发表机构
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了射影双有理态射在 $\eps$-lc 对条件下的有界性，并给出了例外素因子族的有界性及 log canonical thresholds 的一致正下界，解决了 Birkar 的猜想。
AI 中文摘要
对于固定的 $n$ 和 $\eps>0$，我们证明了在 $\C$ 上射影双有理态射 $f\colon X\to\Pj^n$ 的有界性，其中 $(X,B)$ 是 $\eps$-lc 对，$B\ge0$，$K_X+B$ 在 $\Pj^n$ 上是 nef 的，且 $°(f_*B)$ 有界。$B$ 的非零系数不必有正下界，且 $(\Pj^n,f_*B)$ 不必是 log canonical 的。$X$ 上的 $f$-例外素因子，配备其约化诱导概形结构，构成一个有界的射影概形族，包括当它们非正规时。我们还获得了所有有界次数的有效目标除子的拉回的 log canonical thresholds 的一致正下界。本文证明的有界性陈述最初由 Caucher Birkar 猜想。
英文摘要
For fixed $n$ and $ε>0$, we prove boundedness of projective birational morphisms $f\colon X\to\mathbb P^n$ over $\mathbb C$ for which $(X,B)$ is an $ε$-lc pair, $B\geq 0$, $K_X+B$ is nef over $\mathbb P^n$, and $\mathrm{deg}(f_*B)$ is bounded. The nonzero coefficients of $B$ need not have a positive lower bound, and $(\mathbb P^n,f_*B)$ need not be log canonical. The $f$-exceptional prime divisors on $X$, equipped with their reduced induced scheme structures, form a bounded family of projective schemes, including when they are nonnormal. We also obtain a uniform positive lower bound for the log canonical thresholds of pullbacks of all effective target divisors of bounded degree. The boundedness statement proved in this paper was originally conjectured by Caucher Birkar.
Comments32pages, comments welcome
