View source for The abundance conjecture for threefolds in positive characteristic

'''The abundance conjecture for threefolds in positive characteristic''' predicts that, for a projective log canonical threefold pair (''X'', ''B'') over a field of characteristic ''p'' > 0, the divisor ''K''<sub>''X''</sub> + ''B'' is semiample whenever it is nef: some positive multiple is generated by its global sections. Over the complex numbers the threefold case was settled by Miyaoka, Kawamata and Keel–Matsuki–McKernan. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 24 September and 5 October 2026, claim the case of numerical dimension one over algebraically closed fields of characteristic ''p'' > 3, first for terminal threefolds and then for log canonical pairs. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** Over the complex numbers, Miyaoka proved abundance for minimal threefolds of numerical dimension one, Kawamata completed abundance for minimal threefolds, and Keel, Matsuki and McKernan established log abundance for log canonical threefold pairs.
** In positive characteristic, the threefold minimal model program was developed by Hacon–Xu, Birkar and Birkar–Waldron in characteristic greater than five, with log canonical and relative extensions by Waldron and by Hashizume, Nakamura and Tanaka; Hacon and Witaszek treated characteristic five.
** Xu and Zhang proved nonvanishing for terminal minimal threefolds, and Witaszek klt nonvanishing and abundance in nef dimension at most two, in characteristic greater than five. Zheng Xu proved nonvanishing for pseudoeffective log canonical threefold pairs in characteristic greater than three, together with abundance in nef dimension at most two, and later abundance in numerical dimension two. The manuscripts describe the remaining case as numerical dimension one with nef dimension three.

== The original results ==
* '''Claim (terminal threefolds).''' Theorem 1.1 of ''Abundance in numerical dimension one for terminal threefolds in positive characteristic'' (24 September 2026) states that if ''X'' is a normal projective '''Q'''-factorial terminal threefold over an algebraically closed field of characteristic ''p'' > 3 and ''K''<sub>''X''</sub> is nef with ν(''K''<sub>''X''</sub>) = 1, then some positive multiple of ''K''<sub>''X''</sub> has two independent global sections; in particular κ(''X'', ''K''<sub>''X''</sub>) = 1 and ''K''<sub>''X''</sub> is semiample. The manuscript states that no assumption is made on the first Betti number, irregularity or Albanese variety, and that the new step is the second pluricanonical section, the first section and the passage from positive Kodaira dimension to semiampleness being Xu's results.
* ''Method, as described:'' in the hypothetical case of maximal nef dimension, the preparation produces a connected nef Cartier cycle ''D'' on a smooth projective threefold with torsion normal bundle. Strong semistability of the tangent and cotangent bundles for the degree ''c''<sub>1</sub>(−)·''H''·''D'', a three-row jet calculation along ''D'' and a vanishing criterion for Laurent cohomology then make the Picard group of the punctured formal neighbourhood ''p''-divisible by Cartier descent, while intersection with an ample ''H'' gives a nonzero homomorphism from that group to '''Z''', a contradiction.
* '''Claim (log canonical pairs).''' Theorem 1.1 of ''Log abundance in numerical dimension one for threefolds in positive characteristic'' (5 October 2026) states that if (''X'', ''B'') is a projective log canonical threefold pair over an algebraically closed field of characteristic ''p'' > 3, with ''B'' an effective '''Q'''-divisor and ''K''<sub>''X''</sub> + ''B'' '''Q'''-Cartier, nef and of numerical dimension one, then ''K''<sub>''X''</sub> + ''B'' is semiample. Neither terminality nor '''Q'''-factoriality is assumed. The proof constructs from the pair the cycle excluded by the companion manuscript's obstruction theorem, which it states with all its hypotheses (Theorem 2.1), using a separation of the boundary by quantitative bend-and-break, a partial minimal model program, torsion on a Cartier thickening and a finite separable cover.
* '''Sources''' (openai/math repository, commit <code>1581d4c</code>): [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026 Abundance in numerical dimension one for terminal threefolds in positive characteristic] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026 Log abundance in numerical dimension one for threefolds in positive characteristic].
* '''Formalization.''' The collection lists no Lean formalization for these results.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-07 || F. Bernasconi, J. Witaszek and Z. Xu, ''Abundance for threefolds in positive characteristic'', [https://arxiv.org/abs/2610.10439v1 arXiv:2610.10439] || Parallel (independent) ||
* Theorem 1.1 states that if (''X'', ''B'') is a projective log canonical threefold pair over a perfect field of characteristic ''p'' > 42 and ''K''<sub>''X''</sub> + ''B'' is nef, then it is semiample. The paper states that the known cases reduce the theorem to numerical dimension one and nef dimension three, and that its main technical result is a semiampleness criterion (Corollary 6.6) for a prime divisor ''S'' near which (''X'', ''S'') is log canonical, ''K''<sub>''X''</sub> + ''S'' is '''Q'''-linearly a positive multiple of ''S'', and a multiple of ''S'' has trivial restriction to ''S''; the proof uses bounds on the global indices of log Calabi–Yau surfaces from a companion paper. Its range ''p'' > 42 covers every numerical dimension; the OpenAI manuscripts treat numerical dimension one for ''p'' > 3.
* Its AI disclosure states that the project began a couple of years before the advent of AI, that OpenAI's ChatGPT 5.6 was used to solve a very special case of Corollary 6.6 and, interactively, to obtain Proposition 5.6, and that the authors uploaded "this (very) preliminary version" following "the release of OpenAI's list of solved problems in mathematics, which claims to solve the abundance conjecture for threefolds in characteristic ''p'' > 3".
| arXiv preprint, version 1 (57 pages), described by the authors as preliminary; the companion paper with the index bounds is announced for later. The paper mentions the OpenAI release only in its AI disclosure.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_182.json rank_182] (terminal manuscript) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_458.json rank_458] (log canonical manuscript) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap in either manuscript. For the terminal manuscript, the reviewer states that it read the whole source and checked the preparation, slope, jet and Laurent steps in detail; its one minor note is that load-bearing recent inputs (a quantitative bend-and-break bound, Xu's abundance theorem in nef dimension at most two, and nef reduction with a partial minimal model program) could not be checked against their sources, although the manuscript uses them consistently with their stated hypotheses. For the log canonical manuscript, it states that it read the whole paper; its one minor note is that several key steps rest on recent preprints it could not check independently (Xu's partial minimal model program and termination, the quantitative bend-and-break bound, Posva's abundance for semi-log-canonical surfaces), and it states that the companion's obstruction uses only the prepared data that this manuscript constructs.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The characteristic-zero theorems and the earlier positive-characteristic results are due to the authors named in the Background section, as recounted by the manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]