann. Acad. Scient. Fenn., Ser. A. I. 10 (1985), 397–410.

[MW] J. K. Moser and S. M. Webster, Normal forms for real surfaces in ℂ^{2} near complex tangents and hyperbolic surface transformations, Acta Mathematica (3–4) 150 (1983), 255– 296.

[Porteous] I. R. Porteous, Simple singularities of maps, Springer Lecture Notes 192 (1971), 286–307.

[Pragacz] P. Pragacz, Enumerative Geometry of Degeneracy Loci, Ann. scient. Éc. Norm. Sup., 4^{e}, 21 (1988), 413–454.

[Walter] W. Walter, A useful Banach algebra, Elem. Math. (1) 47 (1992), 27–32.

[W_{77}] S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math., 43 (1977), 53–68.

[W_{84}] , Real submanifolds of ℂ^{n} and their complexifications, in Topics in Several Complex Variables, Ramírez de Arellano & Sundararaman, eds., Pitman, Boston, RNM 112, 1985.

[W_{85}] , The Euler and Pontrjagin numbers of an nmanifold in ℂ^{n}, Comment. Math. Helv. 60 (1985), 193–216.

[W_{86}] , On the relation between Chern and Pontrjagin numbers, Contemp. Math. 49 (1986), 135–143.

[W_{94}] , Geometric and dynamical aspects of real submanifolds of complex space, Proceedings of the International Congress of Mathematicians, Vol. 2 (Zürich, 1994), 917– 921, Birkhäuser, Basel, 1995.

[W_{96}] , Double valued reflection in the complex plane, Enseign. Math. (2) 42 (1–2) (1996), 25–48.

[Wells] R. O. Wells, Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann. 179 (1969), 123–129.

[Whitney_{43}] H. Whitney, The general type of singularity of a set of 2n−1 smooth functions of n variables, Duke Math. J. 10 (1943), 161–172.

[Whitney_{44}] , The self-intersections of a smooth n-manifold in 2n-space, Annals of Mathematics (2) 45 (1944), 220–246.