Arithmetic Stein-degree bounds for log Calabi–Yau boundary components
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Can the constants-field degree [k_S:k] of a boundary prime be bounded solely by dimension d and a positive coefficient threshold t? Here (X,B) is an ordinary projective log-canonical Q-pair over a characteristic-zero field, X is normal integral with H⁰(X,O_X)=k, B is effective, K_X+B is Q-linearly trivial, and S has coefficient at least t.
Why it matters and what counts as progress
Why it matters: A constants-field bound adds arithmetic information absent from geometric bounds over algebraically closed fields.
What counts as progress: Prove a uniform N(d,t) bound in this contraction-to-a-point setting or extend it to relative fibrations and generalized pairs.
How the problem developed
- 25 Sept 2025 · Theorem [1]
Caucher Birkar and Santai Qu Proved bounded Stein degree for horizontal boundary divisors on log Calabi–Yau fibrations, in the geometric fibration setting.
- 25 Mar 2026 · Exposition [2]
Caucher Birkar Published an exposition defining Stein degree and outlining the boundedness problem arising from log Calabi–Yau fibrations.
- 25 Sept 2026 · Reported claim [3]
OpenAI The preprint claims a field-independent arithmetic bound for boundary components of ordinary projective log canonical Q-pairs over characteristic-zero fields, proving the contraction-to-a-point formulation.
AI contribution
Arithmetic Stein-degree bounds for log Calabi–Yau pairs
For each dimension d and t>0, the manuscript claims a bound N(d,t) on [k_S:k] for a boundary prime S with coefficient at least t on an ordinary projective log-canonical Q-pair (X,B) over a characteristic-zero field, where X is normal integral, H⁰(X,O_X)=k, B is effective and K_X+B is Q-linearly trivial.
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