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Arithmetic Stein-degree bounds for log Calabi–Yau pairs

OpenAI · September 25, 2026

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Abstract

Fix d ≥ 1 and t > 0. Let $(X,B)$ be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, $H^0(X,\mathcal O_X)=k$, B effective, and $K_X+B\sim_{\mathbb Q}0$. We prove that every prime component S of B with coefficient at least t satisfies $[k_S:k]\leq N(d,t)$, where kS is the relative algebraic closure of k in $k(S)$. This also bounds the Stein degree of S over k, proving the contraction-to-a-point formulation of Birkar's Stein-degree conjecture for ordinary ℚ-pairs.

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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.

Possibly relevant literature (found by the agent, not checked)

  • Caucher Birkar, Santai Qu, Stein degree over fields of characteristic zero (arXiv:2610.16205, 2026-10-12). Shows that the Stein degree of horizontal divisors on generalised log Calabi–Yau fibrations over arbitrary characteristic-zero fields is bounded independently of the ground field, which overlaps with this paper's main theorem over any field of characteristic zero.

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

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026,
  author = {{OpenAI}},
  title = {{Arithmetic Stein-degree bounds for log Calabi--Yau pairs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf}{OAI:Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026}},
  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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