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The normalized-volume gap in dimension four
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Abstract
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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Related prior literature mentioned by others
- Cristiano Spotti and Song Sun, Explicit Gromov–Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds (arXiv:1705.00377, 2017). The AI Proved What? family exposition identifies this work as the source of the ordinary-double-point volume-gap conjecture addressed by both manuscripts. [family-level]
- Chi Li, Minimizing normalized volumes of valuations (arXiv:1511.08164, 2015; revised 2017). The AI Proved What? family exposition cites this work for the normalized-volume invariant used in both manuscripts. [family-level]
- Yuchen Liu, K-stability of cubic fourfolds (arXiv:2007.14320, 2020; revised 2022). The AI Proved What? family exposition identifies Liu's local-complete-intersection case as prior work, and the OpenAI Math Digest fourfold guide explains its use after the claimed embedding-dimension reduction. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Zhiyu Liu, The finite degree formula for normalized volumes (arXiv:2607.27032, July 2026). Proves a finite degree formula for finite surjective crepant morphisms of klt singularities, which relates to the paper's cone reduction (Section 2), where the finite-degree formula of Xu–Zhuang is used to obtain an isolated Gorenstein cone.
Digests and expositions
- OpenAI Math Digest, The normalized-volume gap in dimension four (online exposition, undated). This guide maps the proposed reduction from normalized volume at least 162 to embedding dimension at most five, separating the discrepancy and log-canonical-threshold estimates used for the threefold section. [AI-generated per source]
- AI Proved What?, The mildest singularity? The ordinary-double-point volume gap (episode 037, October 2026). This AI-assisted video and transcript cover both manuscripts, their scope, earlier special cases, and the all-dimensions argument's stabilizer split. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 037
Independent sites; their content is not reviewed here.
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}
}Family 037: The ordinary-double-point volume gap
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.
Other papers in this family:
- The ordinary-double-point gap in every dimension · September 24, 2026