complex projective line

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English

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Noun

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complex projective line (plural complex projective lines)

  1. (analytic geometry, projective geometry) 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".
    • 1985, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, translated by R. G. Burns, Modern Geometry — Methods and Applications: Part II, Springer, page 18:
      By way of an example, we consider in detail the complex projective line . [] Thus via the function the complex projective line 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 is diffeomorphic to the 2-dimensional sphere .
    • 2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:
      Finally, C is interested in the geometry of the complex projective line .
    • 2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups[1], Springer (Birkhäuser), page 142:
      Hence, is an uncountable union of complex projective lines.
    • 2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194:
      As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere ̇provides a conformal model for the complex projective line .

Usage notes

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(complex numbers plus point at infinity):

  • The formal definition above is a version of the definition of a projective space by homogeneous coordinates. (See also Projective space on Wikipedia.Wikipedia )
  • Notations include: .

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