Notes on the elementary geometry of real involutions of complex projective spaces

Adam Coffman Department of Mathematical Sciences Indiana University - Purdue University Fort Wayne Fort Wayne, IN, USA 46805-1499

                   CoffmanA@ipfw.edu

July 10, 2003

Abstract

This approach to real projective geometry takes complex projective geometry as a starting point. ^{1}

1 Introduction

The containment of real projective spaces as subsets of complex projective spaces is considered from an elementary, algebraic point of view. By using complex linear algebra and “conjugate linear” maps, the goal is to construct a model of a real projective space, by taking the set of points to be the fixed point set of a certain type of involution of ℂP^{m}. The lines of the model will be defined as intersections of complex projective lines with this subset of ℂP^{m}. This linear algebraic approach generalizes the special case considered in [C], where the involution of ℂP^{m} was induced by complex conjugation of the coordinates of ℂ^{m+1} with respect to a specific coordinate system.

The main results of the last two Sections find some conditions under which a collineation of ℂP^{m} must be a projective transformation or a projective transformation composed with a conjugation (Theorem 4.6 for m ≠ 1, Theorem 5.12 for m = 1). Corollary 4.7 shows that for m ≠ 1, these are the only collineations that restrict to collineations of a real projective subspace.

2 The complex foundation

Most of the constructions we’ll need are standard in elementary projective geometry, and we briefly recall them here to fix notation.

1MSC 2000 51A05, 51M35, 32V40

Let ℂ be the field of complex numbers, and let m ≥ 0 be an integer, so ℂ^{m+1} is a complex vector space. The complex projective m-space, ℂP^{m}, is the set of one-dimensional subspaces in ℂ^{m+1}. Denote the usual projection π^{m} : ℂ^{m+1} \ {0} → ℂP^{m}, so that a non-zero column vector z spans the onedimensional subspace π^{m}(z). An element z ∈ ℂP^{m} with representative non-zero vector z = (z_{0}, z_{1}, …, z_{m})^{T} will have homogeneous coordinates [z_{0} : z_{1} : …: z_{m}].

For d = 0, …, m, define a d-dimensional complex projective subspace of ℂP^{m} to be the image of some (d + 1)-dimensional complex linear subspace V ⊆ ℂ^{m+1}, V = π^{m}(V \ {0}). So, for example, a complex projective line in ℂP^{m} is, by definition, the image of a 2-dimensional complex linear subspace of ℂ^{m}.

Let f : ℂ^{m+1} → ℂ^{m+1} be any function. Given z ∈ ℂ^{m+1} \ {0}, suppose f has the following two properties: first,

f(z) ≠ 0,  (1)

and second, for any λ ∈ ℂ \ {0}, there exists μ ∈ ℂ \ {0} so that

f(λ · z) = μ · f(z).  (2)

Then f will also have these two properties at every non-zero scalar multiple of z. If U ⊆ ℂ^{m+1} \ {0} is the set of points where f has the two properties, then we will say “f induces a map from ℂP^{m} to ℂP^{m} which is well-defined on the set π^{m}(U),” and we will denote the induced map, which takes π^{m}(z) to π^{m}(f(z)), by f : z ↦ f(z). It should also be mentioned that the map of projective spaces induced by a composition of maps is equal to the composition of the induced maps.

Example 2.1. If f : ℂ^{m+1} → ℂ^{m+1} is complex linear, then f is well-defined on the lines not contained in the kernel of f. If f : ℂ^{m+1} → ℂ^{m+1} is complex linear and invertible, then f is well-defined on all of ℂP^{m}, and also invertible. Let GL(m + 1, ℂ) ⊆ M(m + 1, ℂ) denote the subset of nonsingular matrices in the set of (m + 1) × (m + 1) matrices with entries in ℂ. Let P GL(m + 1, ℂ) denote the set of one-dimensional subspaces of M(m + 1, ℂ) which are subsets of GL(m + 1, ℂ) ∪ {0}. The following construction defines a group action of P GL(m+1, ℂ) on ℂP^{m}. For any nonsingular matrix A, there is a corresponding invertible complex linear transformation, which in turn induces a well-defined map ℂP^{m} → ℂP^{m}, denoted A, and called a projective transformation. Any non-zero scalar multiple of A induces the same map A : ℂP^{m} → ℂP^{m}, so this notation is consistent with the above conventions: a nonsingular matrix A spans a line A ∈ P GL(m + 1, ℂ), and the projective transformation of ℂP^{m} induced by A will be denoted A : z ↦ A(z).

Example 2.2. Suppose ϕ : ℂ → ℂ is a field isomorphism, and define a map:

ϕ̃ : ℂ^{m+1} → ℂ^{m+1} : ϕ̃((z_{0}, …, z_{m})^{T}) = (ϕ(z_{0}), …, ϕ(z_{m}))^{T} .

Since ϕ̃ satisfies properties (1, 2) at every point of ℂ^{m+1} \ {0}, it induces a map

ϕ̃: ℂP^{m} → ℂP^{m} : ϕ̃([z_{0} : …: z_{m}]) = [ϕ(z_{0}) : …: ϕ(z_{m})].

One example of a field isomorphism of ℂ is complex conjugation, κ : z_{0} ↦ z̄_{0}. The induced maps are κ̃ : ℂ^{m+1} → ℂ^{m+1} and κ̃ : ℂP^{m} → ℂP^{m}, and both are involutions.

Definition 2.3. A map g : ℂP^{m} → ℂP^{m} is a “collineation” if it is well-defined on ℂP^{m}, invertible, and it has the property that, for V ⊆ ℂP^{m}, g(V) is a complex projective subspace of ℂP^{m} if and only if V is a complex projective subspace of ℂP^{m}.

The following Proposition is the Fundamental Theorem of Projective Geometry. See [Csikós] §6, [Samuel], [Seidenberg].

Proposition 2.4. For m ≠ 1, the group of projective transformations of ℂP^{m} is a normal subgroup of the group of collineations, and an invertible map g : ℂP^{m} → ℂP^{m} is a collineation if and only if it is of the form g = ϕ̃∘A for some field isomorphism ϕ and projective transformation A.

Example 2.5. A map B : ℂ^{m+1} → ℂ^{m+1} will be called “conjugate linear” if it is additive, and satisfies the identity B(λz) = λ̄B(z) for all z ∈ ℂ^{m+1} and all scalars λ ∈ ℂ. Since κ̃ ∘ B is complex linear, it has some matrix representation A ∈ M(m + 1, ℂ), and we can conclude that every conjugate linear map B is of the form κ̃ ∘ A. An invertible conjugate linear map satisfies properties (1, 2) at every point of ℂ^{m+1} \ {0}, so it induces a map B, and the induced map is a collineation of ℂP^{m}, of the form B = κ̃ ∘ A for some projective transformation A.

