Invariants for Pairs of Almost Complex Structures

Adam Coffman

August 12, 1998

1 Introduction

As an example of a phenomenon to which geometric residue theorems can be applied, Harvey and Lawson ([HL_{1}], [HL_{2}]) considered singular differential forms comparing two almost complex structures on a real vector bundle. The set of vectors in a fiber having the same image under either complex structure operator forms a complex vector subspace; points where this subspace jumps in dimension form the “coincidence locus” of the complex structures.

The formulas describing the cohomology class of the coincidence current are naturally generalized in two ways. First, instead of two structures on one real bundle, the structure on one vector bundle is compared with the structure on another in which the first is mapped injectively. Geometrically, this corresponds to comparing the almost complex structure of a manifold with that on complex tangents inherited from an immersion in an ambient manifold. Second, if the vectors over a coincidence locus form a bundle over a smooth, closed base space, then its chern numbers are given by a universal polynomial formula. Then the relationship between the complex coincidence locus of a map and the CRsingular set of its graph is described, with real-analytic examples. The last sections apply the theory to anticommuting complex structures, and to the coincidence of several complex structures.

This paper represents unpublished lecture notes based on research conducted at the University of Chicago, and presented in a talk in the Geometric Topology seminar, February 1996. Some of these results are included in the author’s 1997 dissertation ([C]), supervised by S. Webster.

2 The Linear Algebra of Coincidence

Consider two complex vector spaces, as real vector spaces together with complex structure operators, V^{r} = (V_{ℝ}^{2r}, J^{V}), F^{n} = (F_{ℝ}^{2n}, J^{F}), and an injective ℝ-linear map α : V_{ℝ} → F_{ℝ}. α is said to be ℂ-linear with respect to J^{V} and J^{F} if

αJ^{V} v⃗ = J^{F} αv⃗  (1)

holds for all vectors v⃗ ∈ V_{ℝ}. Since not all ℝ-linear maps are ℂ-linear, an immediate query about α might regard the nature of the set of vectors satisfying equation (1). Those vectors such that αJ^{V} v⃗ = J^{F} αv⃗ form a real vector space K ⊆ V_{ℝ}, and K is a complex vector space with respect to J^{V}:

v⃗ ∈ K ⇒ αJ^{V} (J^{V} v⃗) = −αv⃗ = J^{F} (J^{F} αv⃗) = J^{F} α(J^{V} v⃗) ⇒ J^{V} v⃗ ∈ K.

The image αK is a complex subspace of F:

αv⃗ ∈ αK ⇒ J^{F} (αv⃗) = αJ^{V} v⃗ ∈ αK.

The subspace K can have complex dimension j, 0 ≤ j ≤ r. The map α restricted to K is a ℂ-linear isomorphism with respect to J^{V} |_{K} and J^{F} |_{αK}.

Example 2.1 If α is the identity map (V_{ℝ} = F_{ℝ}) and the complex structures J^{V} and J^{F} agree, then K, V, and F are ℂ-linearly isomorphic.

Example 2.2 If (V_{ℝ}, J^{V}) = (F_{ℝ}, −J^{F}), then K = {0}. This is the “complex conjugate,” V = F̄, and the complex structures J^{F} and −J^{F} map every v⃗ to different images.

The vector space K can also be interpreted in terms of the kernel of a ℂlinear map of complexifications, V ⊗ ℂ = V_{ℝ} ⊗_{ℝ} ℂ, and F ⊗ ℂ = F_{ℝ} ⊗_{ℝ} ℂ. The map α : V_{ℝ} ↪ F_{ℝ} complexifies as α_{ℂ} : V ⊗ ℂ ↪ F ⊗ ℂ, and the vector spaces break into eigenspaces as follows:

V^{1,0} = {v⃗ ∈ V ⊗ ℂ : J_{ℂ}^{V} v⃗ = iv⃗}
V^{0,1} = {v⃗ ∈ V ⊗ ℂ : J_{ℂ}^{V} v⃗ = −iv⃗}
F^{1,0} = {v⃗ ∈ F ⊗ ℂ : J_{ℂ}^{F} v⃗ = iv⃗}
F^{0,1} = {v⃗ ∈ F ⊗ ℂ : J_{ℂ}^{F} v⃗ = −iv⃗}

Define θ to be the inclusion of V^{0,1} in V ⊗ ℂ and note that φ = \frac{1}{2} − \frac{i}{2}J_{ℂ}^{F} is the projection of F ⊗ ℂ onto F^{1,0}.

Consider the composite map φα_{ℂ}θ : V^{0,1} → F^{1,0}. It is ℂ-linear with respect to i on V^{0,1} and J_{ℂ}^{F} = i on F^{1,0}. θ = \frac{1}{2} + \frac{i}{2}J_{ℂ}^{V} is the identity on V^{0,1}, and the composition can then be expressed as

φα_{ℂ}θ = (\frac{1}{2} − \frac{i}{2}J_{ℂ}^{F})α_{ℂ}(\frac{1}{2} + \frac{i}{2}J_{ℂ}^{V}) = \frac{1}{4}(α_{ℂ} + iα_{ℂ}J_{ℂ}^{V} − iJ_{ℂ}^{F} α_{ℂ} + J_{ℂ}^{F} α_{ℂ}J_{ℂ}^{V}). (2)

The kernel of this map is the preimage by α_{ℂ} of the intersection of α_{ℂ}V^{0,1} with F^{0,1}, the kernel of φ. If α_{ℂ}v⃗ is in this intersection, then

J_{ℂ}^{F} α_{ℂ}v⃗ = −iα_{ℂ}v⃗ = α_{ℂ}(−i)v⃗ = α_{ℂ}J_{ℂ}^{V} v⃗.

Conversely, if J_{ℂ}^{F} α_{ℂ}v⃗ = α_{ℂ}J_{ℂ}^{V} v⃗, then evaluating composition (2) shows that v⃗ is in the kernel. The subspace K also complexifies as K^{1,0} ⊕ K^{0,1}. These remarks show that K^{0,1} = {v⃗ ∈ V^{0,1}|J_{ℂ}^{F} α_{ℂ}θv⃗ = α_{ℂ}J_{ℂ}^{V} θv⃗} is the kernel of φα_{ℂ}θ, and that it is isomorphic to the conjugate K̄ of the coincidence subspace K.

