Infinitesimal structure of log canonical thresholds

TL;DR

This paper characterizes the accumulation behavior of log canonical thresholds in fixed dimension, revealing a structured pattern and connections to other invariants like minimal log discrepancies and canonical thresholds.

Contribution

It provides a detailed description of the infinitesimal structure and accumulation patterns of log canonical thresholds in fixed dimension, including their relation to standard sets.

Findings

Log canonical thresholds in fixed dimension accumulate similarly to standard sets.

Accumulation points include thresholds in lower dimensions (≤ d-2).

The results extend to minimal log discrepancies and canonical thresholds.

Abstract

We show that log canonical thresholds of fixed dimension are standardized. More precisely, we show that any sequence of log canonical thresholds in fixed dimension $d$ accumulates in a way which is i) either similar to how standard and hyperstandard sets accumulate, or ii) to log canonical thresholds in dimension $\leq d-2$. This provides an accurate description on the infinitesimal structure of the set of log canonical thresholds. We also discuss similar behaviors of minimal log discrepancies, canonical thresholds, and K-semistable thresholds.