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complex projective plane

Contents

Contents

Idea

The complex projective plane P 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 2α 2α 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:

P 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.

Referenced

See also

Last revised on August 4, 2020 at 08:33:47. See the history of this page for a list of all contributions to it.