On boundedness of singularities and minimal log discrepancies of Koll\'ar components, II

TL;DR

This paper establishes conditions under which K-semistable log Fano cone singularities are bounded, linking boundedness to local volumes and minimal log discrepancies, and confirms a conjecture in dimension three.

Contribution

It proves the boundedness of K-semistable log Fano cone singularities based on local volume and minimal log discrepancy bounds, confirming a key conjecture in dimension three.

Findings

Boundedness of K-semistable log Fano cone singularities when local volumes are bounded away from zero.

Confirmation of the boundedness conjecture for 3-dimensional K-semistable log Fano cone singularities.

Local volumes of 3-dimensional klt singularities only accumulate at zero.

Abstract

We show that a set of K-semistable log Fano cone singularities is bounded if and only if their local volumes are bounded away from zero, and their minimal log discrepancies of Koll\'ar components are bounded from above. As corollaries, we confirm the boundedness conjecture for K-semistable log Fano cone singularities in dimension three, and show that local volumes of 3-dimensional klt singularities only accumulate at zero.