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Two-step monodromy of special quasi-projective varieties

OpenAI · September 24, 2026

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Abstract

We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.

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Possibly relevant literature (found by the agent, not checked)

Vasily Rogov, Topology of higher Albanese maps and aspherical varieties with nilpotent fundamental group (arXiv:2607.19013, July 2026). Shows that virtually nilpotent fundamental groups of aspherical normal varieties are virtually two-step nilpotent, which relates to the paper's class-two completion theorem (Theorem 1.2) and its remaining ordinary-group question.
Taito Shimoji, On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven (arXiv:2510.09026v3, October 16, 2026; first submitted October 10, 2025). Its class-two bound for torsion-free nilpotent fundamental groups of rank at most seven provides a comparison for the manuscript’s class-two monodromy and unipotent-completion claims, with a rank restriction and without a specialness hypothesis.

Elsewhere

Emergent Mind: family 057

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Cite

@misc{OAI:Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026,
  author = {{OpenAI}},
  title = {{Two-step monodromy of special quasi-projective varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf}{OAI:Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026}},
  year = {2026}
}