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Anticanonical nonvanishing from smooth semipositivity

OpenAI · September 26, 2026

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Abstract

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

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

  • S. Filip, V. Tosatti, Smooth and Rough Positive Currents (arXiv:1709.05385; Ann. Inst. Fourier 68 (2018)). Constructs nef classes on K3 surfaces without smooth semipositive representatives, which bears on the gap between the paper's smooth-semipositivity hypothesis and the nef nonvanishing question it leaves open.
  • J. Cao, A. Höring, Manifolds with nef anticanonical bundle (arXiv:1305.1018, 2013/2017). Studies the Albanese map of compact Kähler manifolds with nef anticanonical bundle, the nef analogue of the Demailly–Peternell–Schneider finite-cover Albanese structure used in the paper's geometric reduction.

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Cite

@misc{OAI:Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026,
  author = {{OpenAI}},
  title = {{Anticanonical nonvanishing from smooth semipositivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf}{OAI:Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026}},
  year = {2026}
}

Family 068: Anticanonical nonvanishing in every dimension

If X is a smooth connected complex projective variety and $-K_X$ admits a smooth Hermitian metric with nonnegative curvature, then $H^0(X,-mK_X)\ne0$ for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

Other papers in this family: