Conditional framed Penrose–Ward correspondence on the round four-sphere

In the chart $w={\lambda }_2\mathrm{/}{\lambda }_1$ on a real incidence fiber, the projected horizontal (0,1) directions are $H_1={\mathrm{\partial }}_{{\bar{z}}_1}-w{\mathrm{\partial }}_{z_2}$ and $H_2={\mathrm{\partial }}_{{\bar{z}}_2}+w{\mathrm{\partial }}_{z_1}$. Indeed these and the vertical ${\mathrm{\partial }}_{\bar{w}}$ annihilate the holomorphic coordinates w, ${\mu }_1=z_1-{\bar{z}}_{2}w$, ${\mu }_2=z_2+{\bar{z}}_{1}w$. They are independent and commute; their coefficients are holomorphic in w. For a unitary connection A define the pulled-back (0,1) operator by the vertical ordinary derivative and ${\mathrm{\nabla }}_A$ along H₁,H₂. Its mixed squares vanish by the displayed commuting generators and holomorphic w coefficients. Its horizontal square is $F_A(H_1,H_2)=F_{\bar{1}\bar{2}}-w(F_{1\bar{1}}+F_{2\bar{2}})+w^2{F}_{12}$, which vanishes for an ASD two-form in the positive orientation: the three coefficients are precisely its self-dual component. Thus the operator squares to zero on this chart, and on the other projective chart by the same tensorial calculation. Apply [F2] to obtain a genuine holomorphic bundle. Its restriction to every real line is the constant bundle $E_x$, with positive reality pairing induced by the unitary metric and framing at infinity. This pairing is holomorphic: its vertical derivative vanishes on the constant fibres, and differentiating horizontally gives zero by the original identity ${\mathrm{\nabla }}_{A}h=0$, with the conjugate pullback at σ interchanging the conjugate horizontal directions.

For any Ward bundle use [F3] to form the analytic vector bundle H over a complex-open neighborhood U of every real line in G=Gr(2,4), with holomorphic evaluation pH≅eE on the universal family. On a graph chart X:A→B, incidence is (X,[λ])↦[λ,Xλ]. The distribution D=ker(de) consists of matrix variations v with vλ=0 and no λ variation. It is a holomorphic involutive rank-two subbundle, since it is tangent to the fibers of the submersion e. The restriction of pΩ¹_U to D has kernel B⊗O(-1) on each projective-line fiber and target D*≅O(1)². The O(-1) vanishings in [F3] give Ω¹_U≅p_D. Equivalently, on the graph chart a linear covector on Hom(A,B) restricts to a section on all kernels vλ=0; the map is injective because those kernels span Hom(A,B), and both coefficient spaces have dimension four. This gives the holomorphic sheaf isomorphism without any extra analytic base-change assumption.

The bundle eE has a canonical flat partial connection δ along D: in a holomorphic frame pulled back from E, differentiate coefficient functions along the fibers of e. Changes of that frame have zero D derivative, so δ is well-defined and flat. Transfer δ to pH by evaluation. For a local holomorphic section s of H, δ(ps) is a section of D⊗pH. Push down using Ω¹_U≅p_D* from step 2.1 to define ∇s. The partial Leibniz rule becomes ∇(fs)=df⊗s+f∇s, so ∇ is a holomorphic full connection. Its pullback restriction to D is δ, first on pulled-back sections and then on all sections by Leibniz. It is unique: the difference of two such connections is an End(H)-valued one-form restricting to zero on D at every λ, hence zero by the same cotangent isomorphism. The curvature restricts on D to δ²=0. Connections on overlapping graph charts agree by uniqueness.

Restrict H and ∇ to the totally real graph locus $M=\mathbb{H}{P}^1$. Its smooth real incidence is diffeomorphic to Z by $(P,[v])\mapsto [v]$, with inverse $[v]\mapsto (\mathrm{span}(v,jv),[v])$; the explicit charts in [F3] verify smoothness. Under this identification dp(D) is exactly the span of H₁,H₂ in step 1.1: write $X_z=((z_1,-{\bar{z}}_2),(z_2,{\bar{z}}_1))$ and solve $v(1,w{)}^T=0$. Therefore the restricted smooth connection has $F(H_1,H_2)=0$ for every w, so all three self-dual coefficients in step 1.1 vanish. Hence it is ASD. Evaluation intertwines the original holomorphic operator on E with the reconstructed vertical/horizontal Dolbeault operator: it is holomorphic vertically, and on D its derivative is exactly the canonical relative derivative δ by step 3.1. These directions generate the (0,1) tangent in step 1.1, proving equality of the full operators, not just of underlying bundles.