Remark: The map α need not be injective for this linear algebra to work— the domain V may have any dimension and α may be singular. Vectors v⃗ in the intersection of the subspaces K and ker α have the property α(J^{V} v⃗) = J^{F} α(v⃗) = 0. However, to simplify the geometric constructions that follow, only injective α will be considered.

3 Grassmannian Constructions

As a universal version of the linear construction, consider the set of all complex r-subspaces of the complex vector space F ⊗ ℂ (subspaces are with respect to multiplication by i as the structure operator). Let V be the tautological complex r-bundle over this grassmannian ℂG(r, F ⊗ ℂ). Then V is a subbundle of the trivial complex 2n-bundle F ⊗ ℂ, and φ is a projection on the trivial n-bundle F^{1,0}. Those planes V ∈ ℂG(r, F ⊗ ℂ) that intersect F^{0,1} in at least j complex dimensions form a subvariety, a degeneracy locus D_{j}, where the bundle map φ|V has a kernel of complex dimension ≥ j. These varieties are singular except for D_{0} and D_{r} ≅ ℂG(r, F^{0,1}). As a partial desingularization of D_{j}, form the complex grassmannian bundle π_{j} : ℂG(j, V) → ℂG(r, F ⊗ ℂ). If U^{j} is the tautological bundle of j-planes in V (candidates for the kernel of φ), then the inclusion of U^{j} in V, followed by the map φ : V → F^{0,1}, defines a section s_{j} of Hom(U^{j}, π_{j}^{∗}F^{1,0}). The zero locus of this section, corresponding to a drop by j in the rank of φ, has real codimension 2jn in the total space ℂG(j, V), and projects to the degeneracy locus D_{j}, which has real codimension 2jn − 2j(r − j) = 2j(n − r + j) in ℂG(r, F ⊗ ℂ).

This can be generalized to allow F to be a complex vector bundle over a smooth base space X; then ℂG(r, F ⊗ ℂ) is a grassmann bundle over X. A real m-bundle T_{ℝ}^{m} → X, with m = 2r, and complex structure J^{T}, and an injective map α : T_{ℝ} → F_{ℝ} determine a “conjugate Gauss map,” γ_{T,α} : X → ℂG(r, F ⊗ ℂ), by γ_{T,α}(x) = (α_{x})_{ℂ}T_{x}^{0,1} ⊆ F_{x} ⊗ ℂ.

\xymatrix{Hom(U^{j}, π_{j}^{∗}F^{1,0}) \ar[d] & V \ar[d] & F \ar[d] \\ ℂG(j, V) \ar[u]^{s_{j}} \ar[r]^{π_{j}} & ℂG(r, F ⊗ ℂ) \ar[r]^{μ} & X \ar[l]^{γ_{T,α}}}

Definition/Lemma 3.1 The “rank j coincidence locus” Q_{j} of the triple (F, T, α) is the set μ(γ_{T,α}(X) ∩ D_{j}) ⊆ X. (F, T, α) is “generic” if γ_{T,α}(X) and C_{j} := D_{j} \D_{j+1} intersect transversely for all 0 ≤ j ≤ r. In this case, Q_{j} \Q_{j+1} is a smooth (possibly empty) submanifold of X, of real codimension 2j(n − r + j).

Lemma 3.2 If N_{j} is the locus of “CR singularities” of the image αT_{ℝ} in F, where αT_{x} ∩ J_{x}^{F} αT_{x} has complex dimension ≥ j, then Q_{j} ⊆ N_{j}.

Proof: N_{j} is defined independently of any complex structure on T. αT_{x} ∩ J^{F} αT_{x} is the largest complex subspace of F_{x} contained in αT_{x}. For T_{x} to contain a j-subspace where the coincidence relation is satisfied, αT_{x} must at least contain a j-subspace of F_{x}.

Over the set Q_{j} \ Q_{j+1}, the set of coincidence subspaces K_{x}^{j}, where α_{x} is ℂ-linear, forms a bundle. In terms of the grassmannian construction, K^{j} is the conjugate of a pullback of U^{j}.

Example 3.3 If T = F (abbreviating α is the identity, J^{T} = J^{F}), then γ_{T,id}(X) ⊆ D_{r}. Unless r = 0, this is not a generic situation.

Example 3.4 T and F can be isomorphic (by α, possibly the identity) as real bundles and have different complex structures so that (T, J^{T}) defines a Gauss map γ_{T,α} transverse to the submanifolds C_{j}. The real codimension of Q_{j} in X is 2j^{2}.

Example 3.5 If T is the conjugate bundle of F, then, since no complex lines in T are complex in F, γ_{T,id}(X) ∩ D_{1} = ∅, and the triple (F, T, id) is generic.

Example 3.6 A “real structure” on the complex bundle F = (F_{ℝ}, J) is a ℝ-linear map C : F → F̄ that is ℂ-linear, i.e., CJ = −JC, and such that C̄C is the identity on F, where C̄ : F̄ → F is the same real-linear map, but considered as ℂ-linear with respect to −J and J. In particular, C is a ℂ-isomorphism of F and F̄, so it is not generic with respect to coincidence. However, the triple (F, F, C) is generic, with Q_{1} = ∅. A bundle F with a real structure must have odd chern classes all zero; this will also follow from the topological results of the next section. In [Wakakuwa], (F_{ℝ}, J, C) is called an “almost complex-product structure of the first kind,” where C is an “almost product” structure. For t ∈ ℝ, the operator J_{t} = \sqrt{1 + t^{2}}J + tC is a complex structure on F_{ℝ}, agreeing with J at t = 0, but coinciding on no vector with either J or −J for t ≠ 0. Similarly, J′_{t} = −J_{−t} = −\sqrt{1 + t^{2}}J + tC agrees with −J at t = 0, but coincides on no vector with either J or −J for t ≠ 0.

