Limits of log canonical thresholds

TL;DR

This paper proves that limits of decreasing sequences of log canonical thresholds in fixed dimension lie in the thresholds of lower dimension, and shows that these sets are closed, rational, and relate to Shokurov's ACC conjecture.

Contribution

It establishes the closedness and rationality of log canonical threshold sets and relates the ACC conjecture to semi-continuity properties, advancing understanding of their structure.

Findings

Limits of decreasing sequences in T_n lie in T_{n-1}.

T_n is a closed subset of real numbers.

Every limit of thresholds on smooth varieties is rational.

Abstract

Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n lies in T_{n-1}, proving in this setting a conjecture of Koll\'{a}r. We also show that T_n is a closed subset in the set of real numbers; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov's ACC Conjecture for all T_n, it is enough to show that 1 is not a point of accumulation from below of any T_n. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.