Boundedness of log Fano cone singularities and discreteness of local volumes

TL;DR

This paper proves boundedness of certain log Fano cone singularities with volume constraints and shows that local volumes of klt singularities in fixed dimensions accumulate only at zero, revealing their discrete nature.

Contribution

It establishes boundedness of K-semistable log Fano cone singularities with volume bounds and demonstrates the discreteness of local volumes of klt singularities.

Findings

Boundedness of K-semistable log Fano cone singularities with volume lower bounds.

Discreteness of local volumes of klt singularities in fixed dimensions.

Zero is the only accumulation point of local volumes.

Abstract

We prove that in any fixed dimension, K-semistable log Fano cone singularities whose volumes are bounded from below by a fixed positive number form a bounded set. As a consequence, we show that the set of local volumes of klt singularities of a fixed dimension has zero as the only accumulation point.