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Smooth threefold rank-two classification

PDF p. 1: Introduction, first paragraph, AF14

Statement

Use the Axiom of Choice and the supplied universe conventions. Let $X$ be a smooth affine threefold over an algebraically closed field $k$ of characteristic different from two. The first two Chern classes give a pointed bijection ${V e c t}_{2} (X) \tilde{\to} Pic (X) \times {C H}^{2} (X), [E] ⟼ (c_{1} (E), c_{2} (E)) .$ Here the first coordinate is $[\det E]$ in $Pic (X)$ , equivalently its usual first Chern class under $Pic (X) ≅ {C H}^{1} (X)$ . Thus every pair of classes is attained by exactly one bundle isomorphism class. No determinant trivialization is included in the classification data.

Statement references

Rank-two Chern map and its fixed-invariant fibers

Asok–Fasel Chern-compatible comparison and lifting input

Facts & Assumptions

Given: A smooth affine $k$ -scheme of finite type of pure dimension three, with $k$ algebraically closed and $char (k) \neq 2$ , and the stated set-theoretic conventions.

Goal: Prove bijectivity for the actual pair of Chern classes and identify the basepoints.

[F1]

The map $C_{X} ([E]) = ([\det E], c_{2} (E))$ is well-defined and takes the trivial rank-two bundle to $([O_{X}], 0)$ . Rank-two Chern map and its fixed-invariant fibers

[F2]

Under exactly these hypotheses, the external Asok–Fasel comparison gives compatible maps $v, t, b$ and exact obstruction-group sequences with zero end groups, together with the nonempty lift-fiber quotient descriptions. It also identifies the determinant coordinate with $c_{1}$ and the second coordinate with the ordinary $c_{2}$ . Asok–Fasel Chern-compatible comparison and lifting input

[F3]

Compatible presentation maps with this lifting and vanishing interface imply singleton Chern fibers and bijectivity. Unique lifting from vanishing groups yields the Chern classification

Proof

1.1givenF1F2

The field and scheme satisfy all hypotheses of [F2]: algebraic closure, characteristic different from two, smoothness, affineness, and dimension three. Apply that input to $X$ . It gives bijections $v$ and $b$ , a surjective $t$ , the equality $b t v = C_{X}$ , and, over every second-stage class, the exact sequence and zero end groups required by [F3]. No assertion about arbitrary singular threefolds or characteristic two is used.

2.1F1F2F3step 1.1∎

Applying [F3] to these actual maps makes $C_{X}$ a bijection, with one bundle isomorphism class for every pair. By [F2] this is precisely the displayed pair of Chern classes, and by [F1] it preserves the specified basepoints. This proves the full stated classification.

Source notes

The original article, Fasel–Yakerson, arXiv:2610.22679v1, p. 1, first paragraph, attributes this result to Asok–Fasel [AF14]. The precise source is Asok–Fasel arXiv:1204.0770v5, Theorem 1, p. 2, with proof at Theorem 6.6, p. 26. The five intermediate prerequisites above author the Chern map, determinant fibers, map-compatible transport, zero-group argument and local lifting application. The substantial motivic representability, truncation and cohomology computation remain explicitly external; no full reconstruction of their proofs is claimed. The native job-00115 authoring obstacle is resolved by this construction, not classified as an error in either paper.

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Sources

Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1, p. 2; §6, pp. 22–26