ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Gts direct sum framing free open gysin

Example

Construct the common free open, coordinate Koszul zero locus, hyperbolic normal complex and Gysin identity; calculate double Euler comparison zero via nowhere-zero tautological section.

Example references

Bojko, Wall-crossing for Calabi–Yau fourfolds: framework, tools, and applications, Assumptions 5.1 and 5.10; Definition 5.3; §6.2; §6.3; Assumption 6.9; Remark 6.10; Example 6.11; Theorem 6.12; Lemma 6.13; Equation (6.14); §7.2.

Bojko, Kuhn, Liu, and Thimm, Wall-crossing for equivariant DT4 invariants, §2.2.8, §3.2, Assumption 4.2.4, and Theorem 1.2.1; used only under their stated hypotheses.

Setup

For a rank-one check take $V_{1} = V_{2} = C$ . Then $P (V_{1} \oplus V_{2}) = P^{1}$ with coordinates $[x : y]$ . The zero-component loci are $J_{1} = [1 : 0]$ and $J_{2} = [0 : 1]$ ; in a simple old-object model the common free locus is all of $P^{1}$ . The sections are $y, x \in H^{0} (P^{1}, O (1))$ , each with a regular coordinate zero. For every point $[x : y]$ , at least one component is nonzero. If a scalar automorphism of the old one-dimensional object and a scalar automorphism of the source line preserve this framing, their scalars must agree; quotienting by the source-line scalar therefore leaves only the identity. Hence the common free locus is all of $P^{1}$ . The normal obstruction pair is ordered $O (1)^{\lor} \oplus O (1) [2]$ .

Facts & Assumptions

Given: the constructed original geometric or dg-quiver framing realization of [F1], the actual obstruction diagrams and compatible original orientations established in [F2], and the stipulated original overlap. The bare framing bundle and a supplier status record alone are not used to identify obstruction theories.

[F1]

Gts weak phase framing moduli datum: exact current interface: “Specify original weak stability, exact phase category, actual ordinary-vs-admissible maps, original concentration opens and exact Fr_k, common-line framing, even emergent class domain, torus linearizations and orientations. No sum closure, cofinal-index availability, or abelian phase category is implicit.” Only these input data are used; the actual obstruction diagrams are supplied separately by [F2].

[F2]

Gts original k direct sum framing oriented gysin supplies the constructed-realization common free open, actual regular zero-component embeddings, hyperbolic orientation and supported identities $i_{i}^{!} [U]^{v i r} = [J_{i}]^{v i r}$ , with zero double Euler cap. The original $J_{i}$ virtual classes and compatible orientations are retained; neither an integral over $U$ nor an arbitrary-enhancement identification is inferred.

Verification

1.1givenF1algebra

Near $J_{1}$ in the chart $x \neq 0$ , the zero section $y$ is the coordinate $y / x$ in a trivialization of $O (1)$ , so its zero ideal is generated by a nonzerodivisor and its normal is $O (1) ∣_{J_{1}}$ . The chart $y \neq 0$ gives the same result for $x$ at $J_{2}$ .

2.1givenF2step 1.1algebra

The two normal halves pair by evaluation. Their determinant $\det (N^{\lor}) \otimes \det (N)$ has the canonical trivialization in the displayed order, so the zero-section Gysin uses the hyperbolic orientation required by [F2].

3.1givenF1F2step 2.1algebra

Since $c_{1} (O (1))$ has degree one, a single cap gives $c_{1} (O (1)) \cap [P^{1}] = [J_{i}]$ . The double cap is $c_{1} (O (1))^{2} \cap [P^{1}] = 0$ because $H^{4} (P^{1}) = 0$ . Equivalently, each own component section is nowhere zero when restricted to its pair locus, so that top Chern class restricts to zero.

4.1givenF2step 3.1algebra∎

This finite model checks regular zero loci, orientation, and the supported single-cap formula. The zero double cap is not injective and says nothing about mixed fixed properness.

Depends on

Used by

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Sources

Y. Bojko, Wall-crossing for Calabi–Yau fourfolds: framework, tools, and applications (standard reference, not scraped)
Bojko, Kuhn, Liu, and Thimm, Wall-crossing for equivariant DT4 invariants (standard reference, not scraped)