Positive incidence geometry for the coupled Penrose transform

Let $S^4=\{X\in {\mathbb{R}}^5:\mathrm{\mid }X\mathrm{\mid }=1\}$ have its unit round metric and the orientation which, on the stereographic chart $S^4\setminus \{\mathrm{\infty }\}\cong {\mathbb{R}}^4$, is $d{x}_0\wedge d{x}_1\wedge d{x}_2\wedge d{x}_3$. Put $z_1=x_0+i{x}_1$, $z_2=x_2+i{x}_3$, and $r^2=\mathrm{\mid }{z}_1{\mathrm{\mid }}^2+\mathrm{\mid }{z}_2{\mathrm{\mid }}^2$; in these coordinates the metric is $g=4(1+r^2{)}^{-2}{\sum }_{\mu =0}^{3}d{x}_{\mu }^2$. Define the Wirtinger derivatives on this chart by

${\mathrm{\partial }}_{z_1}=\frac{1}{2}({\mathrm{\partial }}_{x_0}-i{\mathrm{\partial }}_{x_1}),{\mathrm{\partial }}_{{\bar{z}}_1}=\frac{1}{2}({\mathrm{\partial }}_{x_0}+i{\mathrm{\partial }}_{x_1}),{\mathrm{\partial }}_{z_2}=\frac{1}{2}({\mathrm{\partial }}_{x_2}-i{\mathrm{\partial }}_{x_3}),{\mathrm{\partial }}_{{\bar{z}}_2}=\frac{1}{2}({\mathrm{\partial }}_{x_2}+i{\mathrm{\partial }}_{x_3});$

These vector fields act on the complexification of the real tangent bundle. Fix the antiunitary right quaternionic structure $j:{\mathbb{C}}^4\to {\mathbb{C}}^4$ in these coordinates by

$j({\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2)=(-{\bar{\lambda }}_2,{\bar{\lambda }}_1,-{\bar{\mu }}_2,{\bar{\mu }}_1)\mathrm{.}$

Thus $j$ is conjugate-linear, preserves the standard Hermitian norm, and satisfies $j^2=-I$. Put ${\mathbb{H}}^2={\mathbb{C}}^4$ with this right quaternionic structure, $Z=\mathbb{C}{P}^3={\mathbb{P}}_{\mathbb{C}}({\mathbb{H}}^2)$, and define $\pi :Z\to \mathbb{H}{P}^1=S^4$ by $\pi (\mathrm{\ell })=\mathrm{\ell }+j\mathrm{\ell }$ for each complex line $\mathrm{\ell }\subset {\mathbb{H}}^2$. Its fibres are projective lines $L_x\cong \mathbb{C}{P}^1$. In homogeneous coordinates $[{\lambda }_1:{\lambda }_2:{\mu }_1:{\mu }_2]$, the fibre over finite $z=(z_1,z_2)$ is the line

${\mu }_1=z_1{\lambda }_1-{\bar{z}}_2{\lambda }_2,{\mu }_2=z_2{\lambda }_1+{\bar{z}}_1{\lambda }_2,$

and the fibre over infinity is $L_{\mathrm{\infty }}=\{{\lambda }_1={\lambda }_2=0\}$. Give $Z$ the Fubini–Study metric induced by the standard Hermitian metric on ${\mathbb{C}}^4$, and give ${\mathcal{O}}_Z(-1)$, the tautological holomorphic line bundle, its induced Hermitian metric; ${\mathcal{O}}_Z(-2)={\mathcal{O}}_Z(-1{)}^{\otimes 2}$ has the tensor-square metric. On the incidence chart ${\lambda }_1\mathrm{\ne }0$, set $t={\lambda }_2\mathrm{/}{\lambda }_1$, ${\mu }_1\mathrm{/}{\lambda }_1=z_1-{\bar{z}}_{2}t$, and ${\mu }_2\mathrm{/}{\lambda }_1=z_2+{\bar{z}}_{1}t$. Write $T^{0,1}Z$ for the $-i$ eigensubbundle of the complexified tangent endomorphism determined by the complex structure. The constant-$t$ horizontal complex directions of type $(0,1)$ are spanned by

$H_1={\mathrm{\partial }}_{{\bar{z}}_1}-t{\mathrm{\partial }}_{z_2},H_2={\mathrm{\partial }}_{{\bar{z}}_2}+t{\mathrm{\partial }}_{z_1};$

They annihilate the displayed holomorphic coordinates and, together with ${\mathrm{\partial }}_{\bar{t}}$, form a direct local frame of $T^{0,1}Z$. Define $V^{0,1}=\ker (d\pi )\cap T^{0,1}Z$ and let $H^{0,1}$ be its Fubini–Study orthogonal complement in $T^{0,1}Z$. The complex alpha-plane at $\mathrm{\ell }\in L_x$ is the two-dimensional complex subspace $d{\pi }_{\mathrm{\ell }}(H_{\mathrm{\ell }}^{0,1})\subset T_x{S}^4{\otimes }_{\mathbb{R}}\mathbb{C}$; it is also the image under $d{\pi }_{\mathrm{\ell }}$ of the span of $H_1,H_2$. These constant-$t$ chart lifts project to the same alpha-plane as the global metric-horizontal lifts, although they may differ from them by vertical components. Global horizontal/vertical form decompositions use the Fubini–Study orthogonal splitting; local computations may use the displayed constant-$t$ splitting after retaining the resulting frame and normalization factors. The second projective chart gives the same alpha-plane at $t=\mathrm{\infty }$.

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Rawnsley, §1, pp. 43–46

Wells, §2, pp. 484–488

Donaldson, §§3–4, pp. 4–7

The arrows record only the inclusion of a fibre and the restriction of its projection; they fix the domain and codomain of the twistor fibration used above.