Shokurov's bounded klt complements conjecture for Fano type contractions with finite rational coefficients

Let $f:X\to Z$ be a contraction of normal quasi-projective complex varieties with $X$ of Fano type over $Z$ , and $(X,\Delta)$ an $\epsilon$ -lc pair of dimension $d$ with coefficients in a fixed finite set $I\subset[0,1]\cap\mathbb Q$ and $-(K_X+\Delta)$ nef over $Z$ . A monotone klt $n$ -complement near $z\in Z$ is a boundary $\Delta^+\ge\Delta$ with $(X,\Delta^+)$ klt and $n(K_X+\Delta^+)\sim0$ over a neighbourhood of $z$ . Shokurov introduced complements and conjectured their boundedness; Birkar proved boundedness of lc complements, but a bounded klt index depending only on $d$ , $\epsilon$ and $I$ was known only over curves (Birkar) and surfaces (Chen). In the formulation of B. Chen (2024, Conjecture 1.1): is there $M=M(d,\epsilon,I)$ such that every such contraction has a klt $M$ -complement near every point of $Z$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Birational geometry; complements and the minimal model program
Posed by: V. V. Shokurov (in the precise form stated by Bingyi Chen, Conjecture 1.1)
Year posed: 2000
Years open: 26y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 33 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $d$ and rational $\epsilon>0$ there is $N(d,\epsilon)$ such that every projective contraction $f:X\to Z$ over an algebraically closed field of characteristic zero with $X$ $\epsilon$ -lc of dimension $d$ , $K_X$ $\mathbb Q$ -Cartier and $-K_X$ ample over $Z$ has, near each closed point of $Z$ , a boundary $B$ with $(X,B)$ klt and $N(K_X+B)\sim0$ . Corollary 1.2 (over $\mathbb C$ ): the finite-rational-coefficient form for Fano type pairs with nef anti-log-canonical divisor, with monotone complements. The companion proves that some Cartier divisor through any base point has pulled-back lc threshold at least $\tau(d,\epsilon)$ , also for real boundaries over $\mathbb C$ . Not shown: $\epsilon'$ -lc complements, DCC coefficient sets, or effective values of $N$ .

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, and that some outputs build on earlier model results. 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 family has two manuscripts dated September 25, 2026: 'Bounded klt complements for Fano contractions' (principal) and 'Uniform Cartier sections for Fano type contractions', which proves the equivalent Birkar-Shokurov Cartier-divisor statement by an independent argument.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against Chen's Conjecture 1.1 as both manuscripts quote it. Corollary 1.2 gives exactly the finite-rational-coefficient statement over $\mathbb C$ , and Theorem 1.1 gives klt complements of index $N(d,\epsilon)$ for $\epsilon$ -lc Fano contractions over any algebraically closed field of characteristic zero. Both proofs go through Chen's theorem that bounded klt complements are equivalent to a uniform local threshold statement; the companion proves that statement (the Birkar-Shokurov Cartier-divisor conjecture) independently. There is no Lean formalization. Not addressed: the stronger requirement that the complement itself be $\epsilon'$ -lc (Chen, Remark 1.2(1); Shokurov 2004), and coefficient sets that are DCC or real rather than finite rational.

Shokurov's bounded klt complements conjecture for Fano type contractions with finite rational coefficients

What was actually shown

What the AI did

Verification

Sources

Discussion