nLab complex projective 3-space

Idea

Complex projective 3-space $\mathbb{C}P^3$ is the complex projective space $\mathbb{C}P^n$ for $n = 3$.

Properties

Coset space realization

Complex projective space has the following coset space-realizations:

• special orthogonal group in dimension 5 quotiented by unitary group in complex dimension 2:

  $$\mathbb{C}P^3 \;\simeq\; SO(5)/U(2)$$

• Sp(2) quotiented by Sp(1) times circle group:

  $$\mathbb{C}P^3 \;\simeq\; Sp(2)/ \big( Sp(1)_L \times \mathrm{U}(1)_R \big)$$

  (e.g. Onishchik 60, Таблица 1, Zandi 88, 7, Iriye 90, (3), Gorbatsevich-Onishchik 93, Table 3 Butruille 06, p. 2)

Calabi-Penrose fibration

… twistor space … Calabi-Penrose fibration

Related concepts

• complex projective space

  • complex projective line

  • complex projective plane

References

Complex projective 3-space is conceived in the guise as the twistor space of 4d Minkowski spacetime in

• Roger Penrose, Section VI of: Twistor algebra, Journal of Mathematical Physics 8 (2): 345–366, (1967) (doi:10.1063/1.1705200)

Discussion of $\mathbb{C}P^3$ as the domain of the twistor fibration and with an eye towards Yang-Mills theory:

• Michael Atiyah, Section III.1 of: Geometry of Yang-Mills fields, Pisa, Italy: Sc. Norm. Sup. (1979) 98 (spire:150867)

See also

• A. L. Onishchik, On compact Lie groups transitive on certain manifolds, Dokl. Akad. Nauk SSSR, 135:3 (1960), 531–534 (mathnet:dan24279)

• Ahmad Zandi, Minimal immersion of surfaces in quaternionic projective space, Tsukuba Journal of Mathematics Vol. 12, No. 2 (1988), pp. 423-440 (18 pages) (jstor:43686661)

• Kouyemon Iriye, Manifolds which have two projective space bundle structures from the homotopical point of view, J. Math. Soc. Japan Volume 42, Number 4 (1990), 639-658 (euclid:jmsj/1227108441)

• V. Gorbatsevich, A. L. Onishchik, Compact homogeneous spaces, Chapter 5 in: Lie Groups and Lie Algebras II: Lie Transformation groups, Encyclopaedia of Mathematical Sciences, vol 20. Springer, Spinger 1993 (doi:10.1007/978-3-642-57999-8_11)

• Jean-Baptiste Butruille, Homogeneous nearly Kähler manifolds, in: Vicente Cortés (ed) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, pp 399–423 (arXiv:math/0612655, doi:10.4171/079-1/11)

On the KO-theory of $\mathbb{C}P^3$:

• Michael Atiyah, E. Rees, Vector bundles on projective 3-space, Invent. Math. 35, 131–153 (1976) (pdf)

On symmetries of manifolds of the homotopy type of $\mathbb{C}P^3$:

• Mark Hughes, Symmetries of Homotopy Complex Projective Three Spaces, Transactions of the American Mathematical Society Vol. 337, No. 1 (May, 1993), pp. 291-304 (doi:10.2307/2154323)

On toric symmetries of $\mathbb{C}P^3$:

• Dagan Karp, Dhruv Ranganathan, Paul Riggins, Ursula Whitcher, Toric symmetry of $\mathbb{C}P^3$, Advances in Theoretical and Mathematical Physics, Vol. 16, No. 4, 2012 (arXiv:1109.5157)