Result 048, Algebraic and complex geometry

A characteristic-zero counterexample to Lipman–Zariski

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

Disproof or counterexample

The bigger picture

Why it matters

Can a geometric space have perfectly organized infinitesimal directions and still contain a singularity? The manuscript reports a complex surface where this happens, challenging a proposed way to recognize smooth spaces from their tangent data.

What changes?

The claimed example is a normal affine complex surface: a space of two complex dimensions defined by polynomial equations, satisfying an algebraic regularity condition called normality that still permits singularities. Its tangent sheaf, which records algebraic vector fields, is free of rank two, meaning those fields have a global algebraic basis of two generators. Yet the surface is singular. The manuscript presents this as a counterexample to the Lipman–Zariski conjecture in characteristic zero, specifically over the complex numbers.

What does that help mathematicians do?

If established, this example would rule out using freeness of the tangent sheaf alone as a universal test for smoothness, even for normal affine complex surfaces. The distinction matters because tangent data describe infinitesimal behavior, while smoothness concerns the space itself. Researchers seeking a valid smoothness criterion would need additional hypotheses that exclude this example, rather than relying on normality and tangent-sheaf freeness alone.

Are there practical applications?

The immediate value is foundational in algebraic and complex geometry. The reported construction would provide a test case for proposed links between vector fields and singularities, helping researchers identify which assumptions a smoothness argument actually needs. The supplied abstract describes no computational method or practical deployment; its stated contribution is a counterexample to a geometric criterion.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A singular normal affine surface with free tangent sheaf

September 23, 2026 37 pages

We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026,
  author = {{OpenAI}},
  title = {{A singular normal affine surface with free tangent sheaf}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf}{OAI:A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.