Contents

complex geometry

# Contents

## 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.