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Browse › Algebraic and complex geometry › 034 Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Schnell fiber spaces and good canonical models

OpenAI · September 24, 2026

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Abstract

We prove the Campana–Peternell inequality $\kappa(X)\geq\kappa(D)$ for a smooth connected projective complex variety X and an effective Cartier divisor D whenever $m_0K_X-D$ is pseudo-effective for some positive integer m0. For an algebraic fiber space $f\colon X\to Y$ between smooth connected projective complex varieties, the hypothesis that $m_0K_X-f^*H$ is pseudo-effective with H ample Cartier gives $\kappa(X)=\kappa(F)+\dim Y$ for a very general smooth fiber F, as well as nonzero sections of $mK_X-f^*H$ for all sufficiently large divisible m.

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Digests and expositions

  • AI Proved What?, Always semiample? (Episode 034, October 2026). Explains the family’s abundance claims, proof strategy, and dependencies. [AI-generated per source] [family-level]

Elsewhere

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Cite

@misc{OAI:Schnell-fiber-spaces-and-good-canonical-models-September-24-2026,
  author = {{OpenAI}},
  title = {{Schnell fiber spaces and good canonical models}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf}{OAI:Schnell-fiber-spaces-and-good-canonical-models-September-24-2026}},
  year = {2026}
}

Family 034: Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.

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