As a geometric application, let T be the tangent bundle T X of a smooth mmanifold X, m = 2r, with complex structure J^{T}. If f : X → A is an immersion into another almost complex 2n-manifold, then the tangent bundle (T A, J^{A}) pulls back to (F, J^{F}) = (f^{∗}T A, f^{∗}J^{A}) over X. The map of T into F is the differential map α = f^{∗}df.

Example 3.7 [Audin-Lafontaine] f is a “(J^{T} , J^{A})-holomorphic,” or “pseudoholomorphic” immersion if df ∘ J^{T} = J^{A} ∘ df. Unless X is zero-dimensional, (F, T, f^{∗}df) is not generic with respect to coincidence; γ_{T,f^{∗}df}(X) ⊆ D_{r}.

Example 3.8 The codimension of the locus Q_{1} for a generic df is 2(n − r + 1) = 2n − m + 2. If m ≤ n, then Q_{1} = ∅.

Example 3.9 ([Eells-Lemaire]) If f is a harmonic map from a Riemann surface X to a Kähler manifold, then it is either holomorphic (Q_{1} = X) or Q_{1} is discrete (codimension ≥ 2).

4 Thom-Porteous Formulas and Examples

Theorem 4.1 If X is a compact, oriented, smooth manifold with real dimension 2j(n − r + j), and (F, T, α) is generic, then

∑_{x∈Q_{j}} ind(x) = ∫_{X} Δ_{n−r+j}^{(j)}(c(F − T̄)),

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

Proof: The symbol Δ_{b}^{(a)}, applied to a graded sum c_{0} + c_{1} + c_{2} + …, stands for the determinant of the a × a matrix with p, q entry c_{b−p+q}. In this case, the entries are the graded components of the (formal) quotient c(F −T̄) = (cF)/(cT̄).

Pushing forward the current Zero(s_{j}) and the chern form of the bundle Hom(U^{j}, π_{j}^{∗}F) gives an equation of cohomology classes on ℂG(r, F ⊗ ℂ):

π_{j∗}[Div(s_{j})] = Δ_{n−r+j}^{(j)}(c(F^{1,0} − V)).

Such formulas are due to Giambelli, Thom, and [Porteous_{1}], and are considered, together with useful determinantal identities, in [Fulton] and [HL_{2}].

By the lemma on transversality of generic maps, γ_{T,α}(X) and C_{j} meet transversely at isolated points. The pushforward current π_{j∗}[Zero(s_{j})], when restricted to ℂG(r, F ⊗ ℂ) \ D_{j+1}, is equal to the restriction of the current [D_{j}], since the projection π_{j} is a local diffeomorphism of Zero(s_{j}) over the set C_{j}. The chern class formula pulls back by γ_{T,α} to X by functoriality.

The quotient c(F − T̄) = cF/cT̄ of total chern classes is calculated using the given complex structures: cF = c(F_{ℝ}, J^{F}), and cT̄ = c(T_{ℝ}, −J^{T}). A real vector bundle may admit finitely or infinitely many complex structures with different chern classes, as examples of [Thomas], [Hiller], and [Nash] show. However, there are some relations restricting which chern classes can occur. A given complex structure induces an orientation on the real vector bundle; then the top chern class c_{r}T is equal to the euler class χT of the oriented bundle. The opposite complex structure −J^{T} induces the same orientation as J^{T} if r is even, and reverses the orientation if r is odd. The pontrjagin class of the real bundle does not depend on its orientation. The following familiar relations hold, if cT = 1 + c_{1} + c_{2} + … + c_{r} and p(T_{ℝ}) = 1 + p_{1} + … + p_{r}.

cT̄ = 1 − c_{1} + c_{2} − c_{3} + … ± c_{r}
cT cT̄ = 1 − p_{1} + p_{2} − … ± p_{r}  (3)
= 1 + (2c_{2} − c_{1}^{2}) + (2c_{4} − 2c_{1}c_{3} + c_{2}^{2}) + …

The first few terms of the quotient cF/cT̄ are

\frac{cF}{cT̄} = 1 + (c_{1}F + c_{1}T)
+(c_{1}^{2}T − c_{2}T + c_{1}F c_{1}T + c_{2}F)
+(c_{1}^{3}T − 2c_{1}T c_{2}T + c_{3}T + c_{1}F c_{1}^{2}T − c_{1}F c_{2}T + c_{2}F c_{1}T + c_{3}F)

+(c_{1}^{4}T − 3c_{1}^{2}T + c_{2}^{2}T + 2c_{1}T c_{3}T − c_{4}T + c_{1}F c_{1}^{3}T − 2c_{1}F c_{1}T c_{2}T

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

Example 4.2 ([HL_{2}]) If T_{ℝ} and F_{ℝ} are equal as real (unoriented) bundles, with relatively generic complex structures, the coincidence currents satisfy the cohomological relation

[Q_{j}] = Δ_{j}^{(j)}(c(F − T̄)).

T and F have symmetric roles in this scenario, and the equality Δ_{j}^{(j)}(c(F −T̄)) = Δ_{j}^{(j)}(c(T − F̄)) also follows from a determinantal identity. In this example, diffeomorphisms f : X → X of an almost complex manifold (X, J^{X}) induce a complex structure f^{∗}J^{X}; some of the diffeomorphisms are such that the triple ((T X, J^{X}), (T X, f^{∗}J^{X}), f^{∗}df) is generic. (This example was correctly analyzed, but stated with incorrect conclusion in [HL_{2}].)

Example 4.3 The case where J^{T} = −J^{F} is generic and the numerator and denominator are equal in the quotient cF/cT̄ = 1 + 0 + … + 0. For j > 0, the locus Q_{j} = ∅, and Δ_{j}^{(j)}(c(F − T̄)) = 0 in the cohomology ring.

Example 4.4 Suppose f : X → A is a pseudoholomorphic immersion; then X admits a complex normal bundle ν such that T X ⊕ ν = F = f^{∗}T A. The triple (F, \overline{T X}, df) is generic and Q_{j} = ∅ for j > 0. Computing the coincidence cohomology class,

Δ_{n−r+j}^{(j)}(c(F − T X)) = Δ_{n−r+j}^{(j)}(c(ν)).

