Result 064, Algebraic and complex geometry

Topological triviality of μ-constant surface singularities

Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

Proof

The bigger picture

Why it matters

A complex surface can have an isolated singular point while deforming in a family. The manuscript reports that keeping one numerical invariant, the Milnor number, constant forces its local topology, including the defining function, to stay unchanged.

What changes?

The claim covers every holomorphic one-parameter family of isolated hypersurface singularities in three complex dimensions: surfaces defined locally by one complex analytic equation, singular only at the origin. The Milnor number measures the singularity's complexity. When it stays constant, shrinking the parameter disk and local neighborhoods yields ambient homeomorphisms, meaning coordinate changes continuous in both directions, that preserve the defining functions. They vary jointly continuously with position and parameter, keep the parameter fixed and fix each origin.

What does that help mathematicians do?

This would rule out a local topological change hidden behind a constant Milnor number in these families, settling the surface case of the mu-constant problem. Preserving the defining functions is stronger than merely identifying the singular surfaces: the same coordinate changes also identify nearby level sets with the same function value. Researchers could therefore compare these local structures coherently throughout the parameter disk.

Are there practical applications?

The immediate value is foundational for studying how complex surface singularities deform. Within the stated setting, constancy of the Milnor number would suffice to establish a continuous local topological identification, rather than requiring a separate construction for each family. The claim concerns topology, not analytic equivalence, and does not provide a computational algorithm or a result for arbitrary parameter spaces.

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

Topological triviality of mu-constant families of surface singularities

September 24, 2026 37 pages

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

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Cite (BibTeX) 
@misc{OAI:Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026,
  author = {{OpenAI}},
  title = {{Topological triviality of $\mu$-constant families of surface singularities}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf}{OAI:Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026}},
  year = {2026}
}

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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.