Contents

complex geometry

# Contents

## Idea

The twistor fibration $\mathbb{C}P^3 \to S^4$ (Atyiah-Hitchin-Singer 78, Ex. 2 on p. 438, Atiyah 79, Sec. III.1, see also Bryant 82, ArmstrongSalamon14, ABS 19), also called, in its coset space-version $SO(5)/U(2) \to SO(5)/SO(4)$, the Calabi-Penrose fibration (apparently starting with Lawson 85, Sec. 3, see also, e.g., Loo 89, Seade-Verjovsky 03, 3 for this usage, see Nordstrom 08, Lemma 2.31 for review of Calabi’s construction Calabi 67, Calabi 68) is a fiber bundle-structure on complex projective 3-space over the 4-sphere with 2-sphere (Riemann sphere) fibers:

$\array{ \mathbb{C}P^1 &\longrightarrow& \mathbb{C}P^3 \\ && \big\downarrow^{\mathrlap{p}} \\ && S^4 }$

If one identifies the 4-sphere as the quaternionic projective line $S^4 \simeq \mathbb{H}P^1$, then the fibration $p$ here is given by sending complex lines to the quaternionic lines which they span (Atiyah 79, III (1.1), see also Seade-Verjovsky 03, p. 198):

$\array{ \mathbb{C}P^3 &\overset{p}{\longrightarrow}& \mathbb{H}P^1 \\ \{x \cdot z \vert z \in \mathbb{C}\} &\mapsto& \{x \cdot q \vert q \in \mathbb{H}\} } \,,$

for any $x \in \mathbb{C}^4 \simeq_{\mathbb{R}} \mathbb{H}^2$.

## Generalizations

It is possible to define a twistor fibration over each $S^{2n}$, where the resulting manifold is a complex manifold endowed with a holomorphic? $n$-plane field transverse to the fibers of the mapping. Namely, writing $S^{2n}=SO(2n+1)/SO(2n)$, then, using the inclusion $U(n) \subset SO(2n)$, one has the coset fibration

$Z_n=SO(2n+1)/U(n) \to SO(2n+1)/SO(2n).$

The manifold $Z_n$ canonically has the structure of a complex manifold and is known as the twistor space of $S^{2n}$.

There are generalizations of this picture for each of the so-called ‘inner’ symmetric spaces $G/K$ where $K$ is the fixed subgroup of an involution that is an inner automorphism of $G$. The twistor fibration is of the form $G/U \to G/K$ where $U \subset K$ is a subgroup such that $K/U$ (the typical fiber of the fibration) is an Hermitian symmetric space. There are also other kinds of twistor spaces over $G/K$ that are flag manifolds of the form $G/T$ where $T \subset K$ is a maximal torus. (For these generalizations see Bryant 85.)

## References

• Simon Brain, Giovanni Landi, Differential and Twistor Geometry of the Quantum Hopf Fibration, Commun. Math. Phys. 315 (2012):489-530 (arXiv:1103.0419)

In higher dimensions:

Over $\mathbb{H}P^3$:

Last revised on March 31, 2021 at 05:44:45. See the history of this page for a list of all contributions to it.