# Anticanonical nonvanishing in every dimension > A claimed proof that a smooth nonnegatively curved anticanonical bundle always has a section of some power. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-068/ Last updated: 2026-10-14 **A claimed proof that a smooth nonnegatively curved anticanonical bundle always has a section of some power.** **Trust:** Claimed, not checked · **Papers:** 7 · **First paper:** September 26, 2026 · **Field:** Algebraic and complex geometry · **Entry:** 068 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. Every smooth algebraic variety carries a line of combined tangent directions, its anticanonical bundle. A section is a nonzero holomorphic choice of a value in that line at every point. Yau asked whether a nonnegatively curved anticanonical bundle must have a section of some power. These papers claim yes, in every dimension, when the metric is smooth. The power can matter: an example has no section at the first power but has one at the second. ## At a glance - **First step:** 1994 (Shing-Tung Yau, problem list in Lectures on Differential Geometry) - **Significance:** Major. It claims a positive answer to a question from Yau's problem list in every dimension, where earlier results reached dimension three. - **Trust:** Claimed, not checked - **Field:** Algebraic and complex geometry (7 papers) ## The explanation Take an Enriques surface E, a compact complex surface with a special twist. Its canonical bundle K_E is not trivial, but its square is trivial. K_E has no nonzero section[1]. A section of a line bundle is a holomorphic choice of a value in the line at every point. Now form X = E × P¹, the surface times a sphere. Its anticanonical bundle is the dual of K_E on the E side, times degree 2 on the sphere. The dual of K_E equals K_E, so the E side has no sections. On the sphere, degree 2 has 3 independent sections. The product formula gives 0 × 3 = 0 sections for the first power[2]. The square is trivial on E, with 1 section, and degree 4 on the sphere, with 5. So the second power has 1 × 5 = 5 independent sections. The anticanonical bundle −K_X is the top exterior power of the tangent bundle, one complex line at each point. A smooth Hermitian metric measures lengths in that line. It is semipositive when its curvature is never negative. In the example, a flat metric on E and the round metric on the sphere give one[3]. Such a metric makes −K_X nef, meaning it has degree at least zero on every curve[4]. Nef alone is not known to give a section. Yau asked for a nonzero section of some power −mK_X with m > 0[5]. The paper claims yes whenever −K_X has a smooth semipositive metric[6]. The power matters. In the example, −K_X is not torsion, meaning no power of it is trivial[7]. Yet the first power still has no section. Earlier structure theorems split the universal cover into flat, Ricci-flat and remaining compact pieces[8]. The universal cover is the simply connected space that wraps around X. Ricci-flat means a special kind of zero curvature. In 2023, Lazić, Matsumura, Peternell, Tsakanikas and Xie proved numerical effectivity for log canonical threefold pairs[9]. That is weaker, since a section needs linear effectivity[10]. Müller proved actual nonvanishing in dimension three for a related class[11]. The main obstacle is a leftover symmetry. A section on the compact piece passes down to X only if it respects the deck transformations[12]. These are the symmetries of the universal cover that fold it back onto X. After a finite step, they act on the compact piece through a compact torus, a product of circles. The paper builds an anticanonical section on that piece that the torus fixes[13]. Its product with frames on the other pieces descends to a finite cover of X[14]. A norm map then pushes it down to X itself. The key tool is a finite-volume theorem, proved by induction on dimension[15]. It applies to varieties with no holomorphic forms. Rationally connected varieties, where rational curves join any two points, satisfy that condition[16]. Six companion papers in the release develop related tools. They treat torus actions, klt pairs, metric descent and error terms in twisted sections. How sure is this? There is no Lean scope note for these papers, so no computer proof checker has tested them. The main paper claims the theorem in every dimension and relies on its companions and on published results. No specialist review has been published. **What it doesn't show.** This does not say how large the power m must be, and it does not cover metrics that are only singular. