Arithmetic Stein-degree bounds for log Calabi–Yau pairs

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 $[k_S:k]\le N(d,t)$, where k${}_S$ 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.