3 Hints at reality

Notation 3.1. Given a conjugate linear involution B (so that B ∘ B is the identity map), its fixed point set will be denoted R_{B} ⊆ ℂ^{m+1}. The set of vectors z such that B(z) = −z will be denoted I_{B} ⊆ ℂ^{m+1}.

Lemma 3.2. Given a conjugate linear involution B, these sets are equal:

R_{B} = \{\frac{1}{2}(z + B(z)) : z ∈ ℂ^{m+1}\} = iI_{B},

and every element z ∈ ℂ^{m+1} is uniquely expressible as a sum x + y, for some x ∈ R_{B} and y ∈ I_{B}.

Proof. The claimed equalities of sets are easy to check, using the properties of B. The identity z = \frac{1}{2}(z+B(z))+ \frac{1}{2}(z−B(z)) gives the claimed decomposition. The uniqueness easily follows from the fact that R_{B} and I_{B} intersect only at the origin.

Theorem 3.3. Given a complex linear map A : ℂ^{m+1} → ℂ^{m+1} and conjugate linear involutions B_{1}, B_{2}, if A(R_{B_{1}}) ⊆ R_{B_{2}}, then A = B_{2}∘A∘B_{1}. If, further, A is invertible, then A(R_{B_{1}}) = R_{B_{2}}.

Proof. For any z ∈ R_{B_{1}}, z = B_{1}(z) ⟹ A(z) = A(B_{1}(z)) = B_{2}(A(B_{1}(z))). So, the complex linear maps A and B_{2} ∘ A ∘ B_{1} agree at every element of R_{B_{1}}. It follows from Lemma 3.2 that R_{B_{1}} spans ℂ^{m+1}, so A = B_{2} ∘ A ∘ B_{1}.

To check the claimed equality, it will be enough to show that if z ∈ R_{B_{2}}, then A^{−1}(z) ∈ R_{B_{1}}.

B_{1}(A^{−1}(z)) = B_{1}((B_{2} ∘ A ∘ B_{1})^{−1}(z)) = (B_{1} ∘ B_{1} ∘ A^{−1} ∘ B_{2})(z) = A^{−1}(z).

Theorem 3.4. Given a conjugate linear map B : ℂ^{m+1} → ℂ^{m+1} and conjugate linear involutions B_{1}, B_{2}, if B(R_{B_{1}}) ⊆ R_{B_{2}}, then B = B_{2} ∘B∘B_{1}. If, further, B is invertible, then B(R_{B_{1}}) = R_{B_{2}}.

Proof. The composition B_{2} ∘B is complex linear, and satisfies (B_{2} ∘B)(R_{B_{1}}) ⊆ R_{B_{2}}, so the previous Theorem applies, and B_{2} ∘ B = B_{2} ∘ (B_{2} ∘ B) ∘ B_{1}. It follows that B_{2} ∘ B = B ∘ B_{1}, and B = B_{2} ∘ B ∘ B_{1}.

When B is invertible, the previous Theorem showed (B_{2} ∘ B)(R_{B_{1}}) = R_{B_{2}}, and the conclusion follows from applying B_{2} to both sides.

Definition 3.5. A conjugate linear involution B : ℂ^{m+1} → ℂ^{m+1} induces a map B : ℂP^{m} → ℂP^{m}, which will be called a “real involution.” The fixed point set of a real involution B will be denoted R_{B} ⊆ ℂP^{m}.

Theorem 3.6. Given a conjugate linear involution B, the following subsets of ℂP^{m} are non-empty and equal:

R_{B} = π^{m}(R_{B} \ {0}) = π^{m}(I_{B} \ {0}).

Proof. It follows from m ≥ 0 and Lemma 3.2 that both R_{B} and I_{B} contain non-zero elements, and that they have the same non-empty image under π^{m}. To show π^{m}(R_{B} \ {0}) ⊆ R_{B}, if z = π^{m}(z) for some z ∈ R_{B} \ {0}, then z = B(z) ⟹ z = π^{m}(z) = π^{m}(B(z)) = B(π^{m}(z)) = B(z). For the other direction, suppose z ∈ R_{B}, so B(z) = z = π^{m}(z) for some representative z ≠ 0. It will be enough to show that some non-zero complex scalar multiple of z is an element of R_{B}. B(z) = B(π^{m}(z)) = π^{m}(B(z)) = π^{m}(z) ⟹ B(z) = μz for some non-zero scalar μ. \frac{1}{2}(z + B(z)) = \frac{1}{2}(z + μz) = \frac{1+μ}{2}z ∈ R_{B}, which is enough unless μ = −1, in which case z ∈ I_{B}, and iz ∈ R_{B}.

Theorem 3.7. If B is a conjugate linear involution and V is a complex linear subspace of ℂ^{m+1}, then π^{m}((R_{B} ∩ V) \ {0}) = π^{m}(R_{B} \ {0}) ∩ π^{m}(V \ {0}).

Proof. The inclusion π^{m}((R_{B} ∩ V) \ {0}) ⊆ π^{m}(R_{B} \ {0}) ∩ π^{m}(V \ {0}) is elementary set theory. It remains to show that if z ∈ π^{m}(R_{B}\{0})∩π^{m}(V\{0}), and z has representative z ∈ ℂ^{m+1} \ {0}, then some non-zero complex scalar multiple of z is in (R_{B}∩V)\{0}. From z ∈ π^{m}(V\{0}), we get a representative z ∈ V\{0}, and any non-zero complex scalar multiple of z is still in V\{0}. By Theorem 3.6, any vector z such that π^{m}(z) ∈ π^{m}(R_{B} \ {0}) has some complex scalar multiple in R_{B} \ {0}.

Theorem 3.8. Given a real involution B, and a map A : ℂP^{m} → ℂP^{m} which is either a projective transformation or induced by an invertible conjugate linear map, the composite A ∘ B ∘ A^{−1} is a real involution, with fixed point set R_{A∘B∘A^{−1}} = A(R_{B}).

Proof. If A and B induce A and B, then clearly A ∘ B ∘ A^{−1} is a conjugate linear involution inducing A ∘ B ∘ A^{−1}.

To show R_{A∘B∘A^{−1}} ⊆ A(R_{B}), if z = (A ∘ B ∘ A^{−1})(z), then B(A^{−1}(z)) = A^{−1}(z) ∈ R_{B}, so z is the image under A of the point A^{−1}(z) ∈ R_{B}. To show A(R_{B}) ⊆ R_{A∘B∘A^{−1}}, if z = A(x) for some x ∈ R_{B}, then (A ∘ B ∘ A^{−1})(z) = A(B(A^{−1}(A(x)))) = A(B(x)) = A(x) = z, so z ∈ R_{A∘B∘A^{−1}}.