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major if the argument holds. It spans seven papers with no Lean scope note. No specialist review is published. ## Words used - **Anticanonical bundle:** The top exterior power of the tangent bundle: one complex line of combined tangent directions at each point. - **Section:** A holomorphic choice of a value in a line bundle at every point. - **Semipositive metric:** A smooth way of measuring lengths in a line bundle whose curvature is never negative. - **Nef:** Having degree at least zero on every curve. - **Torsion:** A line bundle some power of which is trivial. - **Enriques surface:** A compact complex surface whose canonical bundle is nontrivial but has trivial square. - **Universal cover:** The simply connected space that wraps around a variety, folding back onto it by deck transformations. ## The original papers (7) - [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf) - September 26, 2026 - [Invariant anticanonical indices and conversion of twisted differentials](https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf) - September 26, 2026 - [Bounded anticanonical metrics on klt pairs and torus quotients](https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf) - September 26, 2026 - [Cohomological transfer and equivariant anticanonical sections](https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf) - September 26, 2026 - [Metric descent and rank-preserving contractions](https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf) - September 26, 2026 - [Integrable metrics and effectivity with controlled boundary](https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf) - September 26, 2026 - [Exact orders and invariant anticanonical linear systems](https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf) - September 26, 2026 ## Quoted lines from the paper (16) [1] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > For a complex Enriques surface \(E\), \(K_E\) is nontrivial of order two and has no section. [2] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > The product formula for sections gives no first-power section, whereas the square has sections. [3] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > The flat metric on the first factor and Fubini--Study metric on the second make it smoothly semipositive. [4] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Such a metric makes the numerical class nef. [5] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > This is the positive-multiple formulation associated with Yau's anticanonical nonvanishing question. [6] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > If \(-K_X\) admits a smooth Hermitian metric of semipositive curvature, then \(H^0(X,-mK_X)\ne0\) for some integer \(m>0\). [7] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Restriction to a \(\mathbb P^1\) fiber shows that this anticanonical line is not torsion. [8] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > The structure results of Demailly--Peternell--Schneider and Campana--Demailly--Peternell describe the universal cover by its flat, compact Ricci-flat, and remaining compact factors. [9] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > In the broader nef setting, Lazi\'c--Matsumura--Peternell--Tsakanikas--Xie proved numerical effectivity for projective log canonical threefold pairs with rational boundary and nef log-anticanonical divisor, under \(\mathbb Q\)-factoriality or rational singularities. [10] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Numerical effectivity means that the divisor class is numerically equivalent to an effective rational divisor; the existence of a section requires rational \emph{linear} effectivity. [11] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > For projective klt \(\mathbb Q\)-pairs with nef log-anticanonical divisor, M\"uller proved actual nonvanishing in dimension three, and in arbitrary dimension when the log-anticanonical divisor of a general fiber of the maximal rationally connected quotient is semiample. [12] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > A section on the compact factor descends only if it is compatible with the deck transformations. [13] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > The flat and Ricci-flat factors supply canonical frames; on the remaining factor we construct an anticanonical section fixed by the torus, for the natural action on that factor's anticanonical bundle. [14] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Its product with those frames descends to a finite \'etale cover, and its finite \'etale norm then gives a section on \(X\). [15] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Theorem … is proved by induction on dimension. [16] From [Anticanonical nonvanishing from smooth semipositivity](https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf): > Rationally connected varieties satisfy the no-forms hypothesis, but the theorem also applies more generally. ## Other sources - [openai/math release notes (2026): model provenance](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” - [S.-T. Yau, “Open problems in geometry”, in Lectures on Differential Geometry (1994)](https://www.intlpress.com/site/pub/files/preview/bookpubs/00000308.pdf) - [V. Lazić, S. Matsumura, T. Peternell, N. Tsakanikas and Z. Xie, “The nonvanishing problem for varieties with nef anticanonical bundle”, Documenta Mathematica (2023)](https://doi.org/10.4171/DM/936) - [N. 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