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Bounded anticanonical metrics on klt pairs and torus quotients

OpenAI · September 26, 2026

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Abstract

Let $(W,D)$ be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of $-(K_W+D)$ has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.

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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)). Compares semiampleness, smooth semipositivity and nefness for classes on K3 surfaces, the distinction the paper uses to set its hypothesis apart from Müller's semiample setting.

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Cite

@misc{OAI:Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026,
  author = {{OpenAI}},
  title = {{Bounded anticanonical metrics on klt pairs and torus quotients}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf}{OAI:Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-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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