Proposition 3.9 ([Csikós] Lemma 6.1.3). Given a basis {z_{0}, …, z_{m}} of ℂ^{m+1}, let q = z_{0} + … + z_{m}. If A ∈ M(m + 1, ℂ) is a matrix such that every vector z_{0}, …, z_{m}, q is an eigenvector of A, then A is a multiple of the identity matrix.

Theorem 3.10. Given a projective transformation A of ℂP^{m}, and conjugate linear involutions B_{1}, B_{2}, if A(R_{B_{1}}) ⊆ R_{B_{2}}, then there exists A ∈ GL(m+1, ℂ) so that A induces A, and A = B_{2} ∘ A ∘ B_{1}.

Proof. There is some invertible matrix A_{0} that induces A. Given a non-zero element z ∈ R_{B_{1}}, z = π^{m}(z) ∈ R_{B_{1}} by Theorem 3.6, and A(z) = A(B_{1}(z)) = B_{2}(A(B_{1}(z))). This implies

A(π^{m}(z)) = π^{m}(A_{0}(z)) = B_{2}(A(B_{1}(π^{m}(z)))) = π^{m}(B_{2}(A_{0}(B_{1}(z)))).

So, there is some complex scalar μ (depending on z) so that B_{2}(A_{0}(B_{1}(z))) = μA_{0}(z), and it follows that every non-zero element z ∈ R_{B_{1}} is an eigenvector of the complex linear map A_{0}^{−1} ∘B_{2} ∘A_{0} ∘B_{1}. Since R_{B_{1}} spans ℂ^{m+1} by Lemma 3.2, and is closed under addition, it contains elements z_{0}, …, z_{m}, q satisfying the hypothesis of Proposition 3.9, which gives a non-zero constant λ ∈ ℂ such that λ · A_{0} = B_{2} ∘ A_{0} ∘ B_{1}. Multiplying both sides by λ̄ gives

λ̄·λ·A_{0} = λ̄·B_{2} ∘A_{0} ∘B_{1} = B_{2} ∘(λ·A_{0})∘B_{1} = B_{2} ∘(B_{2} ∘A_{0} ∘B_{1})∘B_{1} = A_{0},

so λ̄ · λ = 1, and λ = e^{iθ} for some θ = θ̄. Let A = e^{iθ/2} · A_{0}, so that

B_{2}∘A∘B_{1} = B_{2}∘(e^{iθ/2}·A_{0})∘B_{1} = e^{−iθ/2}·B_{2}∘A_{0}∘B_{1} = e^{−iθ/2}·(e^{iθ}A_{0}) = A.

Theorem 3.11. Given a map B : ℂP^{m} → ℂP^{m} which is induced by some invertible conjugate linear map ℂ^{m+1} → ℂ^{m+1}, and conjugate linear involutions B_{1}, B_{2}, if B(R_{B_{1}}) ⊆ R_{B_{2}}, then there exists an invertible conjugate linear map B : ℂ^{m+1} → ℂ^{m+1} so that B induces B, and B = B_{2} ∘ B ∘ B_{1}.

Proof. B is induced by some B_{0}, and B_{2}∘B_{0} is an invertible complex linear map which induces a projective transformation B_{2}∘B such that (B_{2}∘B)(R_{B_{1}}) ⊆ R_{B_{2}}. The previous Theorem applies, so there is some A so that A = B_{2}∘A∘B_{1} and A induces B_{2} ∘ B. Let B = B_{2} ∘ A, so it is an invertible conjugate linear map that induces B_{2}∘(B_{2}∘B) = B, and B_{2}∘B∘B_{1} = B_{2}∘(B_{2}∘A)∘B_{1} = B_{2}∘A = B.

Theorem 3.12. Given a projective transformation A of ℂP^{m}, and conjugate linear involutions B_{1}, B_{2}, if A(R_{B_{1}}) ⊆ R_{B_{2}}, then A(R_{B_{1}}) = R_{B_{2}}.

Proof. Let A be as in Theorem 3.10, so it induces A and A = B_{2} ∘ A ∘ B_{1}. It follows that A = B_{2} ∘ A ∘ B_{1}. It will be enough to show that if z ∈ R_{B_{2}}, then A^{−1}(z) ∈ R_{B_{1}}.

B_{1}(A^{−1}(z)) = B_{1}((B_{2} ∘ A ∘ B_{1})^{−1}(z)) = (B_{1} ∘ B_{1} ∘ A^{−1} ∘ B_{2})(z) = A^{−1}(z).

Theorem 3.13. Given a map B : ℂP^{m} → ℂP^{m} induced by an invertible conjugate linear map, and conjugate linear involutions B_{1}, B_{2}, if B(R_{B_{1}}) ⊆ R_{B_{2}}, then B(R_{B_{1}}) = R_{B_{2}}.

Proof. B is induced by some B, and B_{2} ∘B induces a projective transformation B_{2} ∘ B so that (B_{2} ∘ B)(R_{B_{1}}) ⊆ R_{B_{2}}. The previous Theorem showed (B_{2} ∘ B)(R_{B_{1}}) = R_{B_{2}}, and the conclusion follows from applying B_{2} to both sides.

Theorems 3.12 and 3.13 could also have been proved using Theorems 3.3, 3.4, 3.10, and 3.11.

Theorem 3.14. Given a real involution B and projective transformations A_{1}, A_{2}, if A_{1}(x) = A_{2}(x) for all x ∈ R_{B}, then A_{1} = A_{2}.

Proof. B is induced by some B, and let A_{1}, A_{2} ∈ GL(m + 1, ℂ) induce A_{1}, A_{2}, so A_{2}^{−1}∘A_{1} induces A_{2}^{−1}∘A_{1}, which satisfies (A_{2}^{−1}∘A_{1})(x) = (A_{2}^{−1}∘A_{2})(x) = x for all x ∈ R_{B}. Since any non-zero x ∈ R_{B} satisfies π^{m}((A_{2}^{−1} ∘ A_{1})(x)) = (A_{2}^{−1} ∘ A_{1})(π^{m}(x)) = π^{m}(x), every non-zero element of R_{B} is an eigenvector of A_{2}^{−1} ∘ A_{1}, including elements z_{0}, …, z_{m}, q ∈ R_{B} as in the Proof of Theorem 3.10. It follows from Proposition 3.9 that there is some non-zero λ ∈ ℂ such that A_{1} = λA_{2}, and A_{1} = A_{2}.

Theorem 3.15. Given two real involutions B_{1}, B_{2}, there exists a projective transformation A such that A(R_{B_{1}}) = R_{B_{2}}.

