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Integrable metrics and effectivity with controlled boundary

OpenAI · September 26, 2026

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Abstract

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.

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Cite

@misc{OAI:Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026,
  author = {{OpenAI}},
  title = {{Integrable metrics and effectivity with controlled boundary}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf}{OAI:Integrable-metrics-and-effectivity-with-controlled-boundary-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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