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Math-2586 · mathematics · Algebraic geometry

Two-step linear images of special quasi-projective groups

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every connected smooth special complex quasi-projective variety V, every $d\geq 1$ and every representation $\rho :\pi _{1}(V)\to GL_d(C)$, must its image have a finite-index subgroup nilpotent of class at most two? The assertion concerns representation images, not virtual nilpotence of $\pi _{1}(V)$ itself.

Short English statement

For every connected smooth special complex quasi-projective variety V, every $d\geq 1$ and every representation $\rho :\pi _{1}(V)\to GL_d(C)$, must its image have a finite-index subgroup nilpotent of class at most two? The assertion concerns representation images, not virtual nilpotence of $\pi _{1}(V)$ itself.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2586-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2586

Manuscript 1 (family 057, 2026-09-24): 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.

Two-step linear images of special quasi-projective groups

OpenAI's claim

Every complex linear representation of a smooth special quasi-projective fundamental group is claimed to have virtually two-step-nilpotent image, while the paper also constructs nonspecial general quasi-Albanese fibers. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth complex quasi-projective $V$, specialness requires the following after any proper birational modification: each dominant algebraic fibration onto a normal quasi-projective base $Z$ with $\dim Z>0$ and connected general fiber has $\kappa(Z,\Delta_f)<\dim Z$, where $\Delta_f$ denotes the induced orbifold boundary. The orbifold dimension is computed on smooth compactifications with infimum multiplicities, and removed divisors have infinite multiplicity. For every connected such $V$, every $d\ge1$ and every $\rho:\pi_1(V)\to\operatorname{GL}_d(\mathbb C)$, OpenAI claims \[ \exists H\le\rho(\pi_1V):\quad[\rho(\pi_1V):H]<\infty,\qquad[H,[H,H]]=1. \] Thus every complex linear image is virtually nilpotent of class at most two. Virtual abelianity of the entire open fundamental group is not the stated conclusion.

Outline of the argument

Manuscript 1 (family 057, 2026-09-24): At each canonical nilpotent stage, the fiber Lie algebra equals the commutator algebra. Its abelianization has only weight minus two, making the next conjugation bracket incompatible with the allowed weights; universality then transfers the resulting class-two bound to arbitrary unipotent representations.

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

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