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Cohomological transfer and equivariant anticanonical sections

OpenAI · September 26, 2026

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Abstract

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.

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Cite

@misc{OAI:Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026,
  author = {{OpenAI}},
  title = {{Cohomological transfer and equivariant anticanonical sections}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf}{OAI:Cohomological-transfer-and-equivariant-anticanonical-sections-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.

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