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Exact orders and invariant anticanonical linear systems

OpenAI · September 26, 2026

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Abstract

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.

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Cite

@misc{OAI:Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026,
  author = {{OpenAI}},
  title = {{Exact orders and invariant anticanonical linear systems}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf}{OAI:Exact-orders-and-invariant-anticanonical-linear-systems-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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