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.

Fixed determinant isolates the actual Chern fiber

Statement

For the map $C_{X}$ and subsets just defined, the fiber $C_{X}^{- 1} (L, a)$ is exactly $V_{L, a}$ . The map $C_{X}$ is bijective if and only if for every line-bundle class $L$ and every class $a \in {C H}^{2} (X)$ the set $V_{L, a}$ has exactly one element. This classification forgets every choice of an isomorphism $\det E ≅ L$ .

Statement references

Rank-two Chern map and its fixed-invariant fibers

Facts & Assumptions

Given: $X$ , $C_{X}$ and $V_{L, a}$ from [F1].

Goal: Identify the fibers and the exact existence-and-uniqueness criterion.

[F1]

$C_{X} ([E]) = ([\det E], c_{2} (E))$ , and $V_{L, a}$ imposes these two equalities on bundle isomorphism classes. Rank-two Chern map and its fixed-invariant fibers

Proof

1.1F1given

An element $e$ belongs to $C_{X}^{- 1} (L, a)$ precisely when both coordinates of $C_{X} (e)$ equal the displayed coordinates. The first equality says $e \in V_{L}$ , and the second says $e \in V_{L, a}$ . This proves equality of the two subsets, without adding a determinant isomorphism to their elements.

2.1step 1.1given∎

If all these fibers have one element, every target pair has a preimage and any two elements with equal images lie in the same singleton fiber; hence $C_{X}$ is surjective and injective. Conversely, a bijection has exactly one preimage for each target element. Step 1.1 translates that fact to the asserted sets.

Depends on

Rank-two Chern map and its fixed-invariant fibers

Used by · 2

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