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 .

The existence of the Kähler–Ricci soliton degeneration

“…For fixed , the set of log Fano pairs with , integral and , and are bounded by [Che20] [Jia20] [XZ21]. Applying Proposition 6.2 then completes the proof.…”

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

“…Moreover, such pairs admit a strong ǫ-lc R-complement for some fixed ǫ > 0. This is an improvement of [5].…”

Finite generation for valuations computing stability thresholds and applications to K-stability

“…See also [Che20]. Later a proof which only uses the boundedness result from [HMX14] was given in [XZ20b].…”