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Metric descent and rank-preserving contractions

OpenAI · September 26, 2026

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Abstract

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.

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Cite

@misc{OAI:Metric-descent-and-rank-preserving-contractions-September-26-2026,
  author = {{OpenAI}},
  title = {{Metric descent and rank-preserving contractions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf}{OAI:Metric-descent-and-rank-preserving-contractions-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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