ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

Affine-plane localization from BM classes

Example

Let $T = (C^{\times})^{2}$ act on $X = C^{2}$ by $t \cdot (z_{1}, z_{2}) = (t_{1} z_{1}, t_{2} z_{2})$ , and let $u_{1}, u_{2}$ be the first Chern classes of the two coordinate characters in the product Borel model. The single affine chart $X$ is a finite invariant affine atlas, the fixed locus is the proper point $F = {(0, 0)}$ , and the tangent weights at that point are the two coordinate characters. Compute the intrinsic localized pushforward of the complex-oriented fundamental class; the result is $1 / (u_{1} u_{2})$ .

Example references

[Fixed-point denominator and isolated-point specialization] Edidin–Graham, Section 4

[Borel–Moore fundamental classes] Ginzburg, Section 1

Facts & Assumptions

Given: The diagonal algebraic action of $T = (C^{\times})^{2}$ on $C^{2}$ , rational coefficients, and the product Borel–Moore conventions.

[F1]

$u_{i}$ is the degree-two class of the $i$ th coordinate character line, and $[C^{2}]_{T}$ is the complex-oriented fundamental class of degree $4$ . Localized torus-equivariant Borel–Moore classes

[F2]

For a proper fixed locus, the localized pushforward of $[X]_{T}$ is the sum of proper fixed-component traces divided by the localized normal Euler classes. Noncompact localization with a proper fixed locus

[F3]

At a fixed point, the Borel normal Euler class is the product of the character classes of the tangent representation, with multiplicity. Fixed-embedding BM self-intersection and Euler units

[A1]

AC is inherited through the Borel–Moore suppliers; the coordinate and tangent calculations are choice-free. Localized torus-equivariant Borel–Moore classes

Verification

Given: The action and data in the Example.

1.1A1F1F3givenalgebra

The affine plane itself is an affine $T$ -invariant chart. A point is fixed exactly when both coordinates vanish, so $F = {0}$ is proper. Differentiating the action at $0$ gives the representation $C_{χ_{1}} \oplus C_{χ_{2}}$ ; hence $e_{T} (ν_{F}) = u_{1} u_{2}$ by [F3].

2.1F1F2step 1.1algebra∎

Apply [F2] with the cohomological insertion $1$ . The fixed component is one point, and its proper pushforward sends its degree-zero fundamental class to $1$ . Therefore $p_{*}^{l o c} [C^{2}]_{T} = 1 / (u_{1} u_{2}) \in S_{T}^{- 1} H_{4}^{B M, T} (p t; Q)$ . Here the target means $(S_{T}^{- 1} H_{*}^{B M, T} (p t; Q))_{4} = (S_{T}^{- 1} R_{T})^{- 4}$ , which can be nonzero although the unlocalized group $H_{4}^{B M, T} (p t; Q)$ is zero. The denominator has cohomological degree $4$ , so its inverse has Borel–Moore degree $4$ , matching the fundamental class.

Affine-plane localization from BM classes

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources