Boundedness of 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 the set of hyperstandard multiplicities $\Phi(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $\alpha(X,B)^{d-1}\mathrm{vol}(-(K\_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.