Meaning of complex projective line | Babel Free

Definitions

A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".

analytic

Examples

“By way of an example, we consider in detail the complex projective line #92;C#92;mathbb#123;P#125;¹.[…]Thus via the function w#95;0 the complex projective line #92;C#92;mathbb#123;P#125;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point at infinity"). 2..2.1 Theorem The complex projective line #92;C#92;mathbb#123;P#125;¹ is diffeomorphic to the 2-dimensional sphere S².”

“Finally, C is interested in the geometry of the complex projective line P¹(#92;C).”

“Hence, C(#92;Gamma) is an uncountable union of complex projective lines.”

“As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #92;dot#123;S#125;² ̇provides a conformal model for the complex projective line #92;C#92;mathbb#123;P#125;¹.”

CEFR level