Since ν has complex rank n−r, Δ_{n−r+j}^{(j)}(c(ν)) is a determinant of a matrix with zeroes on and above the diagonal. So, these formulas are not a new obstruction for pseudoholomorphic immersions.

Example 4.5 ([HL_{1}]) If J^{1} and J^{2} are two relatively generic complex structures on a vector bundle over the compact 2-manifold X, then Q_{1} is a finite set, and the theorem gives the count

∑_{x∈Q_{1}} ind(x) = ∫_{X} c_{1}(J^{1}) + c_{1}(J^{2}).

Let X = (ℂ/(ℤ ⊕ iℤ), i = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )); then f(x, y) = (x + y, y) is an

orientation-preserving diffeomorphism of the torus, and c_{1}T X = 0. The dif-

ferential in the x, y coordinates is df = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) . At no point on X does the

equality i ∘ df = df ∘ i hold.

Example 4.6 If F = (T X, J^{1}) and T = (T X, J^{2}) are two relatively generic complex structures on the compact 8-manifold X, then Q_{2} is a finite set, and the theorem gives the formula (again, symmetric in F and T):

∑_{x∈Q_{2}} ind(x) = ∫_{X} Δ_{2}^{(2)}(c(F − T̄))

= ∫_{X} c_{2}^{2}T − c_{3}T c_{1}F + c_{2}T c_{1}^{2}F − c_{1}T c_{3}T + c_{1}T c_{2}T c_{1}F

+c_{2}^{2}F − c_{3}F c_{1}T + c_{2}F c_{1}^{2}T − c_{1}F c_{3}F + c_{1}F c_{2}F c_{1}T − 2c_{2}T c_{2}F.

Theorem 4.7 If X is compact, (F, T, α) is generic and Q_{j+1} = ∅, then Q_{j} is a submanifold of real codimension 2j(n − r + j) in X. The chern numbers of the bundle K^{j} → Q_{j} can be computed by applying the kernel bundle formula of [Pragacz] to the conjugate bundle K^{0,1}: If ∏ c_{i}(K^{0,1})^{β_{i}} = ∑ m_{J}s_{J}(K^{0,1}), where m_{J} ∈ ℤ and ∑ iβ_{i} = \frac{1}{2} dim_{ℝ}(Q_{j}) = d, then

∏_{i} c_{i}(K^{0,1})^{β_{i}}[Q_{j}] = (−1)^{d} ∑_{J} m_{J}s_{j^{n−r+j},J̃}(F − T̄).

J̃ denotes the conjugate multiindex (i_{1}, i_{2}, …), i_{a} = card{h : j_{h} ≥ a}. The subscript j^{n−r+j}, J̃ denotes the concatenation of n−r+j j’s and the multiindex J̃. The symbol s_{(i_{1},... ,i_{k})} is the determinant of a k × k matrix with p, q entry s_{i_{p}−p+q}. In this case, 1 + s_{1} + … denotes the segre class.

The determinants in this formula could immediately be rewritten in terms of chern classes using the determinantal identity s_{I}(E − F) = cĨ(E − F). The segre class is often more natural in enumerative constructions ([Fulton]); in this case, the first few terms of s(F − T̄) are:

s = 1 + s_{1} + s_{2} + s_{3} + …
= 1 + c_{1}F + c_{1}T
+c_{1}^{2}F + c_{1}F c_{1}T + c_{2}T − c_{2}F

+c_{1}^{3}F + c_{1}^{2}F c_{1}T + c_{1}F c_{2}T − 2c_{1}F c_{2}F − c_{1}T c_{2}F + c_{3}T + c_{3}F + …

Example 4.8 Suppose the same real four-plane bundle F_{ℝ}^{4} over the compact 4-manifold X has two complex structures, F^{2} = (F_{ℝ}, J^{F}) and T^{2} = (F_{ℝ}, J^{T}). If the triple (F, T, id) is generic, the coincidence locus Q_{1} is a real surface in X, and its cohomology class [Q_{1}] is equal to the class c_{1}F + c_{1}T by Theorem 4.1. The complex line bundle K^{1} over Q_{1} of vectors where J^{F} = J^{T} has the following chern number:

∫_{Q_{1}} c_{1}K = − ∫_{Q_{1}} c_{1}K^{0,1}
= ∫_{X} s_{1,1}(F − T̄) = ∫_{X} | \begin{matrix}s_{1} & s_{2} \\ 1 & s_{1} \end{matrix} |
= ∫_{X} c_{2}(F − T̄)
= ∫_{X} c_{1}^{2}T − c_{2}T + c_{1}F c_{1}T + c_{2}F

For example, if F = T̄, then Q_{1} = ∅ and the cohomology classes c_{1}F + c_{1}T and c_{1}^{2}T − c_{2}T + c_{1}F c_{1}T + c_{2}F are both zero. So far, this formula does not appear to be symmetric in F and T, however, using relation (3) gives:

∫_{Q_{1}} c_{1}K = ∫_{X} p_{1}F_{ℝ} + c_{2}T + c_{1}F c_{1}T + c_{2}F.  (4)

Example 4.9 Suppose the sphere S^{6} admits a relatively generic pair of complex structures, T = (T S^{6}, J^{T}) and F = (T S^{6}, J^{F}). The chern classes are of the form cT = 1+c_{3}T and cF = 1+c_{3}F, where c_{3}T and c_{3}F are equal to the euler class, up to a sign depending on the orientation induced by J^{T} and J^{F}. Q_{1} is a real 4-submanifold of S^{6}, with ∫_{Q_{1}} c_{1}^{2}K = ∫_{X} c_{3}T + c_{3}F. This number is ±4 if T and F have the same orientation, and 0 if they are oppositely oriented.