Proof. Let B_{1}, B_{2} induce B_{1}, B_{2}. By Lemma 3.2, R_{B_{1}} spans ℂ^{m+1}, and therefore contains a basis {z_{0}, …, z_{m}} of ℂ^{m+1}. Since R_{B_{2}} also spans ℂ^{m+1}, it contains a basis, and there exists some A ∈ GL(m+1, ℂ) taking the first basis to the second. If x ∈ R_{B_{1}}, then it is of the form ∑ c_{p}z_{p} = B_{1}(∑ c_{p}z_{p}) = ∑ c̄_{p}B_{1}(z) = ∑ c̄_{p}z_{p}, so c_{p} = c̄_{p} for p = 0, …, m. It is easy to check A(x) ∈ R_{B_{2}}, so A(R_{B_{1}}) ⊆ R_{B_{2}}, and A(R_{B_{1}}) ⊆ R_{B_{2}}. The equality follows from Theorem 3.12.

Theorem 3.16. The fixed point set of a real involution of ℂP^{m} does not contain any complex projective lines.

Proof. Suppose, toward a contradiction, that B is a conjugate linear involution, and V ⊆ ℂ^{m+1} is a two-dimensional complex linear subspace such that π^{m}(V \ {0}) ⊆ R_{B}. By Theorem 3.6, π^{m}(V \ {0}) ⊆ π^{m}(R_{B} \ {0}), so every element of V is a complex scalar multiple of an element of R_{B}. Let {z_{0}, w_{0}} ⊆ V be a linearly independent (over ℂ) set; there are scalars λ, μ ∈ ℂ so that {z_{1} = λz_{0}, w_{1} = μw_{0}} ⊆ R_{B}, and {z_{1}, w_{1}} is a basis of V. The vector z_{1} + iw_{1} ∈ V is non-zero, and is a non-zero multiple of an element of R_{B}, so for some ν ≠ 0,

ν(z_{1} + iw_{1}) = B(ν(z_{1} + iw_{1})) = ν̄B(z_{1} + iw_{1}) = ν̄(z_{1} − iw_{1}).

Since {z_{1}, w_{1}} is a basis of V, the coefficients must be equal: ν = ν̄ and iν = −iν̄. However, these equations imply ν = 0, a contradiction.

Theorem 3.17. Given two distinct complex projective lines V_{1}, V_{2} in ℂP^{m}, and a real involution B, if V_{1} intersects R_{B} in at least two points, and V_{2} also intersects R_{B} in at least two points, then V_{1} ∩ V_{2} ⊆ R_{B}.

Proof. First, V_{1} ∩ V_{2} could be the empty set, and the claim trivially follows.

Otherwise, denote two distinct points u, v ∈ V_{1}∩R_{B}, with V_{1} = π^{m}(V_{1}\{0}) for some 2-dimensional complex linear subspace V_{1} in ℂ^{m+1}, and let B be induced by some conjugate linear involution B. Similarly, denote two distinct points x, y ∈ V_{2} ∩ R_{B}, with V_{2} = π^{m}(V_{2} \ {0}). It follows from the hypothesis V_{1} ≠ V_{2} that V_{1} ≠ V_{2}, and from the current assumption V_{1} ∩ V_{2} ≠ Ø that V_{1} ∩ V_{2} is a one-dimensional complex linear subspace of ℂ^{m+1}.

By Theorem 3.7, u, v ∈ π^{m}(V_{1}\{0})∩π^{m}(R_{B}\{0}) = π^{m}((V_{1}∩R_{B})\{0}), so there are some representatives u, v ∈ (V_{1} ∩ R_{B}) \ {0} which form a basis of V_{1} (since if they were not independent, then u = v). Similarly, there are x, y ∈ (V_{2} ∩ R_{B}) \ {0} which form a basis of V_{2}. Let z be any representative of z ∈ V_{1} ∩ V_{2}, so z ∈ (V_{1} ∩ V_{2}) \ {0}. There are coefficients α, β ∈ ℂ, not both 0, so that z = αu + βv, and similarly, z = γx + δy. One of the following elements of R_{B} must be non-zero:

\frac{1}{2}(z + B(z)) = \frac{1}{2}(αu + βv + B(αu + βv)) = \frac{1}{2}(α + ᾱ)u + \frac{1}{2}(β + β̄)v
= \frac{1}{2}(γx + δy + B(γx + δy)) = \frac{1}{2}(γ + γ̄)x + \frac{1}{2}(δ + δ̄)y,
\frac{1}{2i}(z − B(z)) = \frac{1}{2i}(αu + βv − B(αu + βv)) = \frac{1}{2i}(α − ᾱ)u + \frac{1}{2i}(β − β̄)v
= \frac{1}{2i}(γx + δy − B(γx + δy)) = \frac{1}{2i}(γ − γ̄)x + \frac{1}{2i}(δ − δ̄)y.

These linear combinations show that there is a non-zero element of R_{B} which is on the one-dimensional complex linear subspace V_{1} ∩ V_{2}, so it must be a complex scalar multiple of z, and z = π^{m}(z) ∈ π^{m}((V_{1} ∩ V_{2} ∩ R_{B}) \ {0}) ⊆ π^{m}(R_{B} \ {0}) = R_{B}.

4 B-lines

At this point, we will take for granted some of the incidence axioms for complex projective geometry. They are straightforward consequences of the linear algebra definition of complex projective subspaces.

1. For any two distinct points in ℂP^{m}, there exists a complex projective line containing them. The fact is true but trivial for m = 0 or 1.

2. If the intersection of a complex projective line L and a complex projective subspace V contains more than one point, then L ⊆ V. (In particular, the line from the previous axiom is unique.)

3. Any two complex projective lines contained in a complex projective plane have a non-empty intersection. Conversely, if two complex projective lines have a non-empty intersection, then they coincide or are contained in a complex projective plane.

Definition 4.1. Given a real involution B, and two distinct points in R_{B}, define the “B-line” through the two points to be the intersection of R_{B} with the unique complex line containing the two given points.

Theorem 4.2. Given a real involution B, and two B-lines k and ℓ, either k = ℓ or k intersects ℓ in at most one point in R_{B}. If k and ℓ are both contained in some complex projective plane U, then k intersects ℓ in at least one point.

