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The Lazić-Peternell Generalised Abundance Conjecture for projective klt pairs

Let (X,B)(X,B) be a projective klt pair with KX+BK_X+B pseudo-effective, and let LL be a nef Cartier divisor such that KX+B+LK_X+B+L is nef. Lazic and Peternell conjectured that KX+B+LK_X+B+L is numerically equivalent to a semiample Q\mathbb Q-divisor (numerical equivalence is needed: a nontorsion degree-zero line bundle on an elliptic curve is nef with no sections). They proved it for surfaces and for threefolds where KX+BK_X+B has positive numerical dimension, and reduced it in general to the minimal model program, abundance and a semiampleness conjecture on Calabi-Yau pairs. Is KX+B+LK_X+B+L always numerically equivalent to a semiample divisor?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Birational geometry; minimal model program
Posed by
Vladimir Lazić and Thomas Peternell (On Generalised Abundance, I)
Year posed
2018
Years open
8y
Solved
2026-10-03
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for a projective klt Q\mathbb Q-pair (X,B)(X,B) over an algebraically closed field of characteristic zero with KX+BK_X+B pseudo-effective and MM nef Q\mathbb Q-Cartier on XX, if KX+B+MK_X+B+M is nef then it is numerically equivalent to a semiample Q\mathbb Q-Cartier divisor. Corollary: on a klt pair with KX+B≡0K_X+B\equiv0 every nef Q\mathbb Q-Cartier divisor is numerically semiample. New ingredient: a nef divisor of full nef dimension on a klt Calabi-Yau pair is big. Kahler companion: for smooth compact Kahler XX, snc BB with coefficients below one and nef MM, c1(KX+B+M)c_1(K_X+B+M) is represented in Bott-Chern cohomology by a semiample line bundle. Not shown: semiampleness of KX+B+MK_X+B+M itself, lc pairs, or nef data living only on a higher model.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is dated October 3, 2026; a compact Kahler companion is dated October 4, 2026. The proof uses two other release manuscripts as inputs: the log abundance theorem and the minimal-model existence theorem for generalized lc pairs.

Verification

No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 of 'Numerical Semiampleness of Nef Adjoint Divisors' read against Lazić-Peternell's conjecture as the paper states it; Theorem 1.1 allows a nef Q\mathbb Q-Cartier summand, so it contains the Cartier case. No Lean formalization. The proof is conditional on two unrefereed release manuscripts, 'Log abundance in characteristic zero' (already in the catalog) and 'Minimal models and Mori fibre spaces for generalized log canonical Q-pairs' (the other entry of this family).

Sources

Changelog1 change

Discussion