On minimal log discrepancies and Koll\'ar components

TL;DR

This paper establishes that in fixed dimension and coefficients, certain singularities with specific blow-ups have minimal log discrepancies from a finite set, leading to a stratification and the ascending chain condition for exceptional singularities.

Contribution

It proves a finiteness result for minimal log discrepancies of singularities admitting an epsilon-plt blow-up, linking geometric properties to boundedness and stratification.

Findings

Minimal log discrepancies belong to a finite set depending on dimension, coefficients, and epsilon.

Proves the ascending chain condition for minimal log discrepancies of exceptional singularities.

Introduces an invariant related to Kollár components and total discrepancy.

Abstract

In this article we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities, with standard coefficients, which admit an $\epsilon$-plt blow-up have minimal log discrepancies belonging to a finite set which only depends on $d,a$ and $\epsilon$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Koll\'ar components.