Coupled Penrose vanishing on the round four-sphere

Assume full AC. Let $S^4$ have the unit round metric and the positive stereographic orientation for $z_1=x_0+i{x}_1$, $z_2=x_2+i{x}_3$. Let $E\to S^4$ be a smooth finite-rank complex vector bundle with a positive Hermitian metric $h$ and a smooth connection ${\mathrm{\nabla }}_A$ preserving $h$. Its curvature is $F_A={\mathrm{\nabla }}_A^2\in {\mathrm{\Omega }}^2(S^4,{\mathrm{End}}_{\mathbb{C}}E)$. For real two-forms $\eta ,\zeta$, let ${\ast }_g$ be characterized by $\eta \wedge {\ast }_g\zeta =⟨\eta ,\zeta {⟩}_{g}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, and extend it complex-linearly on the form factor of $F_A$. Set $F_A^{+}=(F_A+{\ast }_g{F}_A)\mathrm{/}2$ and assume $F_A^{+}=0$, the anti-self-duality condition in this orientation. On $Z=\mathbb{C}{P}^3$ use homogeneous coordinates $[{\lambda }_1:{\lambda }_2:{\mu }_1:{\mu }_2]$ and the incidence equations ${\mu }_1=z_1{\lambda }_1-{\bar{z}}_2{\lambda }_2$, ${\mu }_2=z_2{\lambda }_1+{\bar{z}}_1{\lambda }_2$, with infinity fibre ${\lambda }_1={\lambda }_2=0$; these fibres define the incidence projection $\pi :Z\to S^4$. On ${\lambda }_1\mathrm{\ne }0$, put $t={\lambda }_2\mathrm{/}{\lambda }_1$, $H_1={\mathrm{\partial }}_{{\bar{z}}_1}-t{\mathrm{\partial }}_{z_2}$, and $H_2={\mathrm{\partial }}_{{\bar{z}}_2}+t{\mathrm{\partial }}_{z_1}$. The projected complex alpha-plane at $[\lambda ]\in {\pi }^{-1}(x)$ is ${\mathrm{span}}_{\mathbb{C}}\{d\pi (H_1),d\pi (H_2)\}$; the second projective chart covers $t=\mathrm{\infty }$. Let ${\mathcal{O}}_Z(-1)$ be the tautological holomorphic line bundle and ${\mathcal{O}}_Z(-2)={\mathcal{O}}_Z(-1{)}^{\otimes 2}$. Suppose $F$ is a finite-rank holomorphic bundle (locally trivial with holomorphic transition maps), smoothly identified with ${\pi }^{\ast }E$, and its Dolbeault operator is the coefficientwise ordinary vertical operator and the connection ${\mathrm{\nabla }}_A$ on the projected complex alpha-planes. For a finite-rank holomorphic bundle $G\to Z$, write $H^1(Z,G)$ for analytic sheaf cohomology of its holomorphic-section sheaf; under the analytic Dolbeault comparison it is canonically $\ker ({\bar{\mathrm{\partial }}}_G:{\mathrm{\Omega }}^{0,1}(Z,G)\to {\mathrm{\Omega }}^{0,2}(Z,G))\mathrm{/}\mathrm{im}({\bar{\mathrm{\partial }}}_G:{\mathrm{\Omega }}^{0,0}(Z,G)\to {\mathrm{\Omega }}^{0,1}(Z,G))$, where ${\mathrm{\Omega }}^{0,q}(Z,G)$ denotes the global smooth $G$-valued $(0,q)$-forms. Write $F^{\vee }$ for the fibrewise complex dual. Then

$H^1(Z,F\otimes {\mathcal{O}}_Z(-2))=H^1(Z,F^{\vee }\otimes {\mathcal{O}}_Z(-2))=0.$

This applies to arbitrary reducible and flat summands. It assumes no algebraic-to-analytic GAGA, monad, or inverse ADHM normal form as an independent premise.

The Axiom of Choice

Positive incidence geometry for the coupled Penrose transform

Injective fibre-harmonic edge for coupled Penrose cohomology

Round coupled Penrose correction gives the R-over-six operator

Scalar curvature of the unit round four-sphere

Rawnsley, §§2–3, pp. 46–56

Wells, §§3–4, pp. 488–498

Donaldson, §4, p. 7 (coefficient discrepancy only)

Rawnsley’s theorem on printed p. 49, proved on p. 50, identifies the coupled cohomology with the kernel of $D^{\ast }D+R\mathrm{/}6$; the detailed construction occupies §§2–3, pp. 46–56. His notation calls the sphere bundle $F$ and its twistor pullback $E$, and uses self-duality for his standard orientation. Here the incidence and curvature suppliers specify the orientation in which the connection is anti-self-dual. Rawnsley’s introduction describes positive-charge instanton bundles; the dual, reducible and flat-summand scope asserted here is supplied by the local edge and operator lemmas and the energy argument above. The full cited article is a scan: its article pages have no searchable text layer, so text extraction alone does not display these passages.