Log-canonical thresholds in real and complex dimension 2

TL;DR

This paper investigates the properties of log-canonical thresholds in two-dimensional real and complex spaces, establishing the ascending chain condition and characterizing accumulation points, thereby providing a new proof of a known theorem.

Contribution

It proves the ascending chain condition for log-canonical thresholds in dimension two and characterizes their accumulation points, offering a new proof of Phong-Sturm's theorem.

Findings

Ascending chain condition holds for log-canonical thresholds in dimension two.

Positive accumulation points correspond to thresholds in dimension one.

Provides a new proof of Phong-Sturm's theorem.

Abstract

We study the set of log-canonical thresholds (or critical integrability indices) of holomorphic (resp. real analytic) function germs in $\mathbb{C}^2$ (resp. $\mathbb{R}^2$). In particular, we prove that the ascending chain condition holds, and that the positive accumulation points of decreasing sequences are precisely the integrability indices of holomorphic (resp. real analytic) functions in dimension $1$. This gives a new proof of a theorem of Phong-Sturm.