Proof. Let k = V_{1} ∩ R_{B} and ℓ = V_{2} ∩ R_{B}, for complex projective lines V_{1}, V_{2}, so k ∩ ℓ = (V_{1} ∩ V_{2}) ∩ R_{B}. As a consequence of the incidence axioms, V_{1} ∩ V_{2} is equal to either V_{1}, or a set containing at most one point, and the first claim follows. If k is contained in a complex projective plane U, then V_{1} ∩ U contains at least two points in k, so V_{1} ⊆ U (the incidence axioms again), and similarly if ℓ ⊆ U, then V_{2} ⊆ U. By the third incidence axiom, there is some element z ∈ V_{1} ∩ V_{2}. If V_{1} = V_{2}, then k = ℓ, so k ∩ ℓ contains at least two points. If V_{1} ≠ V_{2}, z is unique, and Theorem 3.17 applies (since k = V_{1} ∩ R_{B} and ℓ = V_{2} ∩ R_{B} each contain at least two points by the definition of B-line), to show that z ∈ (V_{1} ∩ V_{2}) ∩ R_{B} = k ∩ ℓ.

Any R_{B}, together with the set of B-lines contained in R_{B}, can be considered as an abstract projective space satisfying various incidence properties. Without going into the details, we list some of these axioms for a given set R_{B} ⊆ ℂP^{m}.

1. For any two distinct points in R_{B}, there exists exactly one B-line containing them.

2. R_{B} contains at least one point, and there exist infinitely many points on each B-line.

3. For m > 0, R_{B} contains infinitely many points and at least one B-line.

4. For m > 1, R_{B} contains infinitely many B-lines, and not all points are on one B-line.

5. For m > 2, not every pair of B-lines meets.

6. (Pasch-Veblen-Young [Seidenberg]) Given points u, v, x, y ∈ R_{B}, no three of which lie on the same B-line, let k denote the B-line through u, v, and ℓ the B-line through x, y, and p the B-line through u, y, and q the B-line through v, x. If k ∩ ℓ ≠ Ø, then p ∩ q ≠ Ø.

7. The theorems of Pappus and Desargues ([Csikós], [Seidenberg]) hold in R_{B}.

The first above item summarizes Theorem 4.2, and the next few properties are obvious and proofs are omitted. The PVY, Pappus, and Desargues properties follow as a consequence of these properties of ℂP^{m}, together with Theorems 3.17 and 4.2.

Theorem 4.3 ([E] Proposition III.10). Given real involutions B_{1}, B_{2}, a B_{1}-line ℓ, and a B_{2}-line k, if ℓ intersects k in three distinct points, then ℓ = k.

Proof. Let ℓ = V ∩R_{B_{1}}, k = V′∩R_{B_{2}}. Since V′ meets V in distinct points, V′ = V and k = V ∩ R_{B_{2}}. Let x, y, z be distinct points in ℓ ∩ k, with representatives x, y, z ∈ V∩R_{B_{1}} as in Theorem 3.17. Any two of these representatives span V, so, for example, there are non-zero complex constants α, β with z = αx + βy. It follows from z = B_{1}(z) that α = ᾱ and β = β̄.

Since x, y, z ∈ k ⊆ R_{B_{2}}, they are also represented by λx, μy, νz ∈ R_{B_{2}}, for some non-zero complex numbers λ, μ, ν.

νz = B_{2}(νz)
⟹ ν(αx + βy) = B_{2}(ν(αx + βy))
⟹ (ν · α)x + (ν · β)y = B_{2}((ν · α/λ)λx + (ν · β/μ)μy)
= \overline{(\frac{ν · α}{λ})}B_{2}(λx) + \overline{(\frac{ν · β}{μ} )}B_{2}(μy)
= (\frac{ν̄ · ᾱ}{λ̄} · λ) x + (\frac{ν̄ · β̄}{μ̄} · μ) y.

From the equality of the coefficients, the properties α = ᾱ, β = β̄, and the fact that α and β are both non-zero (which is where we are using the assumption that x, y, z are distinct), a little calculation shows that \frac{λ}{λ̄} = \frac{μ}{μ̄} = \frac{ν}{ν̄}.

To show ℓ ⊆ k, let u be any point in ℓ, with representative u = δx + ϵy ∈ R_{B_{1}}, δ = δ̄, ϵ = ϵ̄. Then B_{2}(u) = π^{m}(B_{2}(u)) = π^{m}(B_{2}(δx + ϵy)) = π^{m}(B_{2}((δ/λ)λx + (ϵ/μ)μy)) = π^{m}((δ̄ · \frac{λ}{λ̄})x + (ϵ̄· \frac{μ}{μ̄})y)) = π^{m}(\frac{λ}{λ̄}(δx + ϵy)) = u, so u ∈ V ∩ R_{B_{2}} = k. An analogous argument shows k ⊆ ℓ.

Theorem 4.4. Given a projective transformation A of ℂP^{m}, real involutions B_{1}, B_{2}, and a B_{1}-line ℓ, if there are three distinct points in ℓ whose images under A are elements of R_{B_{2}}, then A(ℓ) is a B_{2}-line.

Proof. ℓ = V ∩R_{B_{1}}, for some complex projective line V, so A(ℓ) = A(V ∩R_{B_{1}}) ⊆ A(V), where A(V) is also a complex projective line. Let k denote the B_{2}-line A(V) ∩ R_{B_{2}}, so A(ℓ) meets k in three distinct points (since A is one-to-one).

The composite map B_{3} = A ∘ B_{1} ∘ A^{−1} is a real involution, by Theorem 3.8, such that R_{B_{3}} = A(R_{B_{1}}). Since A is one-to-one, A(ℓ) = A(V ∩ R_{B_{1}}) = A(V) ∩ A(R_{B_{1}}) = A(V) ∩ R_{B_{3}}. Since A(ℓ) is a B_{3}-line that meets the B_{2}-line k in three distinct points, A(ℓ) = k by Theorem 4.3.

Theorem 4.5. Given a map B : ℂP^{m} → ℂP^{m} induced by an invertible conjugate linear map, real involutions B_{1}, B_{2}, and a B_{1}-line ℓ, if there are three distinct points in ℓ whose images under B are elements of R_{B_{2}}, then B(ℓ) is a B_{2}-line.

Proof. The previous Theorem applies to the projective transformation B_{2} ∘ B, which takes three distinct points in ℓ = V ∩ R_{B_{1}} into R_{B_{2}}. The conclusion is that (B_{2} ∘ B)(ℓ) ⊆ R_{B_{2}} is a B_{2}-line, equal to the set B(ℓ) = B(V) ∩ R_{B_{2}}.

Theorem 4.6. Given m ≠ 1, a collineation g of ℂP^{m}, real involutions B_{1}, B_{2}, and a B_{1}-line ℓ, if g(ℓ) ⊆ R_{B_{2}}, then g is either a projective transformation, or it is induced by some invertible conjugate linear map B : ℂ^{m+1} → ℂ^{m+1}.

Proof. Let x, y be distinct points on ℓ = V ∩ R_{B_{1}} = π^{m}((V ∩ R_{B_{1}}) \ {0}), with complex linearly independent representatives x, y ∈ V ∩ R_{B_{1}} as in the Proof of Theorem 3.17.

