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Anticanonical nonvanishing for smooth projective varieties with smoothly semipositive anticanonical bundle

Let XX be a smooth connected complex projective variety whose anticanonical bundle −KX-K_X carries a smooth Hermitian metric of semipositive curvature (equivalently, by Yau's theorem, a Kahler metric of nonnegative Ricci curvature in that class). Such −KX-K_X is nef. Demailly-Peternell-Schneider and Campana-Demailly-Peternell described the universal cover of such XX, but the existence of anticanonical sections was open in general; for nef log-anticanonical divisors it was known only in dimension three and under fiber semiampleness. The manuscript attaches the question to Yau's Problem 75. Does H0(X,−mKX)≠0H^0(X,-mK_X)\ne0 for some integer m>0m>0?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Complex algebraic geometry; Kahler geometry
Posed by
Shing-Tung Yau, Open problems in geometry (Lectures on Differential Geometry, 1994), Problem 75, as the manuscript cites it
Year posed
1994
Years open
32y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: if XX is smooth connected complex projective and −KX-K_X admits a smooth Hermitian metric of semipositive curvature, then H0(X,−mKX)≠0H^0(X,-mK_X)\ne0 for some m>0m>0. The key intermediate result (Theorem 1.2) gives sections of aL+bDaL+bD on varieties without holomorphic forms from a finite-volume adjoint metric. Not shown: an effective bound on mm, the case m=1m=1 (false in general, e.g. Enriques surface times P1\mathbb P^1), or nonvanishing when −KX-K_X is only nef or carries a singular semipositive metric.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family 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 seven manuscripts, all dated September 26, 2026. The principal one proves the headline from a finite-volume theorem by induction on dimension and a torus-invariant section on the rationally connected factor; companions give invariant indices, klt and torus-quotient versions, metric descent and effectivity with controlled boundary. One cited 2026 paper by other authors (Chen-Filip-Sun-Tosatti-Zhang) records that ChatGPT suggested a preliminary lemma; that is outside this release.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'Anticanonical nonvanishing from smooth semipositivity' were read; the theorem states exactly the displayed question. The posing reference is Yau's 1994 problem list as the manuscript cites it; Yau's text was not obtained, so whether Yau asked for m=1m=1, for a positive multiple, or under nef rather than smooth semipositive hypotheses was not confirmed (the paper's Enriques surface times P1\mathbb P^1 example shows that m=1m=1 can fail). The release has no Lean formalization for this family (lean/docs/068.md does not exist at the pinned commit). Companions were skimmed, not refereed.

Sources

Changelog1 change

Discussion