ACC for local volumes and boundedness of singularities

Abstract

The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities x ∈ ( X , Δ ) x\in (X,\Delta ) satisfies the ACC if the coefficients of Δ \Delta belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of δ \delta -plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.