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

TL;DR

This paper investigates the boundedness of singularities and minimal log discrepancies of Kollár components, providing results in dimension three and proposing conjectures about their general behavior.

Contribution

It establishes boundedness results for minimal log discrepancies of Kollár components in dimension three and conjectures their universal boundedness.

Findings

Bounded local volumes imply bounded minimal log discrepancies in dimension three.

Confirmed the conjecture for dimension three when local volumes are bounded away from zero.

Answered a question relating log canonical thresholds to local volumes.

Abstract

Recent study in K-stability suggests that klt singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true in dimension three, or when the minimal log discrepancies of Koll\'ar components are bounded from above. We conjecture that the minimal log discrepancies of Koll\'ar components are always bounded from above, and verify it in dimension three when the local volumes are bounded away from zero. We also answer a question of Han, Liu and Qi on the relation between log canonical thresholds and local volumes.