Boundedness of minimal modifications

Abstract

For fixed $n$ and $\epsilon>0$, we prove boundedness of projective birational morphisms $f\colon X\to\mathbb P^n$ over $\mathbb C$ for which $(X,B)$ is an $\epsilon$-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.