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

Minimal metrics and interior injectivity for nef adjoints

OpenAI · September 27, 2026

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Abstract

Let $(H,\Theta)$ be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H1-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Digests and expositions

  • AI Proved What?, Always semiample? (Episode 034, October 2026; formalization correction October 15, 2026). Explains the family’s abundance claims, proof strategy, and dependencies, and reports supporting Lean files for this paper's local integrability and gauge lemmas, explicitly excluding the metric and injectivity theorems. [AI-generated per source] [family-level]

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Cite

@misc{OAI:Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026,
  author = {{OpenAI}},
  title = {{Minimal metrics and interior injectivity for nef adjoints}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf}{OAI:Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026}},
  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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