Example 4.10 Consider an immersion f : X^{16} → A^{18}, where X and A are almost complex manifolds (r = 8, n = 9). If f is generic with respect to complex tangency, then H = T X ∩ (f^{∗}J^{A}T X) has j_{0} = 7 complex dimensions, except at isolated points where T X is a complex subspace of k = 8 dimensions in F = f^{∗}T A. If f is generic with respect to complex coincidence, then the smooth locus of Q_{1} is codimension 4 in X, and Q_{2} is a 4-dimensional (codimension 12) submanifold of X. For X compact, the bundle K^{2} over Q_{2} has the same chern numbers as K^{0,1}:

∫_{Q_{2}} c_{2}K^{2} = ∫_{X} | \begin{matrix}s_{2} & s_{3} & s_{4} & s_{5} \\ s_{1} & s_{2} & s_{3} & s_{4} \\ 1 & s_{1} & s_{2} & s_{3} \\ 0 & 1 & s_{1} & s_{2} \end{matrix} | ,

∫_{Q_{2}} c_{1}^{2}K^{2} = ∫_{X} | \begin{matrix}s_{2} & s_{3} & s_{4} & s_{5} \\ s_{1} & s_{2} & s_{3} & s_{4} \\ 1 & s_{1} & s_{2} & s_{3} \\ 0 & 1 & s_{1} & s_{2} \end{matrix} | + | \begin{matrix}s_{2} & s_{3} & s_{4} & s_{5} & s_{6} \\ s_{1} & s_{2} & s_{3} & s_{4} & s_{5} \\ 1 & s_{1} & s_{2} & s_{3} & s_{4} \\ 0 & 0 & 1 & s_{1} & s_{2} \\ 0 & 0 & 0 & 1 & s_{1} \end{matrix} | .

Example 4.11 Tsanov and [Pontecorvo] consider a pair of complex structures on a twistor space Z constructed as follows: Let M = (M^{4}, g) be a compact oriented riemannian manifold. The twistor space Z is the total space of the fiber bundle t : SO(T M, g)/U(2) → M, where local sections correspond to almost complex structures. The tangent space T_{z}Z is split by the metric into a vertical part T_{z}ℂP^{1} and a horizontal part T_{t(z)}M— the horizontal part is given the complex structure operator defined by the point z. The direct sum of the two complex structure operators defines the tautological twistor almost complex structure, (Z, J) which is integrable if and only if g is anti-self-dual. If M is also a complex surface, a different description of Z as the projectivization of a holomorphic bundle gives a different almost complex structure (Z, I), which is integrable without requiring g to be anti-self-dual. The two almost complex structures I and J may have different chern classes. The cohomology ring of Z is a module over H^{∗}(M; ℝ), generated by the cohomology class h over the fiber and subject to the relation 4h^{2} = c_{1}^{2}, where t^{∗}c(T M) = 1 + c_{1} + c_{2}.

c(T Z, I) = (1 + 2h)(1 + c_{1} + c_{2}) = 1 + 2h + c_{1} + c_{2} + 2hc_{1} + 2hc_{2}
c(T Z, J) = (1 + 2h)(1 + 2h + c_{2}) = 1 + 4h + c_{2} + c_{1}^{2} + 2hc_{2}

The invariance relation (3) holds, and c_{3}(I) = c_{3}(J), but the other chern numbers may disagree:

∫_{Z} c_{1}^{3}(I) = ∫_{Z} 8h^{3} + 6hc_{1}^{2} = ∫_{M} 8c_{1}^{2}
∫_{Z} c_{1}c_{2}(I) = ∫_{Z} 2hc_{2} + 2hc_{1}^{2} = ∫_{M} 2(c_{2} + c_{1}^{2})
∫_{Z} c_{1}^{3}(J) = ∫_{Z} 64h^{3} = ∫_{M} 16c_{1}^{2}
∫_{Z} c_{1}c_{2}(J) = ∫_{Z} 4h(c_{2} + c_{1}^{2}) = ∫_{M} 4(c_{2} + c_{1}^{2})

So, if I and J have the same chern classes, the chern numbers c_{1}^{2} and c_{2} of T M must be zero. [Pontecorvo] uses this to conclude that if g is anti-self-dual, then I and J cannot be homotopic.

Perturbing I and J so that they are relatively generic, the locus Q_{2} is expected to be empty, and the locus Q_{1} is expected to be a codimension 2 submanifold in Z. Its current of integration represents the cohomology class c_{1}(I) + c_{1}(J) = 6h + c_{1}. The chern number of the line bundle K over Q_{1} is given by the formula

∫_{Q_{1}} c_{1}^{2}K = ∫_{Z} Δ_{3}^{(1)}(c(J)/c(−I)) = ∫_{Z} s_{111} = ∫_{Z} s_{1}^{3} − 2s_{1}s_{2} + s_{3}
= ∫_{Z} c_{3}(J) + c_{2}(J)c_{1}(I) − c_{1}(J)c_{2}(I) + c_{1}(J)c_{1}^{2}(I)
+c_{3}(I) + c_{1}^{3}(I) − 2c_{1}(I)c_{2}(I)  (5)
= ∫_{Z} 24h^{3} + 12h^{2}c_{1} + 8hc_{1}^{2} − 2hc_{2} + 2c_{1}^{3} − c_{1}c_{2}
= ∫_{M} 14c_{1}^{2} − 2c_{2}

Again, equation (5) is not symmetric in I and J until relation (3) is used:

∫_{Q_{1}} c_{1}^{2}K = ∫_{Z} c_{3}(I) + c_{3}(J) + c_{1}(I)c_{2}(J) + c_{2}(I)c_{1}(J) + (c_{1}(I) + c_{1}(J))p_{1}T Z.

Question: The geometric symmetry of the roles of I and J in the previous example and Example 4.8 does not seem to be exhibited until relation (3) is taken into account. Is there a purely combintorial explanation of this phenomenon, for example, some identity among symmetric functions?

5 Coincidence as CR-Singularities of a Graph

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.

If, in addition to α being generic with respect to coincidence, \underline{α}T is a subbundle of real rank m = 2r in the complex bundle T ⊕ F of complex rank n + r, which is generically included with respect to loci N_{j} of CR-singularities, then a cohomological version of formula (10.5) of [HL_{2}] applies:

[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)) = [Q_{j}].

This shows that, for sufficiently general maps α, Theorem 4.1 can be derived as a corollary to cohomological formulas for CR singularites. (These formulas are investigated in more detail and generalized in [C].)

