we-solve (beta)

Math-2543 · mathematics · Algebraic geometry

Conditional abelianity for special half-weight fourfold pairs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assume the manifold abelianity conjecture for special compact Kähler manifolds in the precise form Assumption ass:manifold of the cited paper. If X is a smooth connected projective complex fourfold and D a nonempty smooth connected reduced divisor whose square-root stack sqrt[2](X,D) is special, must $\pi _{1}(X$ minus D) modulo the normal closure of the square of a positive meridian around D be virtually abelian?

Short English statement

Assume the manifold abelianity conjecture for special compact Kähler manifolds in the precise form Assumption ass:manifold of the cited paper. If X is a smooth connected projective complex fourfold and D a nonempty smooth connected reduced divisor whose square-root stack sqrt[2](X,D) is special, must $\pi _{1}(X$ minus D) modulo the normal closure of the square of a positive meridian around D be virtually abelian?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2543-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2543

Manuscript 1 (family 057, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Conditional abelianity for special half-weight fourfold pairs

OpenAI's claim

Subject to the stated companion abelianity input, virtual abelianity is claimed for the full fundamental group of a special order-two root orbifold from a smooth projective fourfold and one smooth connected divisor. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Assume every smooth connected compact Kähler special manifold has virtually abelian fundamental group in every dimension. Special means there is no line bundle $L$ with a nonzero $L\to\Omega^p$ and $\kappa(L)=p$. Let $X$ be a smooth connected projective complex fourfold and $D$ a reduced divisor that is smooth, nonempty and connected. Require the order-two root orbifold $\sqrt[2]{(X,D)}$ to be special by the same equivariant differential criterion; locally its chart has $z_1=w_1^2$ and sign stabilizer. OpenAI claims \[ \pi_1(X\setminus D,x)/\langle\!\langle\gamma^2\rangle\!\rangle\text{ is virtually abelian}, \] where $\gamma$ is a positively oriented transported meridian. The ordinary-manifold abelianity theorem is an explicit premise; a general orbifold abelianity theorem is not assumed or concluded.

Outline of the argument

Manuscript 1 (family 057, 2026-10-05): A smooth projective eightfold over the root orbifold has genus-one product fibers and submersive lifts over every base divisor. Those lifts detect poles of descended differential tensors, so specialness passes to the source, while even pullback multiplicities give a surjection of its fundamental group onto the orbifold group.

Formal verification

Family 057 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

Go to contribution →