Smooth framed bundle of sections over real twistor lines

Algebraize the holomorphic bundle using [A1]. Its algebraic sheaf is locally free: the holomorphic bundle is dualizable, the exact tensor equivalence transports its evaluation and coevaluation and reflects their triangle identities, and those identities give a finite dual basis on each local algebraic chart, hence a finite projective module. In a splitting C⁴=A⊕B into two-planes, a graph chart of G consists of X:A→B and the incidence is (X,[λ])↦[λ,Xλ], with p a product with P(A). In the library quotient convention it is the projective bundle of the dual of the tautological subbundle. Thus [F1] gives properness, finite presentation and flatness, and e*E is a coherent locally free sheaf flat over G.

Algebraize $\mathcal{E}$ by [A1], obtaining a locally free algebraic sheaf ${\mathcal{E}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ on $Z$, and put ${\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}:=p_{\ast }{e}^{\ast }{\mathcal{E}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$. For a real line $L$, apply absolute projective GAGA to the coherent algebraic sheaf $i_{\ast }({\mathcal{E}}_{\mathrm{a}\mathrm{l}\mathrm{g}}{\mathrm{\mid }}_L)$, where $i:L\hookrightarrow {\mathbb{P}}^3$ is the closed immersion. Exactness of closed-immersion direct image and the local tensor definition of analytification identify its analytic sheaf with $i_{\ast }^{\mathrm{a}\mathrm{n}}(\mathcal{E}{\mathrm{\mid }}_L)$, so its cohomology is that of the trivial analytic restriction. Thus the algebraic fibre cohomology is ${\mathbb{C}}^N$ in degree zero and zero above by [F3]. The degree-one base-change map has zero target, hence is surjective. By F2,(c) it is an isomorphism; the finite stalk of $R^1{p}_{\ast }{e}^{\ast }{\mathcal{E}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ has zero residue fibre, and Nakayama kills it on a neighbourhood. The degree-one local-freeness criterion F2 then gives surjectivity in degree zero. Applying F2–(c) in degree zero makes ${\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ finite locally free of rank $N$ near $L$, with degree-zero base-change isomorphisms there.

The algebraic evaluation morphism $p^{\ast }{\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}\to e^{\ast }{\mathcal{E}}_{\mathrm{a}\mathrm{l}\mathrm{g}}$ is a map of rank-$N$ bundles and is invertible on $L$, since its restriction evaluates all constant sections of the trivial bundle. Its determinant-zero locus is closed. Properness of $p$ makes the image closed in the parameter neighborhood; remove that image to obtain an entire neighboring family on which evaluation is an isomorphism. This is a local neighborhood of every real line, not a claim about all complex lines.

Analytify the evaluation isomorphism from step 3.1. On every local product $V\times {\mathbb{P}}^1$, a holomorphic function is constant on each ${\mathbb{P}}^1$ fibre by the analytic degree-zero scalar assertion in [F3], and its remaining function on $V$ is holomorphic by evaluation at a fixed point. Thus $(p^{\mathrm{a}\mathrm{n}}{)}_{\ast }{\mathcal{O}}_{{\mathcal{L}}^{\mathrm{a}\mathrm{n}}}={\mathcal{O}}_{G^{\mathrm{a}\mathrm{n}}}$ locally. Trivializing ${\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}^{\mathrm{a}\mathrm{n}}$ over $V$ then gives $(p^{\mathrm{a}\mathrm{n}}{)}_{\ast }(p^{\mathrm{a}\mathrm{n}}{)}^{\ast }{\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}^{\mathrm{a}\mathrm{n}}\cong {\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}^{\mathrm{a}\mathrm{n}}$. Pushing forward the analytic evaluation isomorphism therefore identifies ${\mathcal{H}}_{\mathrm{a}\mathrm{l}\mathrm{g}}^{\mathrm{a}\mathrm{n}}$ with $\mathcal{H}=(p^{\mathrm{a}\mathrm{n}}{)}_{\ast }(e^{\mathrm{a}\mathrm{n}}{)}^{\ast }\mathcal{E}$ on this neighbourhood, and identifies their evaluation morphisms. Hence the algebraic degree-zero base-change isomorphisms of step 2.1 give the analytic maps ${\beta }_{g_x}^0$ in the Statement. The same product calculation after any base change identifies the corresponding maps. No circular assumption of analytic base-change surjectivity or broad Grauert theorem is used.

The quaternionic involution on $G$ sends a graph $X$ to $j_{B}X{j}_A^{-1}$; its fixed graph matrices form the real four-dimensional quaternionic subspace, whose complexification is the entire graph space. These charts embed $\mathbb{H}{\mathbb{P}}^1$ smoothly and totally really into $G$. On its incidence family evaluation is a diffeomorphism onto $Z$: the inverse sends $[v]$ to $({\mathrm{span}}_{\mathbb{C}}(v,jv),[v])$, smoothly in projective charts, and $j^2=-1$ excludes dependence of $v,jv$. Restrict $\mathcal{H}$ to this real locus to obtain the smooth rank-$N$ bundle with the claimed fibres and evaluation. The positive pairing induces its Hermitian matrix by evaluating local section frames at a smoothly chosen point on each line; this proves smoothness and positivity, independently of parallelness. At infinity the given unitary frame induces the section-bundle frame. The only use of projective GAGA is the absolute comparison [A1]; all family base-change conclusions are established through [F2].