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 5.2 The graph of a smooth map f : X → A (not necessarily an immersion) defines an embedding \underline{f} : X → X × 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 X × A.

Example 5.3 ([Eells-Wood]) If f is a smooth map between connected, compact, oriented Riemannian surfaces, f : (X, g_{X}) → (A, g_{A}), the degree of f is an integer. Giving X and A complex structures compatible with the metrics, and using [Webster]’s formula for the image of the graph of f in (X × A, J^{⊕}),

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

Reversing the orientation and complex structure on A changes the index sum to χX −(deg f)χA, and similarly, reversing the orientation and complex structure on X gives index sum −χX + (deg f)χA. The sign of the index differs from the [Eells-Wood] formulas, which use the complexified bundle map T^{1,0}X → T^{0,1}A instead of (2).

If f is a generic perturbation of a holomorphic map, with ramification v(q) > 1 at finitely many branch points q ∈ X, then the Riemann-Hurwitz theorem applies ([Griffiths-Harris]),

χX = (deg f)χA − ∑(v(q) − 1),

so the number of points where f is ℂ-linear is 2χX + ∑(v(q) − 1).

It is well-known that in this two-dimensional case, f is conformal (anglepreserving) at points where df is ℂ-linear, and indirectly conformal (anglereversing) at points where df is ℂ-antilinear. ([Ahlfors])

Example 5.4 If X is the almost complex manifold (X_{ℝ}, J) and X̄ = (X_{ℝ}, −J), then the graph of the identity diffeomorphism f : X → X̄ is the diagonal embedding \underline{f} : X → X × X̄. The image is totally real in the “complexification” X × X̄ (cf [Eastwood]); again, for j > 0, this is consistent with the cohomological obstruction to total reality:

Δ_{j}^{(j)}(c(\underline{f}^{∗}T X ⊕ T X̄ − T X ⊗ ℂ)) = 0.

Example 5.5 Something similar holds for the kernel bundle formulas: in the scenario of Example 4.8, equation (4) can be rewritten:

∫_{Q_{1}} c_{1}K = p_{1}F_{ℝ} + c_{2}(T ⊕ F).

This is [Webster]’s formula for the complex tangent locus N_{1} of a real 4-plane subbundle F_{ℝ} of a complex 4-bundle, T ⊕ F.

6 Cartographic Applications

Example 6.1 Considering ℂP^{1} as the sphere S^{2} with the usual complex structure, the graph of the identity diffeomorphism embeds ℂP^{1} as the diagonal complex submanifold of ℂP^{1} × ℂP^{1}, or as a totally real submanifold of ℂP^{1} × \overline{ℂP^{1}}. Perturbing the identity map to a generic map gives a diffeomorphism f : ℂP^{1} → ℂP^{1} with four points of complex coincidence by Example 4.5. Equivalently, the graph of f is a sphere S^{2} with four complex tangents in ℂP^{1} × ℂP^{1}.

The coordinate charts for ℂP^{1} are two complex lines. A smooth map between one of these charts and the unit sphere in ℝ^{3} is given by stereographic projection:

z = \frac{x_{1} + ix_{2}}{1 − x_{3}}
(x_{1}, x_{2}, x_{3}) = (\frac{z + z̄}{zz̄ + 1}, \frac{z − z̄}{i(zz̄ + 1)}, \frac{zz̄ − 1}{zz̄ + 1})

Figure 1.

In particular, the complex structure operator J = i on the complex plane induces the complex structure on S^{2}, and if the diffeomorphism f is written in terms of the local coordinates x, y so that z = x+iy ↦ u(x, y)+iv(x, y), the coincidence relation J ∘ df = df ∘ J is the pair of Cauchy-Riemann equations for complex functions:

df ∘ J = ( \begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix} ) · ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}u_{y} & −u_{x} \\ v_{y} & −v_{x} \end{matrix} )
J ∘ df = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) · ( \begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix} ) = ( \begin{matrix}−v_{x} & −v_{y} \\ u_{x} & u_{y} \end{matrix} )
⟹ u_{x} = v_{y}, u_{y} = −v_{x}.

This condition is equivalent to f being conformal (angle-preserving) [Ahlfors].

The best-known diffeomorphisms of the sphere are conformal at every point. The following composition gives a generic diffeomorphism of the sphere, one expected to satisfy the Cauchy-Riemann equations only on a finite set. First, stretch the unit sphere in ℝ^{3} into the ellipsoid x^{2} + \frac{y^{2}}{a^{2}} + \frac{z^{2}}{b^{2}} = 1 by a linear transformation, with 1 < a < b. (In Figure 2, a = 2, b = 3 are used.)

Figure 2.

Second, a radial projection maps the ellipsoid back onto the unit sphere.

Figure 3.

In terms of the coordinates z = x + iy, the composite f is given by the real analytic equations

u(x, y) = \frac{2x}{\sqrt{b^{2}(x^{2} + y^{2} − 1)^{2} + 4x^{2} + 4a^{2}y^{2}}(1 − \frac{b(x^{2}+y^{2}−1)}{\sqrt{b^{2}(x^{2}+y^{2}−1)^{2}+4x^{2}+4a^{2}y^{2}}} )},

v(x, y) = \frac{2ay}{\sqrt{b^{2}(x^{2} + y^{2} − 1)^{2} + 4x^{2} + 4a^{2}y^{2}}(1 − \frac{b(x^{2}+y^{2}−1)}{\sqrt{b^{2}(x^{2}+y^{2}−1)^{2}+4x^{2}+4a^{2}y^{2}}} )}.

A computer-assisted calculation finds exactly four points (x, y) in the plane where the Cauchy-Riemann relations hold:

(±\frac{\sqrt{b^{2} + a^{2}b^{2} − 2a^{2} ± 2a\sqrt{(b^{2} − 1)(b^{2} − a^{2})}}}{b\sqrt{a^{2} − 1}}, 0).

Corresponding to the symmetry of f, the four-element subset Q_{1} of S^{2} in ℝ^{3} forms a rectangle with sides parallel to the x_{1}- and x_{3}-axes.

