Anticanonical nonvanishing in every dimension

What OpenAI claims

If X is a smooth connected complex projective variety and -K_X admits a smooth Hermitian metric with nonnegative curvature, then H⁰(X,-mK_X) ≠ 0 for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

Papers (7)

1. Anticanonical nonvanishing from smooth semipositivity

   26 September 2026 · Preprint, not peer reviewed · PDF

   Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.

2. Invariant anticanonical indices and conversion of twisted differentials

   26 September 2026 · Preprint, not peer reviewed · PDF

   For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.

3. Bounded anticanonical metrics on klt pairs and torus quotients

   26 September 2026 · Preprint, not peer reviewed · PDF

   Let (W,D) be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of -(K_W+D) has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.

4. Cohomological transfer and equivariant anticanonical sections

   26 September 2026 · Preprint, not peer reviewed · PDF

   Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.

5. Metric descent and rank-preserving contractions

   26 September 2026 · Preprint, not peer reviewed · PDF

   From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.

6. Integrable metrics and effectivity with controlled boundary

   26 September 2026 · Preprint, not peer reviewed · PDF

   Let Y be smooth projective and let C be an effective integral divisor. If -K_Y+C admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive L=-K_X, sections of m_jL-P at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.

7. Exact orders and invariant anticanonical linear systems

   26 September 2026 · Preprint, not peer reviewed · PDF

   On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.