nLab complex projective plane

Idea

The complex projective plane $\mathbb{C}P^2$ 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

$$\array{ d\, \alpha_2 & = 0 \\ d\, \alpha_5 & = \alpha_2 \wedge \alpha_2 \wedge \alpha_2 }$$

• 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:

$$\mathbb{C}P^2 / (-)^* \simeq S^4$$

Blowup to del Pezzo surfaces

The blowup at generic points is a del Pezzo surface.

Related concepts

• complex projective space

  • complex projective line

  • complex projective 3-space

Referenced

See also

• Wikipedia, Complex projective plane