To identify the connection with the first-neighborhood prescription, the line's conormal bundle is O(-1)², by its two independent linear defining equations. The sequence 0→E|L⊗(I/I²)→E|L^(1)→E|L→0 and [F3]'s O(-1) vanishings give a unique extension of every line section to L^(1). Smooth variation is supplied by step 3.1, not merely inferred from this pointwise uniqueness. At a graph X choose a local holomorphic section s of H with specified value and ∇s(X)=0; in a local frame its affine-linear coefficient function can realize any prescribed first jet. Its evaluation derivative in parameter direction v vanishes on vλ=0 by step 3.1, so factors through the normal displacement vλ. These factorizations glue on the two λ charts and give precisely a first-neighborhood extension. Uniqueness identifies it with the preceding extension. Subtracting such a horizontal first jet from an arbitrary section shows that differentiation relative to this extension is exactly ∇. This proves the prescription, its Leibniz rule, chart independence and smooth dependence.

The section metric is the form induced by $b_z(s(z),t(\sigma z))$ in [F1]. It is independent of z on each line because that scalar is holomorphic on P¹, and is smooth by the evaluation frames of [F3]. The preceding construction commutes with duals, conjugate pullback and bundle morphisms: each commutes with the canonical relative e-connection and the cotangent isomorphism, and uniqueness in step 3.1 then forces the full connections to commute. Apply this to the holomorphic reality isomorphism $E\to {\sigma }^{\ast }{\bar{E}}^{\ast }$. On the real line locus, the induced conjugate involution fixes the parameter and its real tangent vectors. Thus the induced map $H\to {\bar{H}}^{\ast }$ intertwines ∇ with the conjugate dual connection, which is precisely $Xh(s,t)=h({\mathrm{\nabla }}_{X}s,t)+h(s,{\mathrm{\nabla }}_{X}t)$ for real X. Positivity from [F1] makes ∇ unitary. Arbitrary independent positive forms would not satisfy this identity and are not used.

Starting from A in step 1.1, its constant real-line sections identify the reconstructed smooth H with E. Along every alpha-plane the recovered connection has the same derivative as A; the cotangent uniqueness in step 3.1 makes the full connections equal. The unitary reality pairing and framing also agree. Starting from a Ward bundle, step 4.1 proves evaluation is a holomorphic isomorphism from the reconstructed pullback to the original bundle, and step 5.2 preserves its reality pairing and metric. These constructions and identities apply to morphisms as well, so descend to inverse maps on based unitary gauge and framed holomorphic isomorphism classes. No irreducibility or absence of parallel sections was used.

The explicit incidence chart fixes vertical complex orientation followed by the positive base orientation to be the complex orientation of Z. The pullback of the positive point Thom class on S⁴ is the positive normal Thom class of its fiber CP¹; [F5] identifies its absolute class as the Poincaré dual of that fiber. A linear CP¹ has dual H²: H restricts with integral one to the line, and the integral projective cohomology ring gives H² as its positive degree-four generator. Hence ${\pi }^{\ast }u=H^2$ for the positive generator u of H⁴(S⁴). Smooth evaluation gives $E_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}}\cong {\pi }^{\ast }H$, so $c_2(E_{\mathrm{t}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}})=k{H}^2$ is equivalent to $c_2(H)[S^4]=k$. By [F4] this is $q_N=-k$; the same calculation verifies the forward map's charge clause. Framing is transported by evaluation at infinity. All constructions use incidence, evaluation and unique relative connections, so commute with the specified projective rotations and framing transformations. They therefore give the claimed T-equivariant bijection and all stated invariants, conditional on the exact absolute GAGA comparison [A1].