With the conformal structure on the ellipsoid induced by the ambient euclidean metric, the stretching map from the sphere preserves angles at the four “umbilic points” of the ellipsoid. ([Boehm-Prautzsch], [Porteous_{2}])

Example 6.2 As another application of Example 5.3, consider the degree 0 map f : S^{2} → ℝ^{2}, given by projecting the unit sphere in ℝ^{3} orthogonally onto a plane. The equations

f(z) = (\frac{z + z̄}{zz̄ + 1}, \frac{z − z̄}{i(zz̄ + 1)})
f(x + iy) = (u(x, y), v(x, y)) = (\frac{2x}{x^{2} + y^{2} + 1}, \frac{2y}{x^{2} + y^{2} + 1})

for f are smooth, and fold along a great circle. Evidently, f is directly conformal at one pole, z = 0, and indirectly conformal at the opposite pole z = ∞. Since a degree 0 map from S^{2} is expected to have two points where it is directly conformal, this projection f is not generic and ([z = 0], 0) is a complex tangent with index 2 in the sphere graphed inside ℂP^{1} × ℂ.

The shear A(s) = ( \begin{matrix}1 & s \\ 0 & 1 \end{matrix} ) is holomorphic ℂ → ℂ only for s = 0. A

simple perturbation of f is A(s) ∘ f, and this composition is conformal when J ∘ A(s) ∘ df = A(s) ∘ df ∘ J:

J ∘ A(s) ∘ df = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) · ( \begin{matrix}1 & s \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix} )
= \frac{2}{(1 + x^{2} + y^{2})^{2}} ( \begin{matrix}2xy & −x^{2} + y^{2} − 1 \\ −x^{2} + y^{2} − 2sxy + 1 & sx^{2} − sy^{2} − 2xy + s \end{matrix} )
A(s) ∘ df ∘ J = ( \begin{matrix}1 & s \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix} ) · ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )
= \frac{2}{(1 + x^{2} + y^{2})^{2}} ( \begin{matrix}sx^{2} − sy^{2} − 2xy + s & x^{2} − y^{2} + 2sxy − 1 \\ x^{2} − y^{2} + 1 & 2xy \end{matrix} ) .

The two real solutions (x, y) of the Cauchy-Riemann equations are

±(\sqrt{\frac{−s(s − \sqrt{s^{2} + 4})}{2(s^{2} + 4)}}, \sqrt{\frac{−2s}{(s^{2} + 4)(s − \sqrt{s^{2} + 4})}}), if s ≥ 0,

and

±(\sqrt{\frac{−s(s + \sqrt{s^{2} + 4})}{2(s^{2} + 4)}}, \sqrt{\frac{−2s}{(s^{2} + 4)(s + \sqrt{s^{2} + 4})}}), if s ≤ 0.

Example 6.3 The unit sphere in ℝ^{3} can also be projected radially onto the x, y-plane from the point (0, 0, 2), again 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 an ellipse. Not too surprisingly, the map is conformal at exactly those points projected onto the foci of the ellipse. The four points are the intersections of the sphere with the lines connecting the poles to the vertex of the cone.

Figure 4.

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 ellipse meets the x-axis at coordinates −r − \frac{4\sqrt{r^{2}+3}}{(r+\sqrt{r^{2}+3})^{2}+3} < 0 and −r + \frac{4\sqrt{r^{2}+3}}{(r−\sqrt{r^{2}+3})^{2}+3} > 0. 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 have two of their foci at the poles of the sphere, and that they are similar (and so conformal) to any parallel ellipse on the cone.

7 Anticommuting Complex Structures

Suppose I and J are two complex structures on a vector space or bundle F that anticommute: IJ + JI = 0. Then their product IJ is a new complex structure, as well as real-linear combinations of the form aI + bJ + cIJ such that a^{2} + b^{2} +c^{2} = 1. F, together with the anticommuting complex structures I, J, could be called “quaternionic;” there are examples of manifolds with anticommuting almost complex structures: [Joyce]. I and J never coincide and so are relatively generic with respect to coincidence.

The geometry of real subspaces of a quaternion vector space is more complicated than the complex vector space situation. ([Dlab-Ringel]) However, the complex subspaces of a quaternion vector space are studied with familiar constructions.

Let F be a complex vector space (F_{ℝ}, I) of even complex dimension n = 2s, and suppose F admits another complex structure J so that I and J anticommute. Let ℂG(r, F) be the grassmannian of I-complex subspaces of complex dimension r in F. Given a plane V ∈ ℂG(r, F), the intersection V ∩ JV is a quaternionic vector space: V ∩ JV is invariant under I and J, and their restrictions form a pair of anticommuting complex structures on V ∩ JV.

Denoting by V the tautological complex r-bundle over ℂG(r, F), the real bundle V_{ℝ} can be considered as a real 2r-subbundle of the trivial complex vector bundle (F_{ℝ}, J). If this inclusion were generic, the CR-singular set N_{r} ⊆ ℂG(r, F), where V_{ℝ} is a J-complex subspace, would have real codimension 2r(n − r), and so be isolated. However, for r even, N_{r} is not isolated; those planes which are simultaneously I-complex and J-complex in (F, I, J) form the quaternionic grassmannian ℍG(r/2, F), of real dimension r(n − r). If further, 2r ≤ n, since the quaternionic subspace V ∩ JV must be at least 4-dimensional, the first-order CR-singular set N_{1} \ N_{2} is empty, although generically of codimension 2(n − 2r + 1).

