# Special-manifold group implication ledger

Family 057: Fundamental groups of special complex varieties and root orbifolds. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Track virtual abelianity for special compact Kähler fundamental groups and separate restrictions on linear representations in the quasi-projective case.

## Applicability and commercial boundary

A virtually nilpotent image of class at most two is not the same as a virtually abelian source group.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Map the geometric and representation hypotheses separately.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The abelianity conjecture for special compact Kähler manifolds](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf).