OpenAI · 034-12 · 2026-09-25 · Algebraic and complex geometry

Arithmetic Stein-degree bounds for log Calabi–Yau pairs

preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026

Family 034. Log abundance and effective Iitaka fibrations.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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.

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, $H^{0} (X, O_{X}) = k$ , B effective, and $K_{X} + B \sim_{Q} 0$ . 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.

Preprint source and Lean scope

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