complex projective plane (changes) in nLab
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Context
Complex geometry
geometry, complex numbers, complex line
Variants
Structures
Examples
-
dim=1dim = 1: Riemann surface, super Riemann surface
-
- dim=2dim = 2: K3 surface
Contents
Idea
The complex projective plane ℂP 2\mathbb{C}P^2 is the complex projective space of complex dimension 2, hence the projective plane over the complex numbers.
Properties
Rational homotopy type
A Sullivan model is given by
dα 2 =0 dα 5 =α 2∧α 2∧α 2 \array{ d\, \alpha_2 & = 0 \\ d\, \alpha_5 & = \alpha_2 \wedge \alpha_2 \wedge \alpha_2 }
- Luc Menichi, Section 5.3 of: Rational homotopy – Sullivan models
Quotient by complex conjugation is 4-sphere
Proposition
(Arnold-Kuiper-Massey theorem)
The 4-sphere is the quotient space of the complex projective plane by the action on homogeneous coordinates of complex conjugation:
ℂP 2/(−) *≃S 4 \mathbb{C}P^2 / (-)^* \simeq S^4
Blowup to del Pezzo surfaces
The blowup at generic points is a del Pezzo surface.
Related concepts
Referenced
See also
- Wikipedia, Complex projective plane
Last revised on August 4, 2020 at 12:33:47. See the history of this page for a list of all contributions to it.