Bounded Complements for Weak Fano Pairs with Alpha-invariants and Volumes Bounded Below

Abstract

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belonging to a fixed DCC set $S$ form a bounded family. Moreover, such pairs admit a strong $\epsilon$-lc $\mathbb{R}$-complement for some fixed $\epsilon >0$.