Let g be induced by some ϕ̃ ∘ A : ℂ^{m+1} → ℂ^{m+1} as in Proposition 2.4. By the hypothesis that g(x) ∈ R_{B_{2}}, g(x) = π^{m}((ϕ̃ ∘ A)(x)) ∈ π^{m}(R_{B_{2}}), so there is some non-zero complex number λ so that λ(ϕ̃ ∘ A)(x) ∈ R_{B_{2}}. Similarly, there is some non-zero μ so that μ(ϕ̃ ∘ A)(y) ∈ R_{B_{2}}. The vectors (ϕ̃ ∘ A)(x), (ϕ̃ ∘ A)(y) are independent (since otherwise, g(x) = g(y), but g is assumed to be one-to-one).

Let ζ be any complex number such that ζ = ζ̄, and consider z = x+ζy. It is easy to check that z ∈ (V∩R_{B_{1}})\{0}, so z = π^{m}(z) ∈ ℓ, g(z) ∈ R_{B_{2}}, and as with x and y, there is some non-zero ν_{ζ} (depending on ζ) so that ν_{ζ}(ϕ̃∘A)(z) ∈ R_{B_{2}}.

We also have the following expression:

ν_{ζ}(ϕ̃ ∘ A)(z) = ν_{ζ}(ϕ̃ ∘ A)(x + ζy)
= ν_{ζ}((ϕ̃ ∘ A)(x) + (ϕ̃ ∘ A)(ζy))
= ν_{ζ}((ϕ̃ ∘ A)(x) + ϕ(ζ)(ϕ̃ ∘ A)(y))
= ν_{ζ}(ϕ̃ ∘ A)(x) + (ν_{ζ} · ϕ(ζ))(ϕ̃ ∘ A)(y)
= (ν_{ζ} · \frac{1}{λ})(λ(ϕ̃ ∘ A)(x)) + (ν_{ζ} · ϕ(ζ) · \frac{1}{μ})(μ(ϕ̃ ∘ A)(y)),

which should be invariant under B_{2}, so:

ν_{ζ}(ϕ̃ ∘ A)(z) = B_{2}(ν_{ζ}(ϕ̃ ∘ A)(z))
= B_{2}(\frac{ν_{ζ}}{λ} (λ(ϕ̃ ∘ A)(x)) + \frac{ν_{ζ} · ϕ(ζ)}{μ}(μ(ϕ̃ ∘ A)(y)))
= \overline{(\frac{ν_{ζ}}{λ} )}B_{2}(λ(ϕ̃ ∘ A)(x)) + \overline{(\frac{ν_{ζ} · ϕ(ζ)}{μ})}B_{2}(μ(ϕ̃ ∘ A)(y))
= \frac{\overline{ν_{ζ}}}{λ̄} (λ(ϕ̃ ∘ A)(x)) + \frac{\overline{ν_{ζ}} · \overline{ϕ(ζ)}}{μ̄}(μ(ϕ̃ ∘ A)(y)).

From the uniqueness of the coefficients of the linearly independent vectors (ϕ̃ ∘ A)(x) and (ϕ̃ ∘ A)(y), we get ν_{ζ} = \overline{ν_{ζ}} · λ/λ̄ and ν_{ζ} · ϕ(ζ) = \overline{ν_{ζ}} · \overline{ϕ(ζ)} · μ/μ̄, and it follows from a short calculation that ϕ(ζ) = \overline{ϕ(ζ)} · \frac{λ̄·μ}{λ·μ̄}. In particular,

that equation holds for ζ = 1, when ϕ(1) = 1 = \overline{ϕ(1)}, so \frac{λ̄·μ}{λ·μ̄} = 1, independent of ζ. The conclusion is that ϕ(ζ) = \overline{ϕ(ζ)} for all ζ such that ζ = ζ̄, but it is known ([Y]) that the only such field isomorphisms with that property are the identity map, so ϕ̃ ∘ A = A, and complex conjugation, ϕ = κ, so κ̃ ∘ A = B is a conjugate linear map.

It also follows from the hypothesis of Theorem 4.6 that g(ℓ) is a B_{2}-line, by Theorems 4.4 and 4.5.

Corollary 4.7. Given m ≠ 1, a collineation g of ℂP^{m}, and conjugate linear involutions B_{1}, B_{2} of ℂ^{m+1}, if g(R_{B_{1}}) ⊆ R_{B_{2}}, then there exists an invertible map A : ℂ^{m+1} → ℂ^{m+1} such that A induces g, A = B_{2} ∘ A ∘ B_{1}, and A is either complex linear or conjugate linear. It also follows that g(R_{B_{1}}) = R_{B_{2}}.

Proof. The m = 0 case is trivial. For m > 1, R_{B_{1}} contains at least two points, and some B_{1}-line ℓ, so the previous Theorem applies. In the first case, where g = A, Theorem 3.10 gives the required complex linear A. In the second case, where g is induced by a conjugate linear map B, Theorem 3.11 gives the required conjugate linear map, which turns out to be some complex scalar multiple of B. The equality of sets follows from Theorems 3.12, 3.13.

Corollary 4.8. Given m ≠ 1, collineations g and h of ℂP^{m}, and a real involution B, if h(x) = g(x) for all x ∈ R_{B}, then either g = h or g = h ∘ B.

Proof. For x ∈ R_{B}, g^{−1}(h(x)) = g^{−1}(g(x)) = x, so g^{−1} ∘ h is a collineation such that (g^{−1} ∘ h)(R_{B}) ⊆ R_{B}.

The previous Corollary applies. In the first case, where g^{−1}∘h is a projective transformation, it agrees with the identity transformation on R_{B}, and g = h by Theorem 3.14. In the second case, g^{−1} ∘ h ∘ B is a projective transformation that agrees with the identity on R_{B}, so g = h ∘ B.

5 The cross ratio

Definition 5.1. Define the cross ratio of four points a = [a_{0} : a_{1}], b = [b_{0} : b_{1}], c = [c_{0} : c_{1}], d = [d_{0} : d_{1}] on the complex projective line by the following rational map.

χ : ℂP^{1} × ℂP^{1} × ℂP^{1} × ℂP^{1} → ℂP^{1} :
(a, b, c, d) ↦ [(a_{1}d_{0} − a_{0}d_{1})(b_{1}c_{0} − b_{0}c_{1}) : (a_{1}c_{0} − c_{1}a_{0})(b_{1}d_{0} − b_{0}d_{1})].

The function χ is well-defined except for those quadruples of points where three coincide.

