Birkar's Stein-degree conjecture for log Calabi-Yau pairs: the contraction-to-a-point case for ordinary pairs

Theorem 1.1: for every field $k$ of characteristic zero and every normal integral projective lc $\mathbb Q$-pair $(X,B)$ of dimension $d$ with $H^0(X,\mathcal O_X)=k$, $B\ge0$ and $K_X+B\sim_{\mathbb Q}0$, every prime component $S$ of $B$ with coefficient at least $t$ satisfies $[k_S:k]\le N(d,t)$, where $k_S$ is the algebraic closure of $k$ in $k(S)$; this bounds the Stein degree of $S$. It covers only the contraction to $\mathrm{Spec}\,k$ and ordinary pairs; the general fibration form of the conjecture and generalized pairs are not treated.

This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it. Theorem 1.1: for every field $k$ of characteristic zero and every normal integral projective lc $\mathbb Q$-pair $(X,B)$ of dimension $d$ with $H^0(X,\mathcal O_X)=k$, $B\ge0$ and $K_X+B\sim_{\mathbb Q}0$, every prime component $S$ of $B$ with coefficient at least $t$ satisfies $[k_S:k]\le N(d,t)$, where $k_S$ is the algebraic closure of $k$ in $k(S)$; this bounds the Stein degree of $S$. It covers only the contraction to $\mathrm{Spec}\,k$ and ordinary pairs; the general fibration form of the conjecture and generalized pairs are not treated.