Boundedness of ($\epsilon, n$)-Complements for projective generalized pairs of Fano type

Abstract

We show the existence of $(\epsilon,n)$-complements for $(\epsilon,\Rr)$-complementary projective generalized pairs of Fano type $(X,B+M)$ when either the coefficients of $B$ and $\mu_j$ belong to a finite set or the coefficients of $B$ belong to a DCC set and $M'\equiv 0$, where $M'=\sum\mu_jM_j'$ and $M_j'$ are b-Cartier nef divisors.