Notation 5.2. Any function f : ℂP^{1} → ℂP^{1} induces the obvious product map, denoted f^{4} : (a, b, c, d) ↦ (f(a), f(b), f(c), f(d)).

The next two facts are well-known and each is proved by a short calculation.

Proposition 5.3. For any d ∈ ℂP^{1}, χ(([0 : 1], [1 : 0], [1 : 1], d)) = d.

Proposition 5.4. For any A ∈ P GL(2, ℂ), χ ∘ A^{4} = χ.

Theorem 5.5. If B : ℂP^{1} → ℂP^{1} is induced by an invertible conjugate linear map B : ℂ^{2} → ℂ^{2}, then χ ∘ B^{4} = κ̃ ∘ χ.

Proof. This follows from B = κ̃ ∘ A for some A, the easily checked identity χ∘κ̃^{4} = κ̃∘χ, and the above Proposition, applied to χ∘(κ̃∘A)^{4} = χ∘κ̃^{4} ∘A^{4}.

Lemma 5.6. Given three distinct points a, b, c ∈ ℂP^{1}, there exists some A ∈ P GL(2, ℂ) such that

A(a) = [0 : 1], A(b) = [1 : 0], A(c) = [1 : 1].

Proof. Consider representative vectors a = (a_{0}, a_{1})^{T}, b = (b_{0}, b_{1})^{T}, c = (c_{0}, c_{1})^{T} ∈ ℂ^{2} \{0} for a, b, and c. Any two of these three representatives are linearly independent because a, b, c are distinct, and there exists some complex linear map A_{1} such that A_{1}(a) = (0, 1)^{T}, A_{1}(b) = (1, 0)^{T}, and A_{1}(c) = (c′_{0}, c′_{1})^{T}, with

c′_{0} ≠ 0 and c′_{1} ≠ 0. Then let A_{2} = ( \begin{matrix}1/c′_{0} & 0 \\ 0 & 1/c′_{1} \end{matrix} ), so that A = A_{2} ∘ A_{1} is the

required map.

The following converse of Proposition 5.4, that maps preserving the cross ratio must be projective transformations, is also well known.

Theorem 5.7. Given any map f : ℂP^{1} → ℂP^{1}, if χ ∘ f^{4} = χ, then f ∈ P GL(2, ℂ).

Proof. First, f must be one-to-one; otherwise, let z, w ∈ ℂP^{1} be points such that z ≠ w and f(z) = f(w), and pick any point x ∈ ℂP^{1} such that x ≠ z. Then, χ((x, z, z, w)) is defined, but χ((f(x), f(z), f(z), f(w))) is not.

Since f is one-to-one, the points a = f([0 : 1]), b = f([1 : 0]), c = f([1 : 1]) are distinct. By Lemma 5.6, there is some A ∈ P GL(2, ℂ) so that A ∘ f fixes each of the points [0 : 1], [1 : 0], [1 : 1]. By Proposition 5.4 and the hypothesis, χ ∘ (A ∘ f)^{4} = χ ∘ A^{4} ∘ f^{4} = χ, so for any point d ∈ ℂP^{1}, expanding (χ ∘ (A ∘ f)^{4})(([0 : 1], [1 : 0], [1 : 1], d)) in two different ways gives:

χ(([0 : 1], [1 : 0], [1 : 1], (A ∘ f)(d))) = χ(([0 : 1], [1 : 0], [1 : 1], d))
= (A ∘ f)(d) = d.

This shows f = A^{−1}.

Corollary 5.8. Given any map f : ℂP^{1} → ℂP^{1}, if χ ∘ f^{4} = κ̃ ∘ χ, then f is induced by some invertible conjugate linear map ℂ^{2} → ℂ^{2}.

Proof. If χ∘f^{4} = κ̃∘χ, then χ∘(κ̃∘f)^{4} = χ∘κ̃^{4} ∘f^{4} = κ̃∘χ∘f^{4} = κ̃∘κ̃∘χ = χ, so the previous Theorem applies to show κ̃ ∘ f ∈ P GL(2, ℂ).

Theorem 5.9. Given four points a, b, c, d ∈ ℂP^{1}, at least three distinct, there exists a real involution B such that {a, b, c, d} ⊆ R_{B} if and only if χ((a, b, c, d)) = (κ̃ ∘ χ)((a, b, c, d)).

Proof. One direction is easy: if B^{4}((a, b, c, d)) = (a, b, c, d), then by Theorem 5.5, χ((a, b, c, d)) = (χ ∘ B^{4})((a, b, c, d)) = (κ̃ ∘ χ)((a, b, c, d)).

Conversely, suppose χ((a, b, c, d)) = (κ̃ ∘ χ)((a, b, c, d)), and since the other cases will be similar, suppose a, b, and c are distinct. Let A ∈ P GL(2, ℂ) be the map from Lemma 5.6, so that

A^{4}((a, b, c, d)) = ([0 : 1], [1 : 0], [1 : 1], [d_{0} : d_{1}]).

Finding the cross ratio of both sides, and then using the hypothesis, gives

(χ ∘ A^{4})((a, b, c, d)) = χ(([0 : 1], [1 : 0], [1 : 1], [d_{0} : d_{1}]))
= χ((a, b, c, d)) = [d_{0} : d_{1}]
= (κ̃ ∘ χ)((a, b, c, d)) = κ̃([d_{0} : d_{1}]).

So, [0 : 1], [1 : 0], [1 : 1], and [d_{0} : d_{1}] are all in the fixed point set Rκ̃. Using Theorem 3.8, a, b, c, d are in A^{−1}(Rκ̃) = R_{A^{−1}∘κ̃∘A}.

To establish the last result, Theorem 5.12, we depart from our attempt to use only complex projective geometry. We will need some facts about real affine geometry, including the following real affine version of the Fundamental Theorem. See [H], [Samuel].

Proposition 5.10. If f : ℝ^{2} → ℝ^{2} is invertible and takes real affine lines to real affine lines, then f is an affine transformation, of the form f(x⃗) = L(x⃗)+t⃗, for some invertible real linear transformation L and vector t⃗.

The next Theorem will make a connection between complex projective geometry and real affine geometry, using the usual identification of ℂ with ℝ^{2}, so that the complex number x+iy corresponds to the ordered pair (x, y). ℂ will also be considered as an affine neighborhood of ℂP^{1}, the complement of the singleton set {[0 : 1]}, so any complex number z corresponds to [1 : z] ∈ ℂP^{1} \ {[0 : 1]}.

Theorem 5.11. Given a real involution B of ℂP^{1}, if [0 : 1] ∈ R_{B}, then the intersection of R_{B} with the affine neighborhood ℂP^{1} \ {[0 : 1]} = ℂ = ℝ^{2} is a real affine line. Conversely, every real affine line is such an intersection.

