ACC for local volumes and boundedness of singularities

TL;DR

This paper proves the ACC conjecture for local volumes of klt singularities under the assumption of analytically bounded ambient germs, advancing understanding of singularity boundedness in algebraic geometry.

Contribution

It establishes the ACC for local volumes in new cases, including analytically bounded germs and low-dimensional singularities, and introduces a related conjecture on $ ext{δ}$-plt blow-ups.

Findings

Proves ACC for local volumes in analytically bounded germs.

Validates the conjecture for dimension 2 and 3-dimensional terminal singularities.

Introduces and confirms a conjecture on $ ext{δ}$-plt blow-ups with positive lower bounds.

Abstract

The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a 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 $\delta$-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.