LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Unique lifting from vanishing groups yields the Chern classification

Statement

Use $C_{X}, V, T$ as defined. Suppose $v : V \to M$ , $b : N \to T$ are bijections and $t : M \to N$ is surjective with $b t v = C_{X}$ . Suppose also that, for every $n \in N$ , there are abelian groups $A_{n}, H_{n}, D_{n}$ exact in a sequence $A_{n} \to H_{n} \to D_{n}$ , with $A_{n} = D_{n} = 0$ , and that the nonempty fiber $t^{- 1} (n)$ is a surjective image of the underlying set of $H_{n}$ . Then every $V_{L, a}$ is a singleton and $C_{X}$ is a bijection.

Statement references

Rank-two Chern map and its fixed-invariant fibers

Fixed determinant isolates the actual Chern fiber

Chern-compatible presentation transports the exact fibers

Vanishing end groups kill the middle obstruction group

Facts & Assumptions

Given: Precisely the sets, maps, compatibility and group hypotheses of the Statement.

Goal: Deduce existence and uniqueness for the actual Chern invariants.

[F1]

Exactness with zero end groups forces the middle group to be zero, and its nonempty surjective images are singletons. Vanishing end groups kill the middle obstruction group

[F2]

The compatible bijections identify $V_{L, a}$ with the corresponding $t$ -fiber. Chern-compatible presentation transports the exact fibers

[F3]

Singleton fixed-invariant fibers are equivalent to bijectivity of $C_{X}$ . Fixed determinant isolates the actual Chern fiber

Proof

1.1givenF1

Fix an arbitrary $y = (L, a) \in T$ . Bijectivity of $b$ gives a unique $n \in N$ over $y$ . Surjectivity of $t$ makes $t^{- 1} (n)$ nonempty. The exact group sequence and zero end groups imply $H_{n} = 0$ ; its asserted surjection onto $t^{- 1} (n)$ and [F1] imply that this fiber is a singleton.

2.1F2F3step 1.1∎

Compatibility identifies that fiber with $V_{L, a}$ by [F2], so $V_{L, a}$ is also a singleton. As $y$ was arbitrary, [F3] gives bijectivity of $C_{X}$ . This argument needs a quotient description only after existence of a lift; it does not select a canonical lift or add determinant framings.

Depends on

Used by · 1

Smooth threefold rank-two classification

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

Sources

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