Complex projective line

The complex projective line is the set of equivalence classes of nonzero vectors (z₀, z₁) in C² under scalar multiplication, written [z₀ : z₁]. Each class corresponds to a one-dimensional complex subspace, and the standard affine chart z₁ ≠ 0 maps [z₀ : z₁] to z = z₀/z₁ in C, with the missing point [1 : 0] representing infinity. This gives a natural bijection with the Riemann sphere, the one-point compactification of the complex plane.¹

As a complex manifold, CP¹ has complex dimension 1 and real dimension 2, and it is diffeomorphic to the 2-sphere S². It carries a unique complex structure up to isomorphism, making it the simplest compact Riemann surface. Its holomorphic automorphism group is the Möbius group PSL(2, C), which acts transitively and is isomorphic to the group of orientation-preserving conformal transformations of the sphere.²

Algebraically, CP¹ is a projective algebraic variety defined by the homogeneous coordinates [z₀ : z₁], and its holomorphic line bundles are classified by integers: the tautological bundle has degree −1, and the hyperplane bundle has degree +1. The tangent bundle has degree 2, reflecting the fact that the Euler characteristic is 2.³

The Fubini–Study metric on CP¹ is the round metric on the sphere, with constant Gaussian curvature 4. This metric is Kähler and gives CP¹ the structure of a Kähler manifold. In algebraic geometry, CP¹ is the base of the Hopf fibration S³ → S², a principal U(1)-bundle that is fundamental in topology and physics.⁴

In complex analysis, the Riemann sphere is the natural domain for meromorphic functions, which are holomorphic maps to CP¹. Rational functions correspond to holomorphic maps from CP¹ to itself, and the sphere's compactness ensures that every meromorphic function on the sphere is rational.⁵

In physics, CP¹ appears as the target space of sigma models in two-dimensional quantum field theory, and its instantons are classified by the degree of the map, corresponding to the winding number. In twistor theory, CP¹ parametrizes the celestial sphere of light rays through a point in Minkowski space, a construction due to Roger Penrose.⁶

Beyond its standard presentations, CP¹ has several subtle features. Its moduli space of holomorphic maps of degree d to CP¹ is a projective space of dimension 2d+1, a fact used in enumerative geometry. The sphere's complex structure is rigid: there is no continuous family of complex structures on S², a result known as the uniformization theorem.⁷

In number theory, CP¹ over finite fields F_q has q+1 points, a simple count that underlies the Weil conjectures. The sphere also appears in the theory of quaternions: the unit quaternions SU(2) double-cover SO(3), and the quotient by the center is PSL(2, C), linking CP¹ to rotations in three dimensions.⁸

Riemann sphere

The complex projective line, viewed as the one-point compactification of the complex plane.

One-point compactification

A topological construction that adds a single point to a space to make it compact.

Möbius group

The group of fractional linear transformations of the Riemann sphere.

Fubini–Study metric

A Kähler metric on complex projective space, which on CP¹ is the round sphere metric.

Hopf fibration

The map S³ → S² with fiber S¹, a principal U(1)-bundle.