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.

Chern-compatible presentation transports the exact fibers

Statement

Let $C_{X} : V \to T = Pic (X) \times {C H}^{2} (X)$ be the defined map. Suppose $M, N$ are sets, $v : V \to M$ and $b : N \to T$ are bijections, and $t : M \to N$ is a map satisfying $b \circ t \circ v = C_{X}$ . For $y = (L, a) \in T$ , let $n$ be its unique preimage under $b$ . The restriction of $v$ is a bijection from $V_{L, a}$ onto $t^{- 1} (n)$ .

Statement references

Rank-two Chern map and its fixed-invariant fibers

Fixed determinant isolates the actual Chern fiber

Facts & Assumptions

Given: The sets and maps in the Statement, including the actual equality $b t v = C_{X}$ .

Goal: Produce the asserted fiber bijection.

[F1]

The fiber of $C_{X}$ over $(L, a)$ is $V_{L, a}$ . Fixed determinant isolates the actual Chern fiber

Proof

1.1F1given

If $e \in V_{L, a}$ , then $b (t (v (e))) = C_{X} (e) = y = b (n)$ . Injectivity of $b$ yields $t (v (e)) = n$ . Conversely, if $m \in t^{- 1} (n)$ , its unique preimage $e$ under $v$ satisfies $C_{X} (e) = b (t (m)) = y$ , so $e \in V_{L, a}$ .

2.1step 1.1given∎

The restrictions of $v$ and its inverse therefore map the two specified subsets into each other. Their composites remain the respective identity maps because they were inverse on the ambient sets. This proves the bijection of fibers. Mere abstract bijections of the ambient sets without the compatibility equality would not imply this conclusion.

Depends on

Used by · 1

Unique lifting from vanishing groups yields the Chern 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