∑math20

Algebraic and complex geometry · Result 037

The ordinary-double-point volume gap

2 papers

Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most \(2(n-1)^n\). Equality holds precisely for an analytic ordinary double point.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of The ordinary-double-point gap in every dimension

The ordinary-double-point gap in every dimension

OpenAI · September 24, 2026

We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most \(2(n-1)^n\), with equality precisely at an analytic ordinary double point.

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@misc{OAI:The-ordinary-double-point-gap-in-every-dimension-September-24-2026,
  author = {{OpenAI}},
  title = {{The ordinary-double-point gap in every dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf}{OAI:The-ordinary-double-point-gap-in-every-dimension-September-24-2026}},
  year = {2026}
}
First page of The normalized-volume gap in dimension four

The normalized-volume gap in dimension four

OpenAI · September 24, 2026

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

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Cite
@misc{OAI:The-normalized-volume-gap-in-dimension-four-September-24-2026,
  author = {{OpenAI}},
  title = {{The normalized-volume gap in dimension four}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf}{OAI:The-normalized-volume-gap-in-dimension-four-September-24-2026}},
  year = {2026}
}

Formal verification

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