On birational boundedness of Fano fibrations

Abstract: We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $\epsilon$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $\epsilon$-Fano type.