Proof. Recall the definition of a real affine line, to be the set of points (x, y) in ℝ^{2} satisfying an equation Ex+F y +G = 0, for E, F, G ∈ ℝ, with E, F not both 0. Converting to the z coordinate system gives E \frac{z+z̄}{2} + F \frac{z−z̄}{2i} + G = \frac{E−iF}{2} z + \frac{E+iF}{2} z̄ + G = 0, a self-conjugate inhomogeneous linear equation. Similarly converting any equation of the form λz + μz̄ + G = 0, with λ̄ = μ = \frac{E+iF}{2} ≠ 0 and Ḡ = G, back to the (x, y) coordinate system gives the equation of a real affine line.

Supposing B is induced by a conjugate linear map, it is of the form B([z_{0} : z_{1}]) = [az̄_{0} +bz̄_{1} : cz̄_{0} +dz̄_{1}], for complex numbers a, b, c, d such that ad−bc ≠ 0. Since B fixes [0 : 1], b = 0, a ≠ 0, d ≠ 0. (B ∘ B)([z_{0} : z_{1}]) = [aāz_{0} : (cā+dc̄)z_{0} +dd̄z_{1}], and since B is an involution, aā = dd̄ and cā+dc̄ = 0. Since B is invertible, its restriction to the complement of {[0 : 1]}, [1 : z] ↦ [a : c+dz̄] is also invertible, and can be written B : ℂ → ℂ : B(z) = \frac{d}{a} z̄ + \frac{c}{a}. |\frac{d}{a}|^{2} = 1, so B can also be written B(z) = −e^{iθ}z̄ + β, where −e^{iθ}β̄ + β = \frac{d}{a} \frac{c̄}{ā} + \frac{c}{a} = 0.

The intersection of R_{B} with the affine neighborhood ℂ is the solution set of z = −e^{iθ}z̄ + β, and multiplying this equation by e^{−iθ/2} gives

e^{−iθ/2}z + e^{iθ/2}z̄ − e^{−iθ/2}β = 0.  (3)

Multiplying the equation −e^{iθ}β̄ + β = 0 by the same quantity gives −e^{iθ/2}β̄ + e^{−iθ/2}β = 0, so e^{−iθ/2}β ∈ ℝ, and (3) is the self-conjugate equation of a real affine line.

Conversely, given any real affine line, with equation λz + λ̄z̄+ G = 0, λ ≠ 0, G = Ḡ, define B : ℂP^{1} → ℂP^{1} by B([z_{0} : z_{1}]) = [−λz̄_{0} : Gz̄_{0} + λ̄z̄_{1}]. B is a real involution that fixes [0 : 1], and restricts to B(z) = −\frac{λ̄}{λ} z̄ − \frac{G}{λ} on ℂ. The equation for the fixed point set in ℂ is z = −\frac{λ̄}{λ} z̄− \frac{G}{λ} ⟺ λz + λ̄z̄+ G = 0.

Theorem 5.12. Given an invertible map f : ℂP^{1} → ℂP^{1}, the following are equivalent.

1. For every real involution B, there exists some real involution B′ such that f(R_{B}) ⊆ R_{B′}.

2. For every quadruple of points (a, b, c, d), if χ((a, b, c, d)) ∈ Rκ̃, then (χ ∘ f^{4})((a, b, c, d)) ∈ Rκ̃.

3. Either f or κ̃ ∘ f is a projective transformation.

Proof. The equivalence of 1 . and 2 . follows from Theorem 5.9. 3 . ⟹ 2 . by Proposition 5.4 and Theorem 5.5, and 3 . ⟹ 1 . by Theorem 3.8. Showing 1 . and 2 . imply 3 . is enough for the claimed equivalence.

As in the Proof of Theorem 5.7, since f is one-to-one, the points a = f([0 : 1]), b = f([1 : 0]), c = f([1 : 1]) are distinct, and by Lemma 5.6, there is some A ∈ P GL(2, ℂ) so that A ∘ f fixes each of the points [0 : 1], [1 : 0], [1 : 1]. A ∘ f also satisfies 1 . and 2 . by Theorem 3.8 and Proposition 5.4. Let g denote the restriction of A ∘ f to the affine neighborhood ℂP^{1} \ {[0 : 1]}, so g is an invertible map ℂ → ℂ.

Given any real affine line in ℂ, there is, by Theorem 5.11, some real involution B such that R_{B} is the union of the line and the point [0 : 1]. By 1 ., (A ∘ f)(R_{B}) ⊆ R_{B′} for some real involution B′, and since A ∘ f is one-to-one and fixes [0 : 1], the restriction g takes the given line to the complement of [0 : 1] in R_{B′}, which by Theorem 5.11 again, is another real affine line. Proposition 5.10 applies, and since g fixes 0 = (0, 0) and 1 = (1, 0) in ℂ = ℝ^{2}, it is of the form g((x, y)) = (x + py, qy), for some p, q ∈ ℝ, with q ≠ 0.

For any α ∈ ℂ such that αᾱ = 1, α ≠ ±1, it is easy to check that the points [1 : 1], [1 : α], [1 : −1], [1 : −α] are distinct fixed points of the real involution B([z_{0} : z_{1}]) = [z̄_{1} : z̄_{0}]. It can also be checked directly that the four points in that order have cross ratio in Rκ̃. In terms of the complex affine neighborhood, the four points are 1, α, −1, and −α, and g fixes 1 and −1. By 2 ., the following cross ratio should also be in Rκ̃:

χ(([1 : 1], [1 : g(α)], [1 : −1], [1 : g(−α)]))
= [(1 + g(α))^{2} : 4g(α)]
= [1 : \frac{4g(α)(1 + \overline{g(α)})^{2}}{(1 + g(α))^{2}(1 + \overline{g(α)})^{2}} ],

which is equivalent to g(α) + |g(α)|^{2}\overline{g(α)} ∈ ℝ. Let α = r + is, with r^{2} + s^{2} = 1, s ≠ 0, so g(α) = r + ps + iqs. The imaginary part of g(α) + |g(α)|^{2}\overline{g(α)} is qs(1 − ((r + ps)^{2} + (qs)^{2})), which is zero if and only if

(r + ps)^{2} + (qs)^{2} = 1.  (4)

This equation holds for α = i, so r = 0, s = 1, and p^{2} + q^{2} = 1. Expanding (4) gives r^{2} + 2rps + (p^{2} + q^{2})s^{2} = 1, which is equivalent to 2rps = 0. Since this holds for some α with non-zero r and s, p must be 0, and it follows that q = ±1.

The conclusion is that g(z) is either the identity function on ℂ, or complex conjugation, so A∘f is either the identity function on ℂP^{1}, or κ̃, and 3 . follows.

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