Example 7.1 The I-complex 2-planes in F = ℍ^{2} form a complex manifold of complex dimension 4. Any plane containing a J-complex line must actually be J-complex, and so a quaternionic line in ℍ^{2}. These planes form the quaternionic projective space ℍP^{1}, a real 4-sphere. This construction appears in twistor geometry ([Eastwood], [Ward-Wells]), where ℂG(2, 4) is the compactified complexified Minkowski space, ℂP^{3} is the projective twistor space, and a projection π is induced by the ℂ-isomorphism ℂ^{4} → ℍ^{2}:

\xymatrix{ℂP^{3} \ar[r]^{π} & ℍP^{1} \ar@{^{(}->}[r] & ℂG(2, ℍ^{2})}

so that π has fiber ℂP^{1} = ℂG(1, ℍ^{1}) and the inclusion is totally real. This is another example of Webster’s formula, where 0 = ∫_{S^{4}} c_{2}(T ℂG(2, 4)|_{S^{4}})+p_{1}T S^{4}. The restriction of the complex bundle to S^{4} is essentially the complexification of T S^{4}, so c_{2} = p_{1} = 0.

Example 7.2 If F is a bundle with anticommuting complex structures I and J, it is known (cf [Vaisman]) that the complex vector bundle (F, I) has all odd chern classes zero. The easy proof is that J is a ℂ-linear isomorphism of F = (F_{ℝ}, I) and F̄ = (F_{ℝ}, −I). The first coincidence current [Q_{1}] is zero, so applying Theorem 4.7 to the chern form c_{1}^{q}K^{1} simplifies to the expression

0 = Δ_{q+1}^{(1)}(\frac{c(F_{ℝ}, I)}{c(F_{ℝ}, −J)}),

for q ≥ 0, so c(F_{ℝ}, I) = c(F_{ℝ}, −J). The same argument holds for the complex structure IJ, which does not coincide with I or J:

c(F_{ℝ}, I) = c(F_{ℝ}, −J) = c(F_{ℝ}, IJ) = c(F_{ℝ}, −I).

More generally, this shows that the odd chern classes of (F_{ℝ}, I) are zero when I and any other two complex structures on F_{ℝ} are mutually non-coincident.

8 Several Complex Structures

The graph of a ℝ-linear map α can be generalized to the graph of finitely many maps α_{i} : T^{m=2r} → (F_{i}, J_{i}) = F_{i}^{n_{i}}, i = 1 … p. Then the subspace of vectors in T where all the maps α_{i} are ℂ-linear corresponds to a complex subspace in the graph \underline{α} of α_{1} ⊕ … ⊕ α_{p} in T ⊕ F_{1} ⊕ … ⊕ F_{p}:

α_{i}J^{T} v⃗ = J_{i}α_{i}v⃗ ∀i ⟺ J^{⊕}\underline{α}v⃗
= (J^{T} v⃗, J_{1}α_{1}v⃗, …, J_{p}α_{p}v⃗)
= (J^{T} v⃗, α_{1}J^{T} v⃗, …, α_{p}J^{T} v⃗)
= \underline{α}J^{T} v⃗ ∈ \underline{α}T.

Q_{j}(α_{1}, …, α_{p}), the locus where all the maps α_{i} are ℂ-linear on the same jsubspace, generically has codimension 2j(n_{1} + … + n_{p} − r + j), and is contained in, but not equal to, Q_{j}(α_{1}) ∩ … ∩ Q_{j}(α_{p}). Using the complex tangent formula gives

[Q_{j}(α_{1}, …, α_{p})] = Δ_{r+n_{1}+...+n_{p}−2r+j}^{(j)}(c(T ⊕ F_{1} ⊕ … ⊕ F_{p}) − c(T ⊗ ℂ))
= Δ_{n_{1}+...+n_{p}−r+j}^{(j)}(cF_{1} · . . . · cF_{p} − cT̄).

Example 8.1 A manifold M of real dimension m = 2r > 0 with three relatively generic almost complex structures has no tangent vectors where all three agree. The expected codimension of Q_{1}(id_{T M}, id_{T M}) is 2(r + r − r + 1) = m + 2.

Example 8.2 A complex line bundle T, with generic maps to two other complex line bundles, α_{i} : T → (F_{i}, J_{i}), will have a coincidence locus Q_{1} of codimension 2(1 + 1 − 1 + 1) = 4. The cohomology formula is

[Q_{1}] = Δ_{2}^{(1)}(cF_{1} · cF_{2} − cT̄)
= c_{1}^{2}T + (c_{1}F_{1} + c_{1}F_{2})c_{1}T + c_{1}F_{1}c_{1}F_{2}.

Example 8.3 Consider the 4-manifold ℂP^{2} of complex lines in ℂ^{3}, with respect to the usual complex structure i. Inside ℂ^{3}, let F_{1} and F_{2} be a pair of real planes, which have their own complex structure but are totally real with respect to i. Also let p_{1} and p_{2} be projections of ℂ^{3} onto these planes:

F_{1} = (\{(x, 0, y, 0, 0, 0)\}, J_{1} =( \begin{matrix}0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 \\ & 0 & & 0 \end{matrix} )), p_{1} =( \begin{matrix}1 \\ & 0 \\ & & 1 \\ & & & 0 \\ & & & & 0 \\ & & & & & 0 \end{matrix} ),

F_{2} = (\{(0, 0, 0, x, 0, y)\}, J_{2} =( \begin{matrix}0 & & 0 \\ & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ & 1 & 0 & 0 \end{matrix} )), p_{2} =( \begin{matrix}0 \\ & 0 \\ & & 0 \\ & & & 1 \\ & & & & 0 \\ & & & & & 1 \end{matrix} ).

The inclusion of the tautological line bundle T in the trivial bundle ℂ^{3} → ℂP^{2}, followed by the projections p_{1} and p_{2}, gives two ℝ-linear maps to the trivial line bundles, T → F_{1} and T → F_{2}. The coincidence locus Q_{1} in ℂP^{2} is the set of complex lines in ℂ^{3} on which both the projections p_{1} and p_{2} are ℂ-linear. The equations p_{1}iv⃗ = J_{1}p_{1}v⃗ and p_{2}iv⃗ = J_{2}p_{2}v⃗ have solution set {(x, y, y, −x, −x, −y)}, which is the complex line ℂ · (1, −i, −1). So, Q_{1} is a point, consistent with the formula

∑_{x∈Q_{1}} ind(x) = ∫_{ℂP^{2}} c_{1}^{2}T = 1.

Of course, this is equivalent to the coincidence geometry of p_{1}⊕p_{2} : T → F_{